\draftcut \section{w-Knots} \label{sec:w-knots} \begin{quote} \small {\bf Section Summary. } \summaryknots \end{quote} {\bf Knots are the wrong objects for study in knot theory,} v-knots are the wrong objects for study in the theory of v-knotted objects and w-knots are the wrong objects for study in the theory of w-knotted objects. Studying uvw-knots on their own is the parallel of studying cakes, cookies and pastries as they come out of the bakery --- we sure want to make them our own, but the theory of desserts is more about the ingredients and how they are put together than about the end products. In algebraic knot theory this reflects through the fact that knots are not finitely generated in any sense (hence, they must be made of some more basic ingredients), and through the fact that there are very few operations defined on knots (connected sums and satellite operations being the main exceptions), and thus, most interesting properties of knots are transcendental, or non-algebraic, when viewed from within the algebra of knots and operations on knots~\cite{Bar-Natan:AKT-CFA}. The right objects for study in knot theory, or v-knot theory or w-knot theory, are thus the ingredients that make up knots and that permit a richer algebraic structure. These are braids, studied in the previous section, and even more so tangles and tangled graphs, studied in \cite{Bar-NatanDancso:WKO2}. Yet tradition has its place and the sweets are tempting, and we can introduce and apply some of the tools we will use in the deeper and healthier study of w-tangles and w-tangled foams in the limited, but tasty, arena of the baked goods of knot theory, the knots themselves. \draftcut \subsection{v-Knots and w-Knots} \label{subsec:VirtualKnots} v-Knots may be understood either as knots drawn on surfaces modulo the addition or removal of empty handles~\cite{Kuperberg:VirtualLink} or as ``Gauss diagrams'' (see Remark~\ref{rem:GD}), or simply ``unembedded but wired together'' crossings modulo the Reidemeister moves (\cite{Kauffman:VirtualKnotTheory, ManturovIlyutko:VirtualKnotsBook, Roukema:GPV} and Section 2 of \cite{Bar-NatanDancso:WKO2}). But right now we forgo the topological and the abstract and give only the ``planar'' (and somewhat less philosophically satisfying) definition of v-knots. \begin{figure}[h] \[ \pstex{VKnot} \] \caption{ A long v-knot diagram with 2 virtual crossings, 2 positive crossings and 2 negative crossings. A positive-negative pair can easily be cancelled using R2, and then a virtual crossing can be cancelled using VR1, and it seems that the rest cannot be simplified any further. } \label{fig:VKnot} \end{figure} \begin{definition} A ``long v-knot diagram'' is an arc smoothly drawn in the plane from $-\infty$ to $+\infty$, with finitely many self-intersections, divided into ``virtual crossings'' $\virtualcrossing$, overcrossings $\overcrossing$ (a.k.a.~positive crossings), and undercrossings $\undercrossing$ (a.k.a.~negative crossings); and regarded up to planar isotopy. A picture is worth more than a more formal definition, and one appears in Figure~\ref{fig:VKnot}. A ``long v-knot'' is an equivalence class of long v-knot diagrams, modulo the equivalence generated by the Reidemeister $1^{\!s}$, 2 and 3 moves (\glost{\Rs}, \glost{R2} and \glost{R3})\footnote{ \Rs\ is the ``spun'' version of R1 --- kinks can be spun around, but not removed outright. See Figure~\ref{fig:VKnotRels}.}, the virtual Reidemeister 1 through 3 moves (\glost{VR1}, \glost{VR2}, \glost{VR3}), and by the mixed relations (\glost{M}); all these are shown in Figure~\ref{fig:VKnotRels}. Finally, ``long w-knots'' are obtained from long v-knots by also dividing by the overcrossings commute (OC) relations, also shown in Figure~\ref{fig:VKnotRels}. Note that we never mod out by the Reidemeister 1 (\glost{R1}) move nor by the undercrossings commute relation (UC). \end{definition} \begin{figure} \[ \pstex{VKnotRels} \] \caption{ The relations defining v-knots and w-knots, along with two relations that are {\em not} imposed. } \label{fig:VKnotRels} \end{figure} \begin{defwarn} A ``circular v-knot'' is like a long v-knot, except parametrized by a circle rather than by a long line. Unlike the case of usual knots, circular v-knots are {\bf not} equivalent to long v-knots \cite{ManturovIlyutko:VirtualKnotsBook}. The same applies to w-knots. \end{defwarn} \begin{defwarn} Long v-knots form a monoid using the concatenation operation $\#$. Unlike the case of usual knots, the resulting monoid is {\bf not} abelian \cite{ManturovIlyutko:VirtualKnotsBook}. % The same applies to w-knots. \end{defwarn} \begin{remark} \label{rem:GD} A ``Gauss diagram'' is a straight ``skeleton line'' along with signed directed chords (signed ``arrows'') marked along it (more at~\cite{Kauffman:VirtualKnotTheory, ManturovIlyutko:VirtualKnotsBook, GoussarovPolyakViro:VirtualKnots}). Gauss diagrams are in obvious bijection with long v-knot diagrams; the skeleton line of a Gauss diagram corresponds to the parameter space of the v-knot, and the arrows correspond to the crossings, with each arrow heading from the upper strand to the lower strand, marked by the sign of the relevant crossing: \[ \pstex{GDExample} \] One may also describe the relations in Figure~\ref{fig:VKnotRels} as well as circular v-knots and other types of v-knots (as we will encounter later) in terms of Gauss diagrams with varying skeletons. \end{remark} \begin{figure} \[ \pstex{Kinks} \] \caption{ The positive and negative under-then-over kinks (left), and the positive and negative over-then-under kinks (right). In each pair the negative kink is the $\#$-inverse of the positive kink, where $\#$ denotes the concatenation operation. \label{fig:Kinks}} \end{figure} \begin{remark}\label{rem:Framing} Since we do not mod out by R1, it is perhaps more appropriate to call our class of v/w-knots ``framed long v/w-knots'', but since we care more about framed v/w-knots than about unframed ones, we reserve the unqualified name for the framed case, and when we do wish to mod out by R1 we will explicitly write ``unframed long v/w-knots''. Recall that in the case of ``usual knots'', or u-knots, dropping the R1 relation altogether also results in a $\bbZ^2$-extension of unframed knot theory, where the two factors of $\bbZ$ are framing and rotation number. If one wants to talk about ``true'' framed knots, one mods out by the spun Reidemeister 1 relation (\Rs\ of Figure~\ref{fig:VKnotRels}), which preserves the blackboard framing but does not preserve the rotation number. We take the analogous approach here, including the \Rs\ relation -- but not R1 -- in the v and w cases. This said, note that the monoid of long v-knots is just a central extension by $\bbZ$ of the monoid of unframed long v-knots, and so studying the framed case is not very different from studying the unframed case. Indeed the four ``kinks'' of Figure~\ref{fig:Kinks} generate a central $\bbZ$ within long v-knots, and it is not hard to show that the sequence \begin{equation} \label{eq:FramedAndUnframed} 1\longrightarrow \bbZ \longrightarrow \{\text{long v-knots}\} \longrightarrow \{\text{unframed long v-knots}\} \longrightarrow 1 \end{equation} is split and exact. The same can be said for w-knots. \end{remark} \begin{exercise} \label{ex:sl} Show that a splitting of the sequence~\eqref{eq:FramedAndUnframed} is given by the ``self-linking'' invariant $\glos{\sl}\colon \{\text{long v-knots}\}\to\bbZ$ defined by \[ \sl(K):=\sum_{\text{crossings}\atop x\text{ in }K}\sign x , \] where $K$ is a v-knot diagram, and the sign of a crossing $x$ is defined so as to agree with the signs in Figure~\ref{fig:Kinks}. \end{exercise} \begin{remark} Note that w-knots are strictly weaker than v-knots --- a notorious example is the Kishino knot (e.g.~\cite{Dye:Kishinos}) which is non-trivial as a v-knot; yet both it and its mirror are trivial as w-knots. Yet ordinary knots inject even into w-knots, as the Wirtinger presentation makes sense for w-knots and therefore w-knots have a ``fundamental quandle'' which generalizes the fundamental quandle of ordinary knots~\cite{Kauffman:VirtualKnotTheory, ManturovIlyutko:VirtualKnotsBook}, and as the fundamental quandle of ordinary knots separates ordinary knots~\cite[Corollary~16.3]{Joyce:TheKnotQuandle}. \end{remark} \subsubsection{A topological construction of Satoh's tubing map} \label{subsubsec:TopTube} Following Satoh~\cite{Satoh:RibbonTorusKnots} and using the same constructions as in Section~\ref{subsubsec:ribbon}, we can map w-knots to (``long'') ribbon tubes in $\bbR^4$ (and the relations in Figure~\ref{fig:VKnotRels} still hold). It is natural to expect that this ``tubing'' map is an isomorphism; in other words, that the theory of w-knots provides a ``Reidemeister framework'' for long ribbon tubes in $\bbR^4$ --- that every long ribbon tube is in the image of this map and that two ``w-knot diagrams'' represent the same long ribbon tube iff they differ by a sequence of moves as in Figure~\ref{fig:VKnotRels}. This remains unproven. Let $\glos{\delta}\colon\{\text{v-knots}\} \to \{\text{Ribbon tori in } \bbR^4\}$ denote the tubing map. In Satoh's~\cite{Satoh:RibbonTorusKnots} $\delta$ is called ``Tube''. It is worthwhile to give a completely ``topological'' definition of $\delta$. To do this we must start with a topological interpretation of v-knots. The standard topological interpretation of v-knots (see e.g.~\cite{Kuperberg:VirtualLink}) is that they are oriented framed knots drawn\footnote{Here and below, ``drawn on $\Sigma$'' means ``embedded in $\Sigma\times[-\epsilon,\epsilon]$''.