VS, TS and TG Algebras

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We introduce VS-, TS- and TG-Algebras; three types of algebraic entities within which some basic equations of knot theory (related to Algebraic Knot Theory) can be written and potentially solved.

Why Bother?

  • These equations are valuable yet not well understood. My hope is to study them in many simpler spaces (i.e., in many simpler VS-, TS- and TG-Algebras) than the ones that naturally occur in knot theory, in the hope that out of many test cases an understanding will emerge.
  • In particular, one day I hope to write (or encourage the writing of) computer programs that will take a VS-, TS-, or TG-algebra "plug-in" and given it, will carry out all the necessary higher-level algebra. This will make it easier to study particular cases computationally. But for this, the notions of VS-, TS- and TG-Algebras must first be completely specified.

VS-Algebras in One Paragraph

A VS-Algebra (Vertical Strands Algebra) is an algebraic object that is endowed with the same operations as the algebras {\mathcal A}_n^{hor} of horizontal chord chord diagrams - multiplication (vertical stacking, a binary operation) and strand permutation, strand addition, strand doubling and strand deletion (all unary operations). It is local if it satisfies the same "locality in time" and "locality in space" relations that {\mathcal A}_n^{hor} satisfies [Bar-Natan_97]. In any local VS-algebra the equations for a Drinfel'd associator (i.e., the pentagon and the hexagon) can be written and potentially be solved, and solutions always lead to braid invariants. Likewise in any local VS-algebra the largely undocumented braidor equations can be written and potentially be solved, and solutions always lead to braid invariants.

TS-Algebras in One Paragraph

A TS-Algebra (Tangled Strands Algebra) is to tangles as a VS-algebra is to braids. Equally cryptically, it is to {\mathcal A}_n as a VS-algebra is to {\mathcal A}_n^{hor}. Thus a TS-algebra has the same unary operations as a VS-algebra along with a fancier collection of "products" that allow for "reversing" and "bending back" strands before they are concatenated. Thus every TS-algebra is in particular a VS-algebra, hence if it is "local" (with the same definition as for a VS-algebra), associators and braidors make sense it it. In a TS-algebra every associator or braidor satisfying some minor further symmetry conditions leads to a knot and link invariant. Furthermore, sufficiently symmetric associators lead to full-fledged Algebraic Knot Theories.

TG-Algebras in One Paragraph

A TG-Algebra (Trivalent Graph Algebra) is to knotted trivalent graphs as a TS-algebra is to tangles. Knotted trivalent graphs are equivalent to tangles, in some topological sense; indeed, given a knotted trivalent graph, pick a maximal tree and contract it until it is just a thick point. What remains, in the complement of that think point, is just a number of knotted edges with no vertices. That is, it is a tangle. It follows that a knotted trivalent graph is merely a tangle with just a bit of extra combinatorial labeling. Likewise the notions of a TG-algebra and of a TS-algebra are nearly equivalent. They differ mostly just by how certain things are labeled.

VS-Algebras in Some Detail

The basic definitions

Definition. For a natural number n let \underline{n} denote the set \{0,1,2,\ldots,n\}. Let {\mathbf{VS}} denote the category whose objects are the natural numbers and whose morphisms are given by


with the obvious composition of morphisms. For brevity we will often specify morphisms/functions by simply listing their values, omitting the value at 0 as it is anyway fixed. Thus for example, \alpha=(1,2,2), or even shorter, \alpha=(122), means \alpha:\underline{3}\to\underline{2} with \alpha(0)=0, \alpha(1)=1, \alpha(2)=2 and \alpha(3)(2) (strictly speaking, the target space of (122) can be any \underline{n} with n\geq 2).

Interpretation. The object "n" stands for "n strands". A morphism \alpha:m\to n means "for any 1\leq k\leq m, strand number k in m looks at strand number \alpha(k) in n if \alpha(k)>0, and looks nowhere if \alpha(k)=0".

Definition. A VS-algebra is a contravariant functor {\mathbf A} from the category {\mathbf{VS}} to the category of algebras over some fixed ring of scalars. We denote {\mathbf A}(n) by A_n and {\mathbf A}(\alpha) by \alpha^\star.

Interpretation. In a TS-algebra we have an algebra for any number of strands, with multiplication corresponding to "stacking two n-stranded objects (imagine braids) one on top the other". We think of an element \Psi of A_n as "n strands each of which carrying some algebraic information". If \alpha:m\to n, then \alpha^\star\Psi has m strands carrying algebraic information, and if \alpha(k)>0, strand number k in \alpha^\star\Psi "reads" its information from strand number \alpha(k) in \Psi. If \alpha(k)=0, strand number k in \alpha^\star\Psi reads its information from nowhere, so it carries some "default" information, presumably "empty".

