Abstract:
In 2015, [1], the author initiated the study of groups denoted by $G_{n}^{k}$, depending on two natural
parameters, $n$ and $k$ and formulated the main principle:
{\em if a dynamical system describing a motion of $n$ particles, is in general position with respect to some property regulated by $n$ particles, then it has topological invariants valued in $G_{n}^{k}$.}
The main examples calculated explicitly [2] led to homomorphisms from the pure braid groups to the groups $G_{n}^{3}$ and
$G_{n}^{4}$.
The group $G_{n}^{k}$ were found to have close connections with Coxeter groups [3], braid groups and other groups.
In the present talk, we address the question:
what happens if the number of particles is not constant but at some moments of time some two particles can get born or get cancelled? The main example here is the extension of topological invariants from braids to knots. A braid can be consiered as a motion of $n$ distinct particles on the plane, whence a knot can be represented by a collection of sections by horizontal planes. In general position, there are finitely many moments, where number of particles changes by two; otherwise, the knot behaves like a braid.
{\em What is the “knot” counterpart of the groups $G_{n}^{k}$-groups considered as analogs of “braids”?}
The most na\"ive approach suggests to consider elements of $G_{n}^{k}$ as $1$-dimensional braid-like objects (“braid” diagrams modulo moves) and to pass to analogous “knot-like” (closed) objects modulo the same moves. However, this approach fails because besides the usual “braid” moves, one should also require some “cobordism-like” moves which make the whole picture almost trivial.
The right approach (at least for $G_{n}^{3}$) is related not to $1$-dimensional formalism, but rather, with a $2$-dimensional formalism.
Then diagrams corresponding to our dynamical system will look like $2$-knot diagrams, and their moves will look like Roseman moves [4].
The first step of this approach is sketched in [5].
This allows one to take the pull-back of invariants of "$2$-knot like objects" as toplogical invariants of dynamical systems of this sort.
The author is partially supported by the Laboratory of Quantum Topology of Chelyabinsk State University (Russian Federation government grant 14.Z50.31.0020)
References:
V.O. Manturov, Non-Reidemeister knot theory and its applications in dynamical systems, geometry and topology. http://arxiv.org/abs/1501.05208.
V.O. Manturov, I.M. Nikonov,On Braids and Groups $G_{n}^{k}$. J. Knot Theory and its Ramifications 24 (2015), no. 13, 1541009, 16 pp.
V.O. Manturov, On Groups $G_{n}^{2}$ and Coxeter Groups. arXiv:1512.09273.
D. Roseman, Reidemeister-type moves for surfaces in four-dimensional space. Knot Theory, Banach Center Publications 42 (1998), Polish Academy of Sciences, Warsaw, 347–380.
V.O. Manturov, A Note on a Map from Knots to 2-Knots. arXiv:1604.06597.