} on an oriented surface $\Sigma$, modulo ``stabilization'', which is the addition and/or removal of empty handles (handles that do not intersect with the knot). We prefer an equivalent, yet even more bare-bones approach. For us, a virtual knot is an oriented framed knot $\gamma$ drawn on a ``virtual surface $\glos{\Sigma}$ for $\gamma$''. More precisely, $\Sigma$ is an oriented surface that may have a boundary, $\gamma$ is drawn on $\Sigma$, and the pair $(\Sigma,\gamma)$ is taken modulo the following relations: \begin{itemize} \item Isotopies of $\gamma$ on $\Sigma$ (meaning, in $\Sigma\times[-\epsilon,\epsilon]$). \item Tearing and puncturing parts of $\Sigma$ away from $\gamma$: \end{itemize} \[ \input{figs/TearingAndPuncturing.pstex_t} \] (We call $\Sigma$ a ``virtual surface'' because tearing and puncturing imply that we only care about it in the immediate vicinity of $\gamma$). We can now define\footnote{Following a private discussion with Dylan Thurston.} a map $\delta$, defined on v-knots and taking values in ribbon tori in $\bbR^4$: given $(\Sigma,\gamma)$, embed $\Sigma$ arbitrarily in $\bbR^3_{xzt}\subset\bbR^4$. Note that the unit normal bundle of $\Sigma$ in $\bbR^4$ is a trivial circle bundle and it has a distinguished trivialization, constructed using its positive-$y$-direction section and the orientation that gives each fibre a linking number $+1$ with the base $\Sigma$. We say that a normal vector to $\Sigma$ in $\bbR^4$ is ``near unit'' if its norm is between $1-\epsilon$ and $1+\epsilon$. The near-unit normal bundle of $\Sigma$ has as fibre an annulus that can be identified with $[-\epsilon,\epsilon]\times S^1$ (identifying the radial direction $[1-\epsilon,1+\epsilon]$ with $[-\epsilon,\epsilon]$ in an orientation-preserving manner), and hence, the near-unit normal bundle of $\Sigma$ defines an embedding of $\Sigma\times[-\epsilon,\epsilon]\times S^1$ into $\bbR^4$. On the other hand, $\gamma$ is embedded in $\Sigma\times[-\epsilon,\epsilon]$ so $\gamma\times S^1$ is embedded in $\Sigma\times[-\epsilon,\epsilon]\times S^1$, and we can let $\delta(\gamma)$ be the composition \[ \gamma\times S^1 \hookrightarrow\Sigma\times[-\epsilon,\epsilon]\times S^1 \hookrightarrow\bbR^4, \] which is a torus in $\bbR^4$, oriented using the given orientation of $\gamma$ and the standard orientation of $S^1$. A framing of a knot (or a v-knot) $\gamma$ can be thought of as a ``nearby companion'' to $\gamma$. Applying the above procedure to a knot and a nearby companion simultaneously, we find that $\delta$ takes framed v-knots to framed ribbon tori in $\bbR^4$, where a framing of a tube in $\bbR^4$ is a continuous up-to-homotopy choice of unit normal vector at every point of the tube. Note that from the perspective of flying rings as in Section~\ref{subsubsec:FlyingRings}, a framing is a ``companion ring'' to a flying ring. In the framing of $\delta(\gamma)$ the companion ring is never linked with the main ring, but can fly parallel inside, outside, above or below it and change these positions, as shown in Figure~\ref{fig:CompanionRing}. \begin{figure} \input figs/CompanionRing.pstex_t \caption{Framing as companion rings.}\label{fig:CompanionRing} \end{figure} We leave it to the reader to verify that $\delta(\gamma)$ is ribbon, that it is independent of the choices made within its construction, that it is invariant under isotopies of $\gamma$ and under tearing and puncturing of $\Sigma$, that it is also invariant under the OC relation of Figure~\ref{fig:VKnotRels} and hence, the true domain of $\delta$ is w-knots, and that it is equivalent to Satoh's tubing map. \draftcut \subsection{Finite Type Invariants of v-Knots and w-Knots} \label{subsec:FTforvwKnots} Much as for v-braids and w-braids (see Section~\ref{subsec:FT4Braids}) and much as for ordinary knots (e.g.~\cite{Bar-Natan:OnVassiliev}) we define finite type invariants for v-knots and for w-knots using an alternation scheme with $\semivirtualover\to\overcrossing-\virtualcrossing$ and $\semivirtualunder\to\virtualcrossing-\undercrossing$. That is, given any invariant of v- or w-knots taking values in an abelian group, we extend the invariant to v- or w-knots also containing ``semi-virtual crossings'' like $\semivirtualover$ and $\semivirtualunder$ using the above assignments, and we declare an invariant to be ``of type $m$'' if it vanishes on v- or w-knots with more than $m$ semi-virtuals. As for v- and w-braids and as for ordinary knots, such invariants have an ``$m$th derivative'', their ``weight system'', which is a linear functional on the space $\calA^{sv}(\uparrow)$ (for v-knots) or $\calA^{sw}(\uparrow)$ (for w-knots). We turn to the definitions of these spaces, following~\cite{GoussarovPolyakViro:VirtualKnots, Bar-NatanHalachevaLeungRoukema:v-Dims}: \begin{definition} \label{def:ArrowDiagrams} An ``arrow diagram'' is a chord diagram along a long line (called ``the skeleton''), in which the chords are oriented (hence ``arrows''). An example is given in Figure~\ref{fig:ADand6T}. Let $\glos{\calD^v(\uparrow)}$ be the space of formal linear combinations of arrow diagrams. Let $\glos{\calA^v(\uparrow)}$ be $\calD^v(\uparrow)$ modulo all ``6T relations''. Here a 6T relation is any (signed) combination of arrow diagrams obtained from the diagrams in Figure~\ref{fig:6T} by placing the 3 vertical strands there along a long line skeleton in any order, and possibly adding some further arrows in between, as shown in Figure~\ref{fig:ADand6T}. Let $\glos{\calA^{sv}(\uparrow)}$ be the further quotient of $\calA^v(\uparrow)$ by the \glost{RI} relation, where the RI (for rotation number independence) relation asserts that an isolated arrow pointing to the right equals an isolated arrow pointing to the left\footnote{ The XII relation of~\cite{Bar-NatanHalachevaLeungRoukema:v-Dims} follows from RI and need not be imposed. }, as shown in Figure~\ref{fig:ADand6T}. \begin{figure} \[ \pstex{ADand6T} \] \caption{ An arrow diagram of degree 6, a 6T relation, and an RI relation. The dotted parts indicate that there may be more arrows on other parts of the skeleton, however these remain the same throughout the relation. } \label{fig:ADand6T} \end{figure} Let $\glos{\calA^w(\uparrow)}$ be the further quotient of $\calA^v(\uparrow)$ by the TC relation, first displayed in Figure~\ref{fig:TCand4T} and reproduced for the case of a long line skeleton in Figure~\ref{fig:TCand4TForKnots}. Likewise, let $\glos{\calA^{sw}(\uparrow)}:=\calA^{sv}(\uparrow)/TC=\calA^w(\uparrow)/RI$. Alternatively, noting that given TC two of the terms in 6T drop out, $\calA^w(\uparrow)$ is the space of formal linear combinations of arrow diagrams modulo TC and $\aft$ relations, displayed in Figures~\ref{fig:TCand4T} and~\ref{fig:TCand4TForKnots}. Likewise, $\calA^{sw}=\calD^v/TC,\aft,RI$. Finally, grade $\calD^v(\uparrow)$ and all of its quotients by declaring that the degree of an arrow diagram is the number of arrows in it. \begin{figure} \[ \pstex{TCand4TForKnots} \] \caption{The TC and the $\protect\aft$ relations for knots.} \label{fig:TCand4TForKnots} \end{figure} \end{definition} As an example, the spaces $\calA^{v,sv,w,sw}(\uparrow)$ (that is, any of the spaces above) restricted to degrees up to 2 are studied in detail in Section~\ref{subsec:ToTwo}. In the same manner as in the theory of finite type invariants of ordinary knots (see especially~\cite[Section~3]{Bar-Natan:OnVassiliev}), the spaces $\calA^{\star}(\uparrow)$ (meaning, all of the spaces above) carry much algebraic structure. The juxtaposition product makes them into graded algebras. The product of two finite type invariants is a finite type invariant (whose type is the sum of the types of the factors); this induces a product on weight systems, and therefore a co-product $\Delta$ on arrow diagrams. In brief (and much the same as in the usual finite type story), the co-product $\Delta D$ of an arrow diagram $D$ is the sum of all ways of dividing the arrows in $D$ between a ``left co-factor'' and a ``right co-factor''. In summary: \begin{proposition} \label{prop:CoarseStructure} $\calA^v(\uparrow)$, $\calA^{sv}(\uparrow)$, $\calA^w(\uparrow)$, and $\calA^{sw}(\uparrow)$ are co-commutative graded bialgebras. \end{proposition} By the Milnor-Moore theorem~\cite[Theorem 6.11]{MilnorMoore:Hopf} we find that $\calA^{v,sv,w,sw}(\uparrow)$ are the universal enveloping algebras of their Lie algebras of primitive elements (that is, elements $D$ such that $\Delta(D)=1\otimes D + D \otimes 1$). Denote these (graded) Lie algebras by $\glos{\calP^{v,sv,w,sw}(\uparrow)}$, respectively. When we grow up we'd like to understand $\calA^v(\uparrow)$ and $\calA^{sv}(\uparrow)$. At the moment we know only very little about these spaces beyond the generalities of Proposition~\ref{prop:CoarseStructure}. In Section \ref{subsec:SomeDimensions} some dimensions of low degree parts of $\calA^{v,sv}(\uparrow)$ are discussed. Also, given a finite dimensional Lie bialgebra and a finite dimensional representation thereof, we know how to construct linear functionals on $\calA^v(\uparrow)$ (one in each degree~\cite{Haviv:DiagrammaticAnalogue, Leung:CombinatorialFormulas}), but not on $\calA^{sv}(\uparrow)$. But we don't even know which degree $m$ linear functionals on $\calA^{sv}(\uparrow)$ are the weight systems of degree $m$ invariants of v-knots (that is, we have not solved the ``Fundamental Problem''~\cite{Bar-NatanStoimenow:Fundamental} for v-knots). As we shall see below, the situation is much brighter for $\calA^{w,sw}(\uparrow)$. \draftcut \subsection{Expansions for w-Knots} \label{subsec:Z4Knots} The notion of ``an expansion'' (or ``a universal finite type invariant'') for w-knots (or v-knots) is defined in complete analogy with the parallel notion for usual knots (see e.g.