You are probably familiar with the notation used elsewhere (in [Bar-Natan_97], for example) when dealing with associators and the pentagon and hexagon equations. Here's a quick dictionary:

Elsewhere In Words Here In Words
(1\otimes\Delta):{\mathcal A}_2\to{\mathcal A}_3 Copy the first strand untouched and double the second strand (122)^\star The first output strand reads from the first input strand, the second and third both read from the second.
s_2:{\mathcal A}_3\to{\mathcal A}_2 Delete the second strand. (13)^\star Nothing reads from the second strand, as 2 isn't in the range of (13).
\Psi\mapsto\Psi^{23} Add an empty strand on the left. (012)^\star The first strand reads from nowhere to it is empty. The second and third output strands read from the first and second input strands respectively.
\Psi\mapsto\Psi^{231} Permute the strands: Install 1 on 2, 2 on 3 and 3 on 1. (312)^\star Permute the strands: Read 1 from 3, 2 from 1 and 3 from 2.

Thus the new notation is the opposite of the old when it comes to permutations: if \sigma is a permutation, \Psi^\sigma is now (\sigma^{-1})^\star\Psi. It is a small price to pay considering the very short description (as above) that now becomes available for VS-algebras.

The VS-Algebra of legs and VS-algebras of animals

One of the most fundamental VS-Algebras is the VS-algebra {\mathbf L} of "legs", defined as follows. Set L_n to be the free associative and commutative algebra generated by "leg symbols" \{l_1,\ldots,l_n\} (think, "l_k is a leg on strand number k"). Then let

(i_1i_2\cdots i_m)^\star l_k := \sum_{\{j\colon i_j=k\}}l_j

(thus strand number j in the output looks at strand number i_j in the input, and if it sees a leg there, it takes a copy). It is a routine exercise to verify that {\mathbf L} is indeed a VS-algebra.

An "animal" is a formal symbol with a fixed number of "legs", and it is bilinear in those legs. Thus for example we may declare that the animal Y has three legs, and so "a Y animal in A_n" will be a symbol of the form Y_{k_1k_2k_3} where l_{k_1}, l_{k_2} and l_{k_3} are legs in L_n (thus 1\leq k_1,k_2,k_3\leq n), and Y animals get pulled back as the sum of all ways of pulling back their legs:

(i_1i_2\cdots i_m)^\star Y_{k_1k_2k_3} := \sum_{\{(j_1,j_2,j_3)\colon\ i_{j_1}=k_1,\ i_{j_2}=k_2, \ i_{j_3}=k_3\}}Y_{j_1j_2j_3}.
*The superscript "nl" stands for "non-local". A local version of this VS-algebra will be introduced after locality is introduced further below.

Once we choose our formal symbols for animals, we may consider the free associative (though not commutative!) algebra A^{nl}_n generated by all such animals with legs in L_n and the resulting collection of algebras and pullback operations will form a VS-algebra*.

The most standard example is the two-legged animal t_{ij} often referred to as "a chord from strand i to strand j"; this animal is also declared to be symmetric - it is declared that t_{ij}=t_{ji} for all i and j. The resulting VS-algebra is the VS-algebra {\mathcal A}^{hor,nl} of (non-local) "horizontal chords".

The pentagon and the hexagons

The Pentagon For Parenthesized Braids.jpg The Hexagons For Parenthesized Braids.jpg
The Pentagon and the Hexagons for Parenthesized Braids

The main thing I'd like to do in a VS-algebra is to write and solve the pentagon and hexagons equations. The unknowns in these equations are an invertible element R\in A_2 and an invertible element \Phi\in A_3, and the equations read:

(1230)^\star\Phi\cdot(1223)^\star\Phi\cdot(0123)^\star\Phi = (1123)^\star\Phi\cdot(1233)^\star\Phi     in     A_4,


(112)^\star(R^{\pm 1}) = \Phi\cdot (012)^\star(R^{\pm 1})\cdot(132)^\star(\Phi^{-1})\cdot(102)^\star(R^{\pm 1})\cdot(231)^\star\Phi     in     A_3.

In old notation, this is:

\Phi^{123}\cdot(1\otimes\Delta\otimes 1)(\Phi)\cdot\Phi^{234}=(\Delta\otimes 1\otimes 1)(\Phi)\cdot(1\otimes 1\otimes\Delta)(\Phi)     in     {\mathcal A}_4,


(\Delta\otimes 1)(R^{\pm 1}) = \Phi^{123}\cdot (R^{\pm 1})^{23}\cdot(\Phi^{-1})^{132}\cdot(R^{\pm 1})^{13}\cdot\Phi^{312}     in     {\mathcal A}_3.


Definition. A VS-algebra {\mathbf A} is called "local" if \Psi_1\Psi_2=\Psi_2\Psi_1 in A_m whenever "\Psi_2 regards the strands that \Psi_1 is supported on as equivalent". More precisely, whenever \Psi_i=\alpha_i^\star\Phi_i for i=1,2, where \Phi_i\in A_{n_i} and \alpha_i:\underline{m}\to\underline{n_i} and where \alpha_2 is constant on the subset of \underline{m} on which \alpha_1 is non-zero.

