\ifpub{This is the first in a series of papers studying w-knots, and more generally,
w-knotted objects (w-braids, w-tangles,
etc.). These are classes of knotted objects which are \underline{w}ider, but
\underline{w}eaker than their ``\underline{u}sual'' counterparts.}
{This is the first in a series of papers studying w-knots, and more generally,
w-knotted objects (w-braids, w-tangles,
etc.). These are classes of knotted objects which are \underline{w}ider, but
\underline{w}eaker than their ``\underline{u}sual'' counterparts. To
get (say) w-knots from usual knots (or u-knots), one has to allow non-planar ``virtual''
knot diagrams, hence enlarging the base set of knots. But then one
imposes a new relation beyond the ordinary collection of Reidemeister moves,
called the ``overcrossings commute'' relation, making w-knotted
objects a bit weaker once again.}
The group of w-braids was studied (under the name
``\underline{w}elded braids'') by Fenn, Rimanyi and
Rourke~\cite{FennRimanyiRourke:BraidPermutation} and was shown to
be isomorphic to the McCool group~\cite{McCool:BasisConjugating}
of ``basis-conjugating'' automorphisms of a free group $F_n$ ---
the smallest subgroup of $\Aut(F_n)$ that contains both braids and
permutations. Brendle and Hatcher~\cite{BrendleHatcher:RingsAndWickets},
in work that traces back to Goldsmith~\cite{Goldsmith:MotionGroups},
have shown this group to be a group of movies of flying rings in
$\bbR^3$. Satoh~\cite{Satoh:RibbonTorusKnots} studied several classes of
w-knotted objects (under the name ``\underline{w}eakly-virtual'') and has
shown them to be closely related to certain classes of knotted surfaces
in $\bbR^4$. So w-knotted objects are algebraically and topologically
interesting.
In this article we study finite type invariants of w-brainds and w-knots.
Following Berceanu and
Papadima~\cite{BerceanuPapadima:BraidPermutation}, we construct
homomorphic universal finite type invariants of w-braids.
We find that the universal finite type invariant of w-knots
is essentially the Alexander polynomial.
Much as the spaces $\calA$ of chord diagrams for ordinary knotted
objects are related to metrized Lie algebras, we find that the spaces
$\calA^w$ of ``arrow diagrams'' for w-knotted objects are related to
not-necessarily-metrized Lie algebras. Many questions concerning w-knotted
objects turn out to be equivalent to questions about Lie algebras, and
in later papers of this series we re-interpret Alekseev-Torossian's~\cite{AlekseevTorossian:KashiwaraVergne}
work on Drinfel'd associators and the Kashiwara-Vergne problem
as a study of w-knotted trivalent graphs.
\ifpub{}{The true value of w-knots, though, is likely to emerge later, for we
expect them to serve as a warmup example for what we expect
will be even more interesting --- the study of virtual
knots, or v-knots. We expect v-knotted objects to provide the global
context whose associated graded structure
will be the Etingof-Kazhdan theory of deformation quantization of Lie
bialgebras~\cite{EtingofKazhdan:BialgebrasI}.}