~\cite{Bar-Natan:OnVassiliev}), except replacing double points $\doublepoint$ with semi-virtual crossings $\semivirtualover$ and $\semivirtualunder$, and replacing chord diagrams by arrow diagrams. Alternatively, it is the same as an expansion for w-braids (as in Definition~\ref{def:vwbraidexpansion}), simply replacing w-braids by w-knots. Just as in the cases of u-knots (i.e., ordinary knots) and/or w-braids, the existence of an expansion $Z\colon \{\text{w-knots}\}\to\calA^{sw}(\uparrow)$ is equivalent to the statement ``every weight system integrates'', i.e., ``every degree $m$ linear functional on $\calA^{sw}(\uparrow)$ is the $m$th derivative of a type $m$ invariant of long w-knots''. \begin{theorem} \label{thm:ExpansionForKnots} There exists an expansion $Z\colon \{\text{w-knots}\}\to\calA^{sw}(\uparrow)$. \end{theorem} \begin{proof} It is best to define $Z$ by an example, and it is best to display the example only as a picture: \[ \pstex{ZwKnotsExample} \] It is clear how to define $Z(K)$ in the general case --- for every crossing in $K$ place an exponential reservoir of arrows (compare with Equation~\eqref{eq:reservoir}) next to that crossing, with the arrows heading from the upper strand to the lower strand, taking positive reservoirs ($e^a$, with $a$ symbolizing the arrow) for positive crossings and negative reservoirs ($e^{-a}$) for negative crossings, and then tug the skeleton until it looks like a straight line. Note that the TC relation in $\calA^{sw}$ is used to show that all reasonable ways of placing an arrow reservoir at a crossing (with its heading and sign fixed) are equivalent: \[ \pstex{FourWays} \] The same proof that shows the invariance of $Z$ in the braid case (see Theorem~\ref{thm:RInvariance}) works here as well\footnote{A tiny bit of extra care is required for invariance under \Rs: it easily follows from RI.}, and the same argument as in the braid case shows the universality of $Z$. \qed \end{proof} \begin{remark} \label{rem:ZwForGD} Using the language of Gauss diagrams (Remark~\ref{rem:GD}) the definition of $Z$ is even simpler. Simply map every positive arrow in a Gauss diagram to a positive ($e^a$) reservoir, and every negative one to a negative ($e^{-a}$) reservoir: \[ \pstex{ZwForGD} \] \end{remark} An expansion (a universal finite type invariant) is as interesting as its target space, for it is just a tool that takes linear functionals on the target space to finite type invariants on its domain space. The purpose of the next section is to find out how interesting are our present target space, $\calA^{sw}(\uparrow)$, and its ``parent'', $\calA^w(\uparrow)$. \draftcut \subsection{Jacobi Diagrams, Trees and Wheels} \label{subsec:Jacobi} In studying $\calA^w(\uparrow)$ we again follow the model set by usual knots: we introduce the space $\calA^{wt}$ of ``w-Jacobi diagrams'' and show that it is isomorphic to $\calA^w$. Major advantages of working with $\calA^{wt}$ are that the co-product, the primitives, and the relationship with Lie algebras are much more natural and easy to describe. Compare the following definitions and theorem with~\cite[Section~3]{Bar-Natan:OnVassiliev}. \begin{definition} \label{def:wJac} A ``w-Jacobi diagram on a long line skeleton''\footnote{What a mouthful! We usually short this to ``w-Jacobi diagram'', or sometimes ``arrow diagram'' or just ``diagram''.} is a connected graph made of the following ingredients: \begin{itemize} \item A ``long'' oriented ``skeleton'' line. We usually draw the skeleton line a bit thicker for emphasis. \item Other directed edges, usually called ``arrows''. \item Trivalent ``skeleton vertices'' in which an arrow starts or ends on the skeleton line. \item Trivalent ``internal vertices'' in which {\em two arrows end and one arrow begins} (this will be important in Section \ref{subsec:LieAlgebras} where we relate these diagrams to Lie algebras). The internal vertices are ``oriented'' --- of the two arrows that end in an internal vertices, one is marked as ``left'' and the other is marked as ``right''. In reality when a diagram is drawn in the plane, we almost never mark ``left'' and ``right'', but instead assume the ``left'' and ``right'' inherited from the plane, as seen from the outgoing arrow from the given vertex. \end{itemize} Note that we allow multiple arrows connecting the same two vertices (though at most two are possible, given connectedness and trivalence) and we allow ``bubbles'' --- arrows that begin and end in the same vertex. Also keep in mind that for the purpose of determining equality of diagrams the skeleton line is distinguished. The ``degree'' of a w-Jacobi diagram is half the number of trivalent vertices in it, including both internal and skeleton vertices. An example of a w-Jacobi diagram is in Figure~\ref{fig:wJacDiag}. \end{definition} \begin{figure} \[ \pstex{wJacDiag} \] \caption{A degree 11 w-Jacobi diagram on a long line skeleton. It has a skeleton line at the bottom, 13 vertices along the skeleton (of which 2 are incoming and 11 are outgoing), 9 internal vertices (with only one explicitly marked with ``left'' ($l$) and ``right'' ($r$)) and one bubble. The five quadrivalent vertices that seem to appear in the diagram are just projection artifacts and graph-theoretically, they don't exist.} \label{fig:wJacDiag} \end{figure} \begin{definition} Let $\glos{\calD^{wt}}(\uparrow)$ be the graded vector space of formal linear combinations\footnote{$\bbQ$-linear, or any other field of characteristic 0.} of w-Jacobi diagrams on a long line skeleton, and let $\glos{\calA^{wt}}(\uparrow)$ be $\calD^{wt}(\uparrow)$ modulo the $\glos{\aSTU_1}$, $\glos{\aSTU_2}$, and TC relations of Figure~\ref{fig:aSTU}. Note that each diagram appearing in each $\aSTU$ relation has a ``central edge'' $e$ which can serve as an ``identifying name'' for that $\aSTU$. Thus, given a diagram $D$ with a marked edge $e$ which is either on the skeleton or which contacts the skeleton, there is an unambiguous $\aSTU$ relation ``around'' or ``along'' the edge $e$. \end{definition} \begin{figure} \[ \pstex{aSTU} \] \caption{The $\protect\aSTU_{1,2}$ and TC relations with their ``central edges'' marked $e$.} \label{fig:aSTU} \end{figure} \begin{figure} \[ \pstex{aIHX} \] \caption{The $\protect\aAS$ and $\protect\aIHX$ relations.} \label{fig:aIHX} \end{figure} We like to call the following theorem ``the bracket-rise theorem'', for it justifies the introduction of internal vertices, and as should be clear from the $\aSTU$ relations and as will become even clearer in Section~\ref{subsec:LieAlgebras}, internal vertices can be viewed as ``brackets''. Two other bracket-rise theorems are Theorem~6 of~\cite{Bar-Natan:OnVassiliev} and Ohtsuki's theorem, i.e., Theorem~4.9 of~\cite{Polyak:ArrowDiagrams}. \begin{theorem}[Bracket-rise] \label{thm:BracketRise} The obvious inclusion $\iota\colon \calD^v(\uparrow)\to\calD^{wt}(\uparrow)$ of arrow diagrams (see Definition~\ref{def:ArrowDiagrams}) into w-Jacobi diagrams descends to the quotient $\calA^w(\uparrow)$ and induces an isomorphism\footnote{At this point a vector space isomorphism, but we'll soon define a bialgebra structure on $\calA^{wt}$ to make it into an isomorphism of bialgebras.} $\bar\iota\colon \calA^w(\uparrow)\stackrel{\sim}{\longrightarrow} \calA^{wt}(\uparrow)$. Furthermore, the $\glos{\aAS}$ and $\glos{\aIHX}$ relations of Figure~\ref{fig:aIHX} hold in $\calA^{wt}(\uparrow)$. \end{theorem} \begin{proof} The proof, joint with D.~Thurston, is modelled after the proof of Theorem~6 of~\cite{Bar-Natan:OnVassiliev}. To show that $\iota$ descends to $\calA^w(\uparrow)$ we just need to show that in $\calA^{wt}(\uparrow)$, $\aft$ follows from $\aSTU_{1,2}$. Indeed, applying $\aSTU_1$ along the edge $e_1$ and $\aSTU_2$ along $e_2$ in the picture below, we get the two sides of $\aft$: \begin{equation} \label{eq:STUto4T} \pstex{STUto4T} \end{equation} The fact that $\bar\iota$ is surjective is easy: indeed, for diagrams in $\calA^{wt}(\uparrow)$ that have no internal vertices there is nothing to show, for they are really in $\calA^w(\uparrow)$. Further, by repeated use of $\aSTU_{1,2}$ relations, all internal vertices in any diagram in $\calA^{wt}(\uparrow)$ can be removed (remember that the diagrams in $\calA^{wt}(\uparrow)$ are always connected, and in particular, if they have an internal vertex they must have an internal vertex connected by an edge to the long line skeleton, and the latter vertex can be removed first). To complete the proof that $\bar\iota$ is an isomorphism it is enough to show that the ``elimination of internal vertices'' procedure of the last paragraph is well-defined -- that its output is independent of the order in which $\aSTU_{1,2}$ relations are applied in order to eliminate internal vertices. Indeed, this done, the elimination map would by definition satisfy the $\aSTU_{1,2}$ relations and thus, descend to a well-defined inverse for $\bar\iota$. On diagrams with just one internal vertex, Equation~\eqref{eq:STUto4T} shows that all ways of eliminating that vertex are equivalent modulo $\aft$ relations, and hence, the elimination map is well-defined on such diagrams. We proceed by induction on the number of internal vertices. We have shown that $\bar\iota$ is well-defined if there is only one internal vertex. Now assume that we have shown that the elimination map is well defined on all diagrams with at most $k$ internal vertices for some positive integer $k\geq 1$, and let $D$ be a diagram with $(k+1)$ internal vertices. Let $e$ and $e'$ be edges in $D$ that connect the skeleton of $D$ to an internal vertex. We need to show that any elimination process that begins with eliminating $e$ yields the same answer, modulo $\aft$, as any elimination process that begins with eliminating $e'$. There are several cases to consider. \parpic[r]{$\pstex{CaseI}$} {\bf Case I.