It is an excellent idea to pause briefly and try to see why the informal part of the above definition matches with it formal part. Also, if you know about "locality in space" and "locality in scale" as they are described in [Bar-Natan_97], it is an excellent idea to verify that our single notion of locality is equivalent to the conjunction of the two localities of that paper. Finally, it is clear that every VS-algebra can be made local by imposing the relation \Psi_1\Psi_2=\Psi_2\Psi_1 whenever necessary. Thus the VS-algebras "of animals" above also have local versions. We leave it as an exercise to our readers to verify that the local version {\mathcal A}^{hor} of the non-local VS-algebra {\mathcal A}^{hor,nl} of horizontal chords, defined above, is the well known algebra of horizontal chord diagrams modulo 4T relations of the theory of finite type invariants of braids.

Theorem. Any solution (R,\Phi) of the pentagon and hexagon equations in a local VS-algebra leads to an invariant of braids.

Of course, this theorem is not quite well-formulated; strictly speaking one can always take the invariant of braids in the theorem to be the trivial invariant. So the theorem fully makes sense only after its proof is given. But we have no intention of providing a proof here - the theorem and its proof are quite obvious if you think in the spirit of [Bar-Natan_97].

Solutions in the graded case

If a local VS-algebra {\mathbf A} is graded in the obvious sense and if every A_n has an identity element in degree 0 one can run exactly the same iterative degree-by-degree procedure as in [Bar-Natan_97] in attempt to find solutions to the pentagon and hexagon. The existence and uniqueness of such solutions is then goverened by exactly the same homology theory as in [Bar-Natan_97], with obvious adjustments made to accomodate for the more general setup. However, for different VS-algebras the corresponding homology groups may or may not be trivial and hence solutions may or may not exist. Thus comes our main question:

Question. Find examples of local VS-algebras to illuminate the various existence/uniqueness possibilities. Examples in which a pair (R,\Phi) exists and is unique, examples in which a solution isn't unique but the different solutions can be enumerated in a reasonable way, etc. Also, find examples of local VS-algebras in which solutions (R,\Phi) can be written in "closed form".

My paperlet Associators with Frozen Feet is about the simplest non-trivial local VS-algebra I am aware off in which closed-form solutions seem to exist. A similar example is implicitly in [Kurlin_04].

TS-Algebras in Some Detail

Enumerating r-ary operations and multicategories

In the previous section we've defined VS-algebras using contravariant functors on the category {\mathbf{VS}}. For the purpose of our game, {\mathbf{VS}} is "that thing which enumerates the spaces and operations in a VS-algebras"; a VS-algebra has one space for each object of {\mathbf{VS}} (i.e., for each integer) and one operation for each morphism in {\mathbf{VS}}. More precisely, {\mathbf{VS}} does not enumerate all the operations in a VS-algebra - only the unary ones. As is the case for the set of braids, a VS-algebra has some, though very few, r-ary operations as well, corresponding to "vertical stacking". These we have included into the structure by insisting that each A_n space in a VS-algebra be an algebra, hence equipped with a binary operation. Higher r-ary operations are simply iterated compositions of the binary product.

As said before, a TS-algebra is to tangles as a VS-algebra is to braids. But tangles can be composed in many more ways than braids and so in enumerating all the operations in a TS-algebra we will need a more sophisticated approach. No longer will it be enough to enumerate unary operations using a category and then to throw in a single binary operation almost as an afterthought. We now need to genuinely enumerate all r-ary operations for all r.

The formal tool for enumerating r-ary operations is a certain cross of a category and an operad called a "multicategory". In short, a multicategory has objects and identity morphisms like in an ordinary category but its sets of morphisms (which really should be called "multimorphisms") depend on r "input objects" and one "output object". Such morphisms can be composed much like ordinary r-ary operations, and a certain associativity of the composition maps is required. In a multicategory {\mathbf M} , we will denote the set of multimorphisms whose r input objects are {\mathcal O}_1,\ldots,{\mathcal O}_r and whose output object is {\mathcal O} by \operatorname{mor}_{\mathbf M}({\mathcal O}_1\times\dots\times{\mathcal O_r},{\mathcal O}).

The Multicategory TS

Let {\mathbf{TS}} be the multicategory whose objects are the natural numbers and whose morphism are

TG-Algebras is Some Detail

This section will surely wait a while to be written; at the moment I don't have an immediate reason to write it nor do I know exactly what I want written.


[Bar-Natan_97] ^  Dror Bar-Natan, Non-Associative Tangles, in Geometric topology, proceedings of the Georgia international topology conference, W. H. Kazez, ed., 139-183, Amer. Math. Soc. and International Press, Providence, 1997.

[Kurlin_04] ^  Vitaliy Kurlin, Compressed Drinfel'd Associators, arXiv:math.GT/0408398.