} The edges $e$ and $e'$ connect the skeleton to {\em different} internal vertices of $D$. In this case, after eliminating $e$ we get a signed sum of two diagrams with exactly 7 internal vertices, and since the elimination process is well-defined on such diagrams, we may as well continue by eliminating $e'$ in each of those, getting a signed sum of 4 diagrams with 6 internal vertices each. On the other hand, if we start by eliminating $e'$ we can continue by eliminating $e$, and we get the {\em same} signed sum of 4 diagrams with 6 internal vertices. \parpic[r]{$\pstex{CaseII}$} {\bf Case II.} The edges $e$ and $e'$ are connected to the same internal vertex $v$ of $D$, yet some other edge $e''$ exists in $D$ that connects the skeleton of $D$ to some other internal vertex $v'$ in $D$. In that case, use the previous case and the transitivity of equality: (elimination starting with $e$)=(elimination starting with $e''$)=(elimination starting with $e'$). \parpic[r]{$\pstex{CaseIII}$} {\bf Case III.} This is what remains if neither Case I nor Case II hold. In that case, $D$ must have a schematic form as on the right, with the ``blob'' not connected to the skeleton other than via $e$ or $e'$, yet further arrows may exist outside of the blob. Let $f$ denote the edge connecting the blob to $e$ and $e'$. The ``two in one out'' rule for vertices implies that any part of a diagram must have an excess of incoming edges over outgoing edges, equal to the total number of vertices in that diagram part. Applying this principle to the blob, we find that it must contain exactly one vertex, as shown on the right below. Then by the ``two in one out'' rule $f$ must be oriented upwards, and hence, by the ``two in one out'' rule again, $e$ and $e'$ must be oriented upwards as well. \parpic[r]{$\pstex{CaseIIIa}$} We leave it to the reader to verify that in this case the two ways of applying the elimination procedure, $e$ and then $f$ or $e'$ and then $f$, yield the same answer modulo $\aft$ (in fact, that answer is $0$). We also leave it to the reader to verify that $\aSTU_1$ implies $\aAS$ and $\aIHX$. In Section \ref{subsec:LieAlgebras} we'll describe the relationship between $\calA^{wt}$ and Lie algebras. Algebraically, the relations $\aSTU_1$, $\aAS$ and $\aIHX$ are restatements of the anti-symmetry of the bracket and of Jacobi's identity: if $[x,y]:=xy-yx$, then $0=[x,y]+[y,x]$ and $[x,[y,z]]=[[x,y],z]-[[x,z],y]$. \qed \end{proof} Note that $\calA^{wt}(\uparrow)$ inherits algebraic structure from $\calA^w(\uparrow)$: it is an algebra by concatenation of diagrams, and a co-algebra with $\Delta(D)$, for $D\in\calD^{wt}(\uparrow)$, being the sum of all ways of dividing $D$ between a ``left co-factor'' and a ``right co-factor'' so that connected components of $D-S$ are kept intact, where $S$ is the skeleton line of $D$ (compare with~\cite[Definition~3.7]{Bar-Natan:OnVassiliev}). As $\calA^w(\uparrow)$ and $\calA^{wt}(\uparrow)$ are canonically isomorphic, from this point on we will not keep the distinction between the two spaces. One may add the RI relation to the definition of $\calA^{wt}(\uparrow)$ to get a space $\calA^{swt}(\uparrow)$. For an unframed version one may add the stronger framing independence (FI) relation, setting $D_L=D_R=0$, with $D_L$ and $D_R$ the single arrows as in Figure~\ref{fig:AwGenerators}. The resulting space is called $\calA^{rwt}(\uparrow)$. The statement and proof of the bracket rise theorem adapt with no difficulty, and we find that $\calA^{sw}(\uparrow)\cong\calA^{swt}(\uparrow)$ and $\calA^{rw}(\uparrow)\cong\calA^{rwt}(\uparrow)$. In the future we'll drop the $t$ from all superscripts. The advantages of allowing internal trivalent vertices are already apparent (for example, note that there is a nice description of primitive elements: they are the arrow diagrams which remain connected if the skeleton is removed). Further advantages will emerge in Section \ref{subsec:LieAlgebras}. \begin{figure} \[ \pstex{AwGenerators} \] \caption{The left-arrow diagram $D_L$, the right-arrow diagram $D_R$ and the $k$-wheel $w_k$.} \label{fig:AwGenerators} \end{figure} \begin{theorem} \label{thm:Aw} The bialgebra $\calA^w(\uparrow)$ is the bialgebra of polynomials in the diagrams $\glos{D_L}$, $\glos{D_R}$ and $\glos{w_k}$ (for $k\geq 1$) shown in Figure~\ref{fig:AwGenerators}, where $\deg D_L=\deg D_R=1$ and $\deg w_k=k$, subject to the one relation $w_1=D_L-D_R$. Thus, $\calA^w(\uparrow)$ has two generators in degree 1 and one generator in every degree greater than 1, as stated in Section~\ref{subsec:SomeDimensions}. \end{theorem} \begin{proof} (sketch). Readers familiar with the diagrammatic PBW theorem~\cite[Theorem~8]{Bar-Natan:OnVassiliev} will note that it has a direct analogue for the $\calA^w(\uparrow)$ case, and that the proof in~\cite{Bar-Natan:OnVassiliev} carries through almost verbatim. Namely, the space $\calA^w(\uparrow)$ is isomorphic to a space $\glos{\calB^w}$ of ``unitrivalent diagrams'' with symmetrized univalent ends modulo $\aAS$ and $\aIHX$. Given the ``two in one out'' rule for arrow diagrams in $\calA^w(\uparrow)$ (and hence, in $\calB^w$) the connected components of diagrams in $\calB^w$ can only be ``trees'' or ``wheels''. A tree is a unitrivalent diagram with no cycles (oriented or not). A wheel is a single oriented cycle with some number of incoming ``spokes'' (see $w_k$ in Figure \ref{fig:AwGenerators} and remove the skeleton line). The reader might object that there are also ``wheels of trees'': trees attached to an oriented cycle, but these can be reduced to linear combinations of wheels using the $\aIHX$ relation. Trees vanish if they have more than one leaf, as their leafs are symmetric while their internal vertices are anti-symmetric, so $\calB^w$ is generated by wheels and by the one-leaf-one-root tree, which is simply a single arrow. Wheels map to the $w_k$ in $\calA^w(\uparrow)$ under the isomorphism, and the arrow maps to the average of $D_L$ and $D_R$. The relation $w_1=D_L-D_R$ is then easily verified using $\aSTU_2$. One may also argue directly, without using $\calB^w$. In short, let $D$ be a diagram in $\calA^w(\uparrow)$ and $S$ is its skeleton. Then $D-S$ may have several connected components, whose ``legs'' are intermingled along $S$. Using the $\aSTU$ relations these legs can be sorted (at a cost of diagrams with fewer connected components, which could have been treated earlier in an inductive proof). At the end of the sorting procedure one can see that the only diagrams that remain are our declared generators. It remains to show that our generators are linearly independent (apart from the relation $w_1=D_L-D_R$). For the generators in degree 1, simply write everything out explicitly in the spirit of Section~\ref{subsubsec:DegreeOne}. In higher degrees there is only one primitive diagram in each degree, so it is enough to show that $w_k\neq 0$ for every $k$. This can be done ``by hand'', but it is more easily done using Lie algebraic tools in Section~\ref{subsec:LieAlgebras}. \qed \end{proof} \begin{exercise} \label{exe:Asw} Show that the bialgebra $\calA^{rw}(\uparrow)$ (see Section~\ref{subsec:SomeDimensions}) is the bialgebra of polynomials in the wheel diagrams $w_k$ ($k\geq 2$), and that $\calA^{sw}(\uparrow)$ is the bialgebra of polynomials in the same wheel diagrams and an additional generator $\glos{D_A}:=D_L=D_R$. \end{exercise} \begin{proposition} \label{prop:AwCirc} In $\calA^w(\bigcirc)$ all wheels vanish, and hence, the bialgebra $\calA^w(\bigcirc)$ is the bialgebra of polynomials in a single variable $D_L=D_R$. \end{proposition} \begin{proof} This is Lemma~2.7 of~\cite{Naot:BF}. In short, a wheel in $\calA^w(\bigcirc)$ can be reduced using $\aSTU_2$ to a difference of trees, as shown in Figure \ref{fig:WheelInCircle}. One of these trees has two adjoining leafs, and hence, it is 0 by TC and $\aAS$. In the other two of the leafs can be commuted ``around the circle'' using TC until they are adjoining and hence vanish by TC and $\aAS$. \qed \end{proof} \begin{figure} \input{figs/WheelInCircle.pstex_t} \caption{Wheels in a circle vanish.}\label{fig:WheelInCircle} \end{figure} \begin{exercise} Show that $\calA^{sw}(\bigcirc)\cong\calA^w(\bigcirc)$ yet $\calA^{rw}(\bigcirc)$ vanishes except in degree $0$. \end{exercise} The following two exercises may help the reader to develop a better ``feel'' for $\calA^w(\uparrow)$ and will be needed, within the discussion of the Alexander polynomial (especially within Definition~\ref{def:InterpretationMap}). \parpic[r]{\raisebox{-12mm}{$\pstex{CC}$}} \begin{exercise} Show that the ``commutators commute'' (\glost{CC}) relation, shown on the right, holds in $\calA^w(\uparrow)$. (Interpreted in terms of Lie algebras as in the next section, this relation becomes $[[x,y], [z,w]]=0$, and hence the name ``commutators commute''). Note that the proof of CC depends on the skeleton having a single component; later, when we will work with $\calA^w$-spaces with more complicated skeleta, the CC relation will not hold. \end{exercise} \parpic[r]{\raisebox{-2mm}{$\pstex{Hair}$}} \begin{exercise} \label{ex:Hair} Show that ``detached wheels'' and ``hairy $Y$ s'' make sense in $\calA^w(\uparrow)$. As on the right, a detached wheel is a wheel with a number of spokes, and a hairy $Y$ is a combinatorial $Y$ shape (three arrows meeting at a single internal vertex) with further ``hair'' on its trunk (its outgoing arrow). It is specified where the trunk and the leafs of the $Y$ connect to the skeleton, but it is not specified where the spokes of the wheel and where the hair on the $Y$ connect to the skeleton. The content of the exercise is to show that modulo the relations of $\calA^w(\uparrow)$, it is not necessary to specify this further information: all ways of connecting the spokes and the hair to the skeleton are equivalent. Like the previous exercise, this result depends on the skeleton having a single component. \end{exercise} \begin{remark} In the case of usual knots and usual chord diagrams, Jacobi diagrams have a topological interpretation using the Goussarov-Habiro calculus of claspers~\cite{Goussarov:3Manifolds, Habiro:Claspers}. In the w case a similar such calculus was developed by Watanabe in~\cite{Watanabe:ClasperMoves}. Various related results are at~\cite{HabiroKanenobuShima:R2K, HabiroShima:R2KII}. \end{remark} \draftcut \subsection{The Relation with Lie Algebras} \label{subsec:LieAlgebras} The theory of finite type invariants of knots is related to the theory of metrized Lie algebras via the space $\calA$ of chord diagrams, as explained in~\cite[Theorem~4 and Exercise~5.1]{Bar-Natan:OnVassiliev}. In a similar manner the theory of finite type invariants of w-knots is related to arbitrary finite-dimensional Lie algebras (or equivalently, to doubles of co-commutative Lie bialgebras, as explained below) via the space $\calA^w(\uparrow)$ of arrow diagrams. \subsubsection{Preliminaries} Given a finite dimensional Lie algebra\footnote{Over $\bbQ$, or another field of characteristic 0.} $\glos{\frakg}$ let $\glos{I\frakg}:=\frakg^\ast\rtimes\frakg$ be the semi-direct product of the dual $\frakg^\ast$ of $\frakg$ with $\frakg$, with $\frakg^\ast$ taken as an abelian algebra and with $\frakg$ acting on $\frakg^\ast$ by the usual coadjoint action. In formulae, \[ I\frakg=\{(\varphi, x)\colon \,\varphi\in\frakg^\ast,\,x\in\frakg\}, \] \[ [(\varphi_1,x_1), (\varphi_2,x_2)] = (x_1\varphi_2-x_2\varphi_1, [x_1,x_2]). \] In the case where $\frakg$ is the algebra $\mathfrak{so}(3)$ of infinitesimal symmetries of $\bbR^3$, its dual $\frakg^\ast$ is $\bbR^3$ itself with the usual action of $\mathfrak{so}(3)$ on it, and $I\frakg$ is the algebra $\bbR^3\rtimes \mathfrak{so}(3)$ of infinitesimal affine isometries of $\bbR^3$. This is the Lie algebra of the Euclidean group of isometries of $\bbR^3$, which is often denoted $ISO(3)$. This explains our choice of the name $I\frakg$. Note that, if $\frakg$ is a co-commutative Lie bialgebra, then $I\frakg$ is the ``double'' of $\frakg$~\cite{Drinfeld:QuantumGroups}. This is a significant observation, for it is a part of the relationship between this paper and the Etingof-Kazhdan theory of quantization of Lie bialgebras~\cite{EtingofKazhdan:BialgebrasI}. Yet we will make no explicit use of this observation below. In the construction that follows we are going to define a map from $\calA^w$ to $\glos{\calU}(I\frakg)$, the universal enveloping algebra of $I\frakg$. Note that a map ${\calA}^w \to \calU(I\frakg)$ is ``almost the same'' as a map $\calA^{sw} \to \calU(I\frakg)$, in the following sense. The quotient map $p\colon {\calA}^w \to \calA^{sw}$ has a one-sided inverse $F\colon \calA^{sw} \to {\calA}^w$ defined by \[ F(D)= \sum_{k=0}^\infty \frac{(-1)^k}{k!} S_L^k(D)\cdot w_1^k. \] Here $S_L$ denotes the map that sends an arrow diagram to the sum of all ways of deleting a left-going arrow, $S_L^k$ is $S_L$ applied $k$ times, and $w_1$ denotes the 1-wheel, as shown in Figure~\ref{fig:AwGenerators}. The reader can verify that $F$ is well-defined, an algebra- and co-algebra homomorphism, and that $p\circ F= id_{\calA^{sw}}$. \subsubsection{The Construction} Fixing a finite dimensional Lie algebra $\frakg$, we construct a map $\glos{\calT^w_\frakg}\colon {\calA}^w\to\calU(I\frakg)$ which assigns to every arrow diagram $D$ an element of the universal enveloping algebra $\calU(I\frakg)$. As is often the case in our subject, a picture of a typical example is worth more than a formal definition: \[ \def\I{{$I$}} \def\B{{$B$}} \def\g{{$\frakg$}} \def\d{{$\frakg^\ast$}} \def\p{{$\frakg^\ast\otimes\frakg^\ast\otimes\frakg\otimes\frakg \otimes\frakg^\ast\otimes\frakg^\ast$}} \def\u{{$\calU(I\frakg)$}} \pstex{Twg} \] In short, we break up the diagram $D$ into its constituent pieces and assign a copy of the structure constants tensor $B\in\frakg^\ast\otimes\frakg^\ast\otimes\frakg$ to each internal vertex $v$ of $D$ (keeping an association between the tensor factors in $\frakg^\ast\otimes\frakg^\ast\otimes\frakg$ and the edges emanating from $v$, as dictated by the orientations of the edges and of the vertex $v$ itself). We assign the identity tensor in $\frakg^\ast\otimes\frakg$ to every arrow in $D$ that is not connected to an internal vertex, and contract any pair of factors connected by a fully internal arrow. The remaining tensor factors ($\frakg^\ast\otimes\frakg^\ast\otimes\frakg\otimes\frakg \otimes\frakg^\ast\otimes\frakg^\ast$ in our examples) are all along the skeleton and can thus be ordered by the skeleton. We then multiply these factors to get an output $\calT^w_\frakg(D)$ in $\calU(I\frakg)$. It is also useful to restate this construction given a choice of a basis. Let $\glos{(x_j)}$ be a basis of $\frakg$ and let $\glos{(\varphi^i)}$ be the dual basis of $\frakg^\ast$, so that $\varphi^i(x_j)=\delta^i_j$, and let $\glos{b_{ij}^k}$ denote the structure constants of $\frakg$ in the chosen basis: $[x_i,x_j]=\sum b_{ij}^kx_k$. Mark every arrow in $D$ with lower case Latin letter from within\footnote{The supply of these can be made inexhaustible by the addition of numerical subscripts.} $\{i,j,k,\dots\}$. Form a product $P_D$ by taking one $b_{\alpha\beta}^\gamma$ factor for each internal vertex $v$ of $D$ using the letters marking the edges around $v$ for $\alpha$, $\beta$ and $\gamma$ and by taking one $x_\alpha$ or $\varphi^\beta$ factor for each skeleton vertex of $D$, taken in the order that they appear along the long line skeleton, with the indices $\alpha$ and $\beta$ dictated by the edge markings and with the choice between factors in $\frakg$ and factors in $\frakg^\ast$ dictated by the orientations of the edges. Finally let $\calT^w_\frakg(D)$ be the sum of $P_D$ over the indices $i,j,k,\dots$ running from $1$ to $\dim\frakg$: \begin{equation} \label{eq:Twb} \def\P{{$\displaystyle \sum_{i,j,k,l,m,n=1}^{\dim\frakg} \hspace{-4mm} b_{ij}^kb_{kl}^m \varphi^i\varphi^jx_nx_m\varphi^l\in\calU(I\frakg) $}} \pstex{Twb} \end{equation} The next proposition is easy to verify (compare with~\cite[Theorem~4 and Exercise~5.1]{Bar-Natan:OnVassiliev}): \begin{proposition} The above two definitions of $T^w_\frakg$ agree, are independent of the choices made within them, and respect all the relations defining ${\calA}^w$. \qed \end{proposition} While we do not provide a proof of this proposition here, it is worthwhile to state the correspondence between the relations defining ${\calA}^w$ and the Lie algebraic information in $\calU(I\frakg)$: $\aAS$ is the antisymmetry of the bracket of $\frakg$, $\aIHX$ is the Jacobi identity of $\frakg$, $\aSTU_1$ and $\aSTU_2$ are the relations $[x_i,x_j]=x_ix_j-x_jx_i$ and $[\varphi^i,x_j]=\varphi^ix_j-x_j\varphi^i$ in $\calU(I\frakg)$, $TC$ is the fact that $\frakg^\star$ is taken as an abelian algebra, and $\aft$ is the fact that the identity tensor in $\frakg^\ast\otimes\frakg$ is $\frakg$-invariant. \subsubsection{Example: The 2 Dimensional Non-Abelian Lie Algebra} Let $\frakg$ be the Lie algebra with two generators $x_{1,2}$ satisfying $[x_1,x_2]=x_2$, so that the only non-vanishing structure constants $b_{ij}^k$ of $\frakg$ are $b_{12}^2=-b_{21}^2=1$. Let $\varphi^i\in\frakg^\ast$ be the dual basis of $x_i$; by an easy calculation, we find that in $I\frakg$ the element $\varphi^1$ is central, while $[x_1,\varphi^2]=-\varphi^2$ and $[x_2,\varphi^2]=\varphi^1$. We calculate $\calT^w_\frakg(D_L)$, $\calT^w_\frakg(D_R)$ and $\calT^w_\frakg(w_k)$ using the ``in basis'' technique of Equation~\eqref{eq:Twb}. The outputs of these calculations lie in $\calU(I\frakg)$; we display these results in a PBW basis in which the elements of $\frakg^\ast$ precede the elements of $\frakg$: \begin{eqnarray} \calT^w_\frakg(D_L) &=& x_1\varphi^1+x_2\varphi^2 = \varphi^1x_1+\varphi^2x_2+[x_2,\varphi^2] = \varphi^1x_1+\varphi^2x_2+\varphi_1, \notag \\ \calT^w_\frakg(D_R) &=& \varphi^1x_1+\varphi^2x_2, \label{eq:2DExample} \\ \calT^w_\frakg(w_k) &=& (\varphi^1)^k. \notag \end{eqnarray} \parpic[r]{$\pstex{4wheel}$} For the last assertion above, note that all non-vanishing structure constants $b_{ij}^k$ in our case have $k=2$, and therefore all indices corresponding to edges that exit an internal vertex must be set equal to $2$. This forces the ``hub'' of $w_k$ to be marked $2$ and therefore the legs to be marked $1$, and therefore $w_k$ is mapped to $(\varphi^1)^k$. Note that the calculations in~\eqref{eq:2DExample} are consistent with the relation $D_L-D_R=w_1$ of Theorem~\ref{thm:Aw} and that they show that other than that relation, the generators of ${\calA}^w$ are linearly independent. \draftcut \subsection{The Alexander Polynomial} \label{subsec:Alexander} Let $K$ be a long w-knot, and let $Z(K)$ be the invariant of Theorem~\ref{thm:ExpansionForKnots}. Theorem~\ref{thm:Alexander} below asserts that apart from self-linking, $Z(K)$ contains precisely the same information as the Alexander polynomial $A(K)$ of $K$ (recalled below). But we have to start with some definitions. \begin{figure} \begin{center} $\pstex{8-17}$ \qquad \raisebox{-18mm}{\includegraphics[height=40mm]{figs/SandersonsGarden_640.ps}} \end{center} \caption{ A long $8_{17}$, with the span of crossing $\#3$ marked. The projection is as in Brian Sanderson's garden. See~\cite{WKO}/\href{http://www.math.toronto.edu/~drorbn/papers/WKO/SandersonsGarden.html}{\tt SandersonsGarden.html}. } \label{fig:817} \end{figure} \begin{definition} \label{def:STA} Enumerate the crossings of $K$ from $1$ to $n$ in some arbitrary order. For \linebreak $1\leq j\leq n$, the ``span'' of crossing $\#i$ is the connected open interval along the line parametrizing $K$ between the two times $K$ ``visits'' crossing $\#i$ (see Figure~\ref{fig:817}). Form a matrix $T=\glos{T(K)}$ with $T_{ij}$ the indicator function of ``the lower strand of crossing $\#j$ is within the span of crossing $\#i$'' (so $T_{ij}$ is $1$ if for a given $i,j$ the quoted statement is true, and $0$ otherwise). Let $\glos{s_i}$ be the sign of crossing $\#i$ (recall that $\overcrossing$ is positive, $\undercrossing$ is negative; $(-,-,-,-,+,+,+,+)$ for Figure~\ref{fig:817}), let $\glos{d_i}$ be $+1$ if $K$ visits the ``over'' strand of crossing $\#i$ before visiting the ``under'' strand of that crossing, and let $d_i=-1$ otherwise ($(-,+,-,+,-,+,-,+$) for Figure~\ref{fig:817}). Let $S=\glos{S(K)}$ be the diagonal matrix with $S_{ii}=s_id_i$, and for an indeterminate $\glos{X}$, let $X^{-S}$ denote the diagonal matrix with diagonal entries $X^{-s_id_i}$. Finally, let $\glos{A(K)}$ be the Laurent polynomial in $\bbZ[X,X^{-1}]$ given by \begin{equation} \label{eq:AKDef} A(K)(X) := \det\left(I+T(I-X^{-S})\right). \end{equation} \end{definition} \begin{example} For the knot diagram in Figure~\ref{fig:817}, \[ \scriptstyle T = \left(\begin{smallmatrix} 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \end{smallmatrix}\right), \quad S = \left(\begin{smallmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{smallmatrix}\right), \quad\text{and}\quad A = \left|\begin{smallmatrix} 1 & 1-X & 1-X^{-1} & 1-X & 1-X & 0 & 1-X & 0 \\ 0 & 1 & 1-X^{-1} & 0 & 1-X & 0 & 0 & 0 \\ 0 & 1-X & 1 & 0 & 1-X & 0 & 0 & 0 \\ 0 & 1-X & 0 & 1 & 1-X & 0 & 1-X & 0 \\ 0 & 1-X & 0 & 1-X & 1 & 1-X^{-1} & 1-X & 1-X^{-1} \\ 0 & 1-X & 0 & 1-X & 0 & 1 & 1-X & 0 \\ 0 & 0 & 0 & 1-X & 0 & 1-X^{-1} & 1 & 0 \\ 0 & 0 & 0 & 1-X & 0 & 1-X^{-1} & 0 & 1 \end{smallmatrix}\right|. \] The last determinant equals $-X^3+4X^2-8X+11-8X^{-1}+4X^{-2}-X^{-3}$, the Alexander polynomial of the knot $8_{17}$ (see e.g.~\cite{Rolfsen:KnotsAndLinks}). \end{example} \begin{theorem} \label{thm:AlexanderFormula} (Lee,~\cite[Theorem~1]{Lee:AlexanderInvariant}) For any (classical) knot $K$, $A(K)$ is equal to the normalized Alexander polynomial~\cite{Rolfsen:KnotsAndLinks} of $K$. \qed \end{theorem} The Mathematica notebook~\cite[``wA'']{WKO} verifies Theorem~\ref{thm:AlexanderFormula} for all prime knots with up to 11 crossings. The following theorem asserts that $Z(K)$ can be computed from $A(K)$ (see Equation~\eqref{eq:AtoZ}) and that modulo a certain additional relation and with the appropriate identifications in place, $Z(K)$ {\em is} $A(K)$ (see Equation~\eqref{eq:ZisA}). \begin{theorem} \label{thm:Alexander} (Proof in Section~\ref{subsec:AlexanderProof}). Let $x$ be an indeterminate, let $\sl$ be self-linking as in Exercise~\ref{ex:sl}, let $D_A:=D_L=D_R$ and $w_k$ be as in Figure~\ref{fig:AwGenerators}, and let $\glos{w}\colon \bbQ\llbracket x\rrbracket \to\calA^w$ be the linear map defined by $x^k\mapsto w_k$. Then for a long w-knot $K$, \begin{equation} \label{eq:AtoZ} Z(K) = \underbrace{ \exp_{\calA^{sw}}\left(\sl(K)D_A\right) }_\text{$\sl$ coded in arrows} \cdot \underbrace{ \exp_{\calA^{sw}}\left(-w\left(\log_{\bbQ\llbracket x\rrbracket} A(K)(e^x) \right)\right) }_\text{main part: Alexander coded in wheels}, \end{equation} where the logarithm and inner exponentiation are computed by formal power series in $\bbQ\llbracket x\rrbracket$ and the outer exponentiations are likewise computed in $\calA^{sw}$. \end{theorem} \parpic[r]{$\pstex{wkl}$} Let $\calA^\text{reduced}$ be $\calA^{sw}$ modulo the additional relations $D_A=0$ and $w_kw_l=w_{k+l}$ for $k,l\neq 1$. The quotient $\calA^\text{reduced}$ can be identified with the vector space of (infinite) linear combinations of the $w_k$ (with $k\neq 1$). Identifying the $k$-wheel $w_k$ with $x^k$, we see that $\calA^\text{reduced}$ is the space of power series in $x$ having no linear terms. Note by inspecting Equation~\eqref{eq:AKDef} that $A(K)(e^x)$ never has a term linear in $x$, and that modulo $w_kw_l=w_{k+l}$, the exponential and the logarithm in Equation~\eqref{eq:AtoZ} cancel each other out. Hence, within $\calA^\text{reduced}$, \begin{equation} \label{eq:ZisA} Z(K) = A^{-1}(K)(e^x). \end{equation} \begin{remark} In~\cite{HabiroKanenobuShima:R2K} Habiro, Kanenobu, and Shima show that all coefficients of the Alexander polynomial are finite type invariants of w-knots, and in~\cite{HabiroShima:R2KII} Habiro and Shima show that all finite type invariants of w-knots are polynomials in the coefficients of the Alexander polynomial. Thus, Theorem~\ref{thm:Alexander} is merely an ``explicit form'' of these earlier results. \end{remark} \draftcut \subsection{Proof of Theorem~\ref{thm:Alexander}} \label{subsec:AlexanderProof} We start with a sketch. The proof of Theorem~\ref{thm:Alexander} can be divided into three parts: differentiation, bulk management, and computation. \noindent{\bf Differentiation.} Both sides of our goal, that is, Equation~\eqref{eq:AtoZ}, are exponential in nature. When seeking to show an equality of exponentials it is often beneficial to compare their derivatives\footnote{Thanks, Dylan.}. In our case the useful ``derivatives'' to use are the ``Euler operator'' $\glos{E}$ (``multiply every term by its degree'', an analogue of $f\mapsto xf'$, defined in Section~\ref{subsubsec:Euler}), and the ``normalized Euler operator'' $Z\mapsto\glos{\tilE} Z:=Z^{-1}EZ$, which is a variant of the logarithmic derivative $f\mapsto x(\log f)'=xf'/f$. Since $\tilE$ is one to one (see Section~\ref{subsubsec:Euler}) and since we know how to apply $\tilE$ to the right hand side of Equation~\eqref{eq:AtoZ} (see Section~\ref{subsubsec:Euler}), it is enough to show that with $\glos{B}:=T(\exp(-xS)-I)$ and suppressing the fixed w-knot $K$ from the notation, \begin{equation} \label{eq:EofAtoZ} EZ = Z\cdot\left( \sl\cdot D_A-w\!\left[x\tr\left( (I-B)^{-1}TS\exp(-xS) \right)\right] \right) \qquad \text{ in }\calA^{sw}. \end{equation} \noindent{\bf Bulk Management.} Next we seek to understand the left hand side of Equation~\eqref{eq:EofAtoZ}. $Z$ is made up of ``quantities in bulk'': arrows that come in exponential ``reservoirs''. As it turns out, $EZ$ is made up of the same bulk quantities, but also allowing for a single non-bulk ``excitation'', which we often highlight in red. (compare with $Ee^x={\red x}\cdot e^x$; the ``bulk'' $e^x$ remains, and single ``excited red'' $\red x$ gets created). We wish to manipulate and simplify that red excitation. This is best done by introducing a certain module, $\glos{\IAM_K}$, the ``Infinitesimal Alexander Module'' of $K$ (see Section~\ref{subsubsec:IAM}). The elements of $\IAM_K$ can be thought of as names for ``bulk objects with a red excitation'', and hence, there is an ``interpretation map'' $\glos{\iota}\colon \IAM_K\to\calA^{sw}$, which maps every ``name'' into the object it represents. There are three special elements in $\IAM_K$: an element $\glos{\lambda}$, which is the name of $EZ$ (that is, $\iota(\lambda)=EZ$), the element $\glos{\delta_A}$ which is the name of $D_A\cdot Z$ (so $\iota(\delta_A)=D_A\cdot Z$), and an element $\glos{\omega_1}$ which is the name of a ``detached'' 1-wheel that is appended to $Z$. The latter can take a coefficient which is a power of $x$, with $\iota(x^k\omega_1)=w(x^{k+1})\cdot Z=(Z\text{ times a } (k+1)\text{-wheel})$. Thus, it is enough to show that in $\IAM_K$, \begin{equation} \label{eq:GoalInIAM} \lambda = \sl\cdot\delta_A - \tr\left((I-B)^{-1}TSX^{-S}\right)\omega_1, \quad\text{with}\quad X=e^x. \end{equation} Indeed, applying $\iota$ to both sides of the above equation, we get Equation~\eqref{eq:EofAtoZ} back again. \noindent{\bf Computation.} Last, we show in Section~\ref{sec:ComputeLambda} that Equation~\eqref{eq:GoalInIAM} holds true. This is a computation that happens entirely in $\IAM_K$ and does not mention finite type invariants, expansions or arrow diagrams in any way. \subsubsection{The Euler Operator} \label{subsubsec:Euler} Let $A$ be a completed, graded algebra with unit, in which all degrees are $\geq 0$. Define a continuous linear operator $E\colon A\to A$ by setting $Ea=(\deg a)a$ for homogeneous $a\in A$. In the case $A=\bbQ\llbracket x\rrbracket$, we have $Ef=xf'$, the standard ``Euler operator'': indeed, for each $n$, $Ex^n=nx^n=x\cdot(x^n)'$. Hence, we adopt the name $E$ for this operator in general. We say that $Z\in A$ is a ``perturbation of the identity'' if its degree 0 piece is 1. Such a $Z$ is always invertible. For such a $Z$, set $\tilE Z:=Z^{-1}\cdot EZ$, and call the thus (partially) defined operator $\tilE \colon A\to A$ the ``normalized Euler operator''. From this point on when we write $\tilE Z$ for some $Z\in A$, we automatically assume that $Z$ is a perturbation of the identity or that it is trivial to show that $Z$ is a perturbation of the identity. Note that for $f\in\bbQ\llbracket x\rrbracket$, we have $\tilE f=x(\log f)'$, so $\tilE$ is a variant of the logarithmic derivative. \begin{claim} $\tilE$ is one to one. \end{claim} \begin{proof} Assume $Z_1\neq Z_2$ and let $d$ be the smallest degree in which they differ. Then $d>0$ and in degree $d$ the difference $\tilE Z_1-\tilE Z_2$ is $d$ times the difference $Z_1-Z_2$, and hence, $\tilE Z_1\neq\tilE Z_2$. \qed \end{proof} Thus, in order to prove our goal, that is, Equation~\eqref{eq:AtoZ}, it is enough to compute $\tilE$ of both sides and to show the equality then. We start with the right hand side of Equation~\eqref{eq:AtoZ}; but first, we need some simple properties of $E$ and $\tilE$. The proofs of these properties are routine, and hence, they are omitted. \begin{proposition} The following hold true: \begin{enumerate} \item $E$ is a derivation: $E(fg)=(Ef)g+f(Eg)$. \item If $Z_1$ commutes with $Z_2$, then $\tilE(Z_1Z_2)=\tilE Z_1+\tilE Z_2$. \item If $z$ commutes with $Ez$, then $Ee^z=e^z(Ez)$ and $\tilE e^z=Ez$. \item If $w\colon A\to\calA$ is a morphism of graded algebras, then it commutes with $E$ and $\tilE$. \qed \end{enumerate} \end{proposition} Let us denote the right hand side of Equation~\eqref{eq:AtoZ} by $Z_1(K)$. Then, by the above proposition, remembering (see Theorem~\ref{thm:Aw}) that $\calA^{sw}$ is commutative and that $\deg D_A=1$, we have \[ \tilE Z_1(K) = \sl\cdot D_A-w(E\log A(K)(e^x)) = \sl\cdot D_A-w\left(x\frac{d}{dx}\log A(K)(e^x)\right). \] The rest is an exercise in matrices and differentiation. $A(K)$ is a determinant, see Equation~\eqref{eq:AKDef}, and in general, $\frac{d}{dx}\log\det(M) = \tr\left(M^{-1}\frac{d}{dx}M\right)$. So with $B=T(e^{-xS}-I)$ (so $M=I-B$), we have \[ \tilE Z_1(K) = \sl\cdot D_A + w\left(x\tr\left((I-B)^{-1}\frac{d}{dx}B\right)\right) = \sl\cdot D_A - w\left(x\tr\left((I-B)^{-1}TSe^{-xS}\right)\right), \] as promised in Equation~\eqref{eq:EofAtoZ}. \subsubsection{The Infinitesimal Alexander Module} \label{subsubsec:IAM} Let $K$ be a w-knot diagram. The ``Infinitesimal Alexander Module'' $\IAM_K$ of $K$, which is defined in detail below, is a certain module made from a certain space $\glos{\IAM^0_K}$ of pictures ``annotating'' $K$ with ``red excitations'' modulo some pictorial relations that indicate how the red excitations can be moved around. The space $\IAM^0_K$ in itself is made of three pieces, or ``sectors''. The ``A sector'' in which the excitations are red arrows, the ``Y sector'' in which the excitations are ``red hairy Y-diagrams'', and a rank 1 ``W sector'' for ``red hairy wheels''. There is an ``interpretation map'' $\glos{\iota}\colon \IAM^0_K\to\calA^w$ which descends to a well-defined (and homonymous) $\iota\colon \IAM_K\to\calA^w$. Finally, there are some special elements $\lambda$ and $\delta_A$ that live in the A sector of $\IAM^0_K$ and $\omega_1$ that lives in the W sector. In principle, the description of $\IAM^0_K$ and of $\IAM_K$ can be given independently of the interpretation map $\iota$, and there are some good questions to ask about $\IAM_K$ (and the special elements in it) that are completely independent of the interpretation of the elements of $\IAM_K$ as ``perturbed bulk quantities'' within $\calA^{sw}$. Yet $\IAM_K$ is a complicated object and we fear its definition will appear completely artificial without its interpretation. Hence, below the two definitions will be woven together. $\IAM_K$ and $\iota$ may equally well be described in terms of $K$ or in terms of the Gauss diagram of $K$ (see Remark~\ref{rem:GD}). For pictorial simplicity, we choose to use the latter; so let $G=G(K)$ be the Gauss diagram of $K$. It is best to read the following definition while at the same time studying Figure~\ref{fig:IAM0Def}. \begin{figure} \[ \pstex{IAM0Def} \] \caption{ A sample w-knot $K$, its Gauss diagram $G$, and one generator from each of the A, Y, and W sectors of $\IAM^0_K$. Red parts are marked with the letter ``r''. } \label{fig:IAM0Def} \end{figure} \begin{definition} Let $\glos{R}$ be the ring $\bbZ[X,X^{-1}]$ of Laurent polynomials in a variable $X$ with integer coefficients\footnote{Later, $X$ is interpreted in $\calA^w$ as a formal exponential $e^x$. So within $\IAM$ we can restrict to coefficients in $\bbZ$, yet in $\calA^w$ we must allow coefficients in $\bbQ$.}, and let $\glos{R_1}$ be the subring of polynomials that vanish at $X=1$ (i.e., whose sum of coefficients is $0$)\footnote{$R_1$ is only very lightly needed, and only within Definition~\ref{def:InterpretationMap}. In particular, all that we say about $\IAM_K$ that does not concern the interpretation map $\iota$ is equally valid with $R$ replacing $R_1$.}. Let $\IAM^0_K$ be the direct sum of the following three modules (which for the purpose of taking the direct sum, are all regarded as $\bbZ$-modules): \begin{enumerate} \item The ``A sector'' is the free $\bbZ$-module generated by all diagrams made from $G$ by the addition of a single unmarked ``red excitation'' arrow, whose endpoints are on the long line skeleton of $G$ and are distinct from each other and from all other endpoints of arrows in $G$. Such diagrams are considered combinatorially --- so two are equivalent iff they differ only by an orientation preserving diffeomorphism of the skeleton. Let us count: if $K$ has $n$ crossings, then $G$ has $n$ arrows and the skeleton of $G$ gets subdivided into $m:=2n+1$ arcs. An A sector diagram is specified by the choice of an arc for the tail of the red arrow and an arc for the head ($m^2$ choices), except if the head and the tail fall within the same arc, then their relative ordering has to be specified as well ($m$ further choices). So the rank of the A sector over $\bbZ$ is $m(m+1)$. \item The ``Y sector'' is the free $R_1$-module generated by all diagrams made from $G$ by the addition of a single ``red excitation'' $Y$-shape single-vertex graph, with two incoming edges (``tails'') and one outgoing (``head''), modulo anti-symmetry for the two incoming edges (again, considered combinatorially). Counting is more elaborate: when the three edges of the $Y$ end in distinct arcs in the skeleton of $G$, we have $\frac12m(m-1)(m-2)$ possibilities ($\frac12$ for the antisymmetry). When the two tails of the $Y$ lie on the same arc, we get $0$ by anti-symmetry, unless they are separated by the head of that $Y$ ($m$ possibilities). The remaining possibility is to have the head and one tail on one arc (order matters!) and the other tail on another, at $2m(m-1)$ possibilities. So the rank of the Y sector over $R_1$ is $\frac12m^2(m+1)$. \item The ``W sector'' is the rank 1 free $R$-module with a single generator $w_1$. It is not necessary for $w_1$ to have a pictorial representation, yet one, involving a single ``red'' 1-wheel, is shown in Figure~\ref{fig:IAM0Def}. This pictorial representation is consistent with the interpretation in the definition below of $\omega_1$ as a detached 1-wheel. \end{enumerate} \end{definition} \begin{definition} \label{def:InterpretationMap} The ``interpretation map'' $\iota\colon \IAM^0_K\to\calA^w$ is defined by sending the arrows (marked $+$ or $-$) of a diagram in $\IAM^0_K$ to $(e^{\pm a})$-exponential reservoirs of arrows, as in the definition of $Z$ (see Remark~\ref{rem:ZwForGD}). In addition, the red excitations of diagrams in $\IAM^0_K$ are interpreted as follows: \begin{enumerate} \item In the A sector, the red arrow is simply mapped to itself, with the colour red suppressed. \item In the Y sector diagrams have red $Y$~s and coefficients $f\in R_1$. Substitute $X=e^x$ in $f$, expand in powers of $x$, and interpret $x^kY$ as a ``hairy $Y$ with $k-1$ hairs'' as in Exercise~\ref{ex:Hair}. Note that $f(1)=0$, so only positive powers of $x$ occur. So we never need to worry about ``$Y$~s with $-1$ hairs''. This is the only point where the condition $f\in R_1$ (as opposed to $f\in R$) is needed. \item In the W sector treat the coefficients as above, but interpret $x^kw_1$ as a detached $w_{k+1}$. I.e., as a detached wheel with $k+1$ spokes, as in Exercise~\ref{ex:Hair}. \end{enumerate} \end{definition} As stated above, $\IAM_K$ is the quotient of $\IAM^0_K$ by some set of relations. The best way to think of this set of relations is as ``everything that's obviously annihilated by $\iota$''. Here's the same thing, in a more formal language: \begin{figure} \[ \pstex{IAMRelations} \] \caption{The relations $\calR$ making $\IAM_K$.} \label{fig:IAMRelations} \end{figure} \begin{definition} Let $\glos{\IAM_K}:=\IAM^0_K/\calR$, where $\glos{\calR}$ is the linear span of the relations depicted in Figure~\ref{fig:IAMRelations}. The top 8 relations are about moving a leg of the red excitation across an arrow head or an arrow tail in $G$. Since the red excitation may be either an arrow $A$ or a $Y$, its leg in motion may be either a tail or a head, and it may be moving either past a tail or past a head, there are 8 relations of that type. The $A_w$ relation corresponds to $D_L-D_R=w_1=0$. The $Y_w$ relation indicates the ``price'' (always a red $w_1$) of commuting a red head across a red tail. As per custom, in each case only the changing part of the diagrams involved is shown. Further, the red excitations are marked with the letter ``r'' and the sign of an arrow in $G$ is marked $s$; so always $s\in\{\pm 1\}$. The ``$A$'' relations in Figure~\ref{fig:IAMRelations} ($A_{tt}$, $A_{th}$, $A_{ht}$, $A_{hh}$, $A_w$) may be multiplied by a scalar in $\bbZ$, while the ``$Y$'' relations may be multiplied by a scalar in $R$. Hence, for example, $x^0w_1=0$ by $A_w$, yet $x^kw_1\neq 0$ for $k>0$. \end{definition} \begin{proposition} The interpretation map $\iota$ indeed annihilates all the relations in $\calR$. \end{proposition} \begin{proof} Both $\iota A_{tt}$ and $\iota Y_{tt}$ follow immediately from the TC relation. The formal identity $e^{\ad b}(a)=e^bae^{-b}$ (here $\ad$ denotes the adjoint representation) implies $e^{\ad b}(a)e^b=e^ba$, and hence, $ae^b-e^ba=(1-e^{\ad b})(a)e^b$. With $a$ interpreted as ``red head'', $b$ as ``black head'', and $\ad b$ as ``hair'' (justified by the $\iota$-meaning of hair and by the $\aSTU_1$ relation, see Figure~\ref{fig:aSTU}), the last equality becomes a proof of $\iota Y_{hh}$. Further pushing that same equality, we get $ae^b-e^ba=\frac{1-e^{\ad b}}{\ad b}([b,a])$, where $\frac{1-e^{\ad b}}{\ad b}$ is first interpreted as a power series $\frac{1-e^y}{y}$ involving only non-negative powers of $y$, and then the substitution $y=\ad b$ is made. But that's $\iota A_{hh}$, when one remembers that $\iota$ on the Y sector automatically contains a single ``$\frac{1}{\text{hair}}$'' factor. Similar arguments, though using $\aSTU_2$ instead of $\aSTU_1$, prove that $Y_{ht}$, $Y_{th}$, $A_{ht}$, and $A_{th}$ are all in $\ker\iota$. Finally, $\iota A_w$ is RI, and $\iota Y_w$ is a direct consequence of $\aSTU_2$. \qed \end{proof} Finally, we come to the special elements $\lambda$, $\delta_A$, and $\omega_1$. \begin{figure} \[ \pstex{SpecialElements} \] \caption{The special elements $\omega_1$, $\delta_A$, and $\lambda$ in $\IAM_G$, for a sample 3-arrow Gauss diagram $G$. } \label{fig:SpecialElements} \end{figure} \begin{definition} Within $\IAM_G$, let $\omega_1$ be, as before, the generator of the W sector. Let $\delta_A$ be a ``short'' red arrow, as in the $A_w$ relation (exercise: modulo $\calR$, this is independent of the placement of the short arrows within $G$). Finally, let $\lambda$ be the signed sum of exciting each of the (black) arrows in $G$ in turn. The picture says all, and it is Figure~\ref{fig:SpecialElements}. \end{definition} \begin{lemma} In $\calA^{sw}(\uparrow)$, the special elements of $\IAM_G$ are interpreted as follows: \linebreak $\iota(\omega_1)=Zw_1$, $\iota(\delta_A)=ZD_A$, and most interestingly, $\iota(\lambda)=EZ$. Therefore, Equation~\eqref{eq:GoalInIAM} (if true) implies Equation~\eqref{eq:EofAtoZ} and hence, it implies our goal, Theorem~\ref{thm:Alexander}. \end{lemma} \begin{proof} For the proof of this lemma, the only thing that isn't done yet and isn't trivial is the assertion $\iota(\lambda)=EZ$. But this assertion is a consequence of $Ee^{\pm a}=\pm ae^{\pm a}$ and of a Leibniz law for the derivation $E$, appropriately generalized to a context where $Z$ can be thought of as a ``product'' of ``arrow reservoirs''. The details are left to the reader. \qed \end{proof} \subsubsection{The Computation of $\lambda$} \label{sec:ComputeLambda} Naturally, our next task is to prove Equation~\eqref{eq:GoalInIAM}. This is done entirely algebraically within the finite rank module $\IAM_G$. To read this section one need not know about $\calA^{sw}(\uparrow)$, or $\iota$, or $Z$, but we do need to lay out some notation. Start by marking the arrows of $G$ with $a_1$ through $a_n$ in some order. Let $\epsilon$ stand for the informal yet useful quantity ``a little''. Let $\lambda_{ij}$ denote the difference $\lambda'_{ij}-\lambda''_{ij}$ of red excitations in the A sector of $\IAM_G$, where $\lambda'_{ij}$ is the diagram with a red arrow whose tail is $\epsilon$ to the right of the left end of $a_i$ and whose head is $\frac12\epsilon$ away from the head of $a_j$ in the direction of the tail of $a_j$, and where $\lambda''_{ij}$ has a red arrow whose tail is $\epsilon$ to the left of the right end of $a_i$ and whose head is as before, $\frac12\epsilon$ away from head of $a_j$ in the direction of the tail of $a_j$. Let $\Lambda=(\lambda_{ij})$ be the matrix whose entries are the $\lambda_{ij}$, as shown in Figure~\ref{fig:LambdaAndY}. \begin{figure} \[ \pstex{LambdaAndY} \] \caption{The matrices $\Lambda$ and $Y$ for a sample 2-arrow Gauss diagram (the signs on $a_1$ and $a_2$ are suppressed, and so are the $r$ marks). The twists in $y_{11}$ and $y_{22}$ may be replaced by minus signs. } \label{fig:LambdaAndY} \end{figure} Similarly, let $y_{ij}$ denote the element in the Y sector of $\IAM_G$ whose red Y has its head $\frac12\epsilon$ away from head of $a_j$ in the direction of the tail of $a_j$, its right tail (as seen from the head) $\epsilon$ to the left of the right end of $a_i$ and its left tail $\epsilon$ to the right of the left end of $a_i$. Let $Y=(y_{ij})$ be the matrix whose entries are the $y_{ij}$, as shown in Figure~\ref{fig:LambdaAndY}. \begin{lemma} \label{lem:IAMStructure} With $S$ and $T$ as in Definition~\ref{def:STA}, and with $B=T(X^{-S}-I)$ and $\lambda$ as above, the following identities between elements of $\IAM_G$ and matrices with entries in $\IAM_G$ hold true: \begin{eqnarray} \lambda-\sl\cdot D_A &=& \tr S\Lambda \label{eq:lambda}, \\ \Lambda &=& -BY-TX^{-S}w_1 \label{eq:Lambda}, \\ Y &=& BY + TX^{-S}w_1 \label{eq:Y}. \end{eqnarray} \end{lemma} \noindent{\em Proof of Equation~\eqref{eq:GoalInIAM} given Lemma~\ref{lem:IAMStructure}.} The last of the equalities above implies that $Y=(I-B)^{-1}TX^{-S}w_1$. Thus, \begin{align*} \lambda-\sl\cdot D_A = \tr S\Lambda = -\tr S(BY+TX^{-S}w_1) &= -\tr S(B(I-B)^{-1}TX^{-S}+TX^{-S})w_1 \\ &= -\tr\left((I-B)^{-1}TSX^{-S}\right)w_1, \end{align*} and this is exactly Equation~\eqref{eq:GoalInIAM}. \qed \noindent{\em Proof of Lemma~\ref{lem:IAMStructure}.} Equation~\eqref{eq:lambda} is trivial. The proofs of Equations~\eqref{eq:Lambda} and~\eqref{eq:Y} both have the same simple cores, that have to be supplemented by highly unpleasant tracking of signs and conventions and powers of $X$. Let us start from the cores. To prove Equation~\eqref{eq:Lambda} we wish to ``compute'' $\lambda_{ik}=\lambda'_{ik}-\lambda''_{ik}$. As $\lambda'_{ik}$ and $\lambda''_{ik}$ have their heads in the same place, we can compute their difference by gradually sliding the tail of $\lambda'_{ik}$ from its original position near the left end of $a_i$ towards the right end of $a_i$, where it would be cancelled by $\lambda''_{ik}$. As the tail slides we pick up a $y_{jk}$ term each time it crosses a head of an $a_j$ (relation $A_{th}$), we pick up a vanishing term each time it crosses a tail (relation $A_{tt}$), and we pick up a $w_1$ term if the tail needs to cross over its own head (relation $A_w$). Ignoring signs and $(X^{\pm 1}-1)$ factors, the sum of the $y_{jk}$-terms should be proportional to $TY$, for indeed, the matrix $T$ has non-zero entries precisely when the head of an $a_j$ falls within the span of an $a_i$. Unignoring these signs and factors, we get $-BY$ (recall that $B=T(X^{-S}-I)$ is just $T$ with added $(X^{\pm 1}-1)$ factors). Similarly, a $w_1$ term arises in this process when a tail has to cross over its own head, that is, when the head of $a_k$ is within the span of $a_i$. Thus, the $w_1$ term should be proportional to $Tw_1$, and we claim it is $-TX^{-S}w_1$. The core of the proof of Equation~\eqref{eq:Y} is more or less the same. We wish to ``compute'' $y_{ik}$ by sliding its left leg, starting near the left end of $a_i$, towards its right leg, which is stationary near the right end of $a_i$. When the two legs come together, we get 0 because of the anti-symmetry of Y excitations. Along the way we pick up further Y terms from the $Y_{th}$ relations, and sometimes a $w_1$ term from the $Y_w$ relation. When all signs and $(X^{\pm 1}-1)$ factors are accounted for, we get Equation~\eqref{eq:Y}. We leave it to the reader to complete the details in the above proofs. It is a major headache, and we would not have trusted ourselves had we not written a computer program to manipulate quantities in $\IAM_G$ by a brute force application of the relations in $\calR$. Everything checks; see~\cite[``The Infinitesimal Alexander Module'']{WKO}. \qed This concludes the proof of Theorem~\ref{thm:Alexander}. \qed \begin{remark} We chose the name ``Infinitesimal Alexander Module'' as in our mind there is some similarity between $\IAM_K$ and the ``Alexander Module'' of $K$. Yet beyond the above, we did not embark on any serious study of $\IAM_K$. In particular, we do not know if $\IAM_K$ in itself is an invariant of $K$ (though we suspect it wouldn't be hard to show that it is), we do not know if $\IAM_K$ contains any further information beyond $\sl$ and the Alexander polynomial, and we do not know if there is any formal relationship between $\IAM_K$ and the Alexander module of $K$. \end{remark} \begin{remark} The logarithmic derivative of the Alexander polynomial also appears in Lescop's work, see~\cite{Lescop:EquivariantLinking, Lescop:Cube}. We don't know if its appearances there are related to its appearance here. \end{remark} \draftcut \subsection{The Relationship with u-Knots} \label{subsec:RelWithKont} Unlike in the case of braids, there is a canonical universal finite type invariant of $u$-knots: the Kontsevich integral $\glos{Z^u}$. So it makes sense to ask how it is related to the expansion $Z^w$. \parpic[l]{$\xymatrix{ \calK^u(\uparrow) \ar[r]^{Z^u} \ar[d]^a & \glos{\calA^u}(\uparrow) \ar[d]^\alpha \\ \calK^w(\uparrow) \ar[r]^{Z^{w}} & \calA^{sw}(\uparrow) }$} We claim that the square on the left commutes, where $\glos{\calK^u}(\uparrow)$ stands for long $u$-knots (knottings of an oriented line), and similarly $\calK^w(\uparrow)$ denotes long $w$-knots. As before, $a$ is the composition of the maps $u$-knots $\to$ $v$-knots $\to$ $w$-knots, and $\alpha$ is the induced map on the associated graded spaces, mapping each chord to the sum of the two ways to direct it. Recall that $\alpha$ kills everything but wheels and arrows (hence $Z^w$ is much weaker, but also easier to handle, than the Kontsevich integral). We are going to use the formula for the ``wheel part'' of the Kontsevich integral as stated in \cite{Kricker:Kontsevich}. Let $K$ be a 0-framed long knot, and let $A(K)$ denote the Alexander polynomial. Then by \cite{Kricker:Kontsevich}, $$Z^u(K)= \exp_{\calA^u}\left(-\frac{1}{2} \log A(K)(e^h)|_{h^{2n}\to w^u_{2n}}\right)+\text{ ``loopy terms''},$$ where $w^u_{2n}$ stands for the unoriented wheel with $2n$ spokes; and ``loopy terms'' means terms that contain diagrams with more than one loop, which are killed by $\alpha$. Note that by the symmetry $A(z)=A(z^{-1})$ of the Alexander polynomial, $A(K)(e^h)$ contains only even powers of $h$, as suggested by the formula. We need to understand how $\alpha$ acts on wheels. Due to the two-in-one-out rule, a wheel is zero unless all the ``spokes'' are oriented inward, and the cycle oriented in one direction. In other words, there are two ways to orient an unoriented wheel: clockwise or counterclockwise. Due to the anti-symmetry of chord vertices, we get that for odd wheels $\alpha(w^u_{2h+1})=0$ and for even wheels $\alpha(w^u_{2h})=2w^w_{2h}$. As a result, $$\alpha Z^u(K)=\exp_{\calA^{sw}}\left(-\frac{1}{2} \log A(K)(e^h)|_{h^{2n}\to 2w_{2n}}\right) =\exp_{\calA^{sw}}\left(-\log A(K)(e^h)|_{h^{2n}\to w_{2n}}\right)$$ which agrees with the Formula (\ref{eq:AtoZ}) of Theorem \ref{thm:Alexander}. Note that since $K$ is 0-framed, the first part (``$\sl$ coded in arrows'') of Formula~(\ref{eq:AtoZ}) is trivial.