Topology Atlas | Conferences

Braids Groups, Higher G_n^k Groups, and Imaginary Generators
by
Vassily Olegovich Manturov
Bauman Moscow State Technical University
Coauthors: (Partially) Seongjeong Kim

(To J.S.Carter, on the occasion of this 60th birthday.)

Given a braid-word in standard generators of the Artin braid group.

Our goal is to read between letters. Namely, for a given braid-word we are going to construct a word in a larger set of generators such that when deleting new generators, we shall get the initial word and the rule of "splicing" the new letters is algebraically meaningful.

Algebraically, we wish to construct a monomorphism from a smaller group to a larger group such that its composition with the obvious forgetful map (deleting new letters) is the identical map. The new generators (letters) are not visible in the initial work, we call them imaginary generators.

Topologically, this construction comes from the following argument. In 2015, the author defined the two-parametric family of groups G_n^k, and formulated the main principle:

If a dynamical system of n moving particles possesses a nice codimension one property governed by some k particles then this dynamical system has an invariant valued in the group G_n^k.

The usual Artin presentation is spiritually of G_n² nature: we can consider a dynamics of points on the line and whenever some of them coincide, we put a generator. When dealing with the genuine braid group, we can think of some n distinct points moving on the plane, and we put a generator of the Artin braid group when some two points have the same x coordinate.

However, there is a more interesting way of looking at dynamics of n points on the plane: we can associate a generator of the group G_n³ with the situation when some three points are collinear. This was done in 2015 in a joint work with I.M.Nikonov.

Now, one can mention that the former G_n² nature of usual Artin's generators is just a partial case of the new G_n³-approach if we just add some infinite point. Indeed, if add an "infinite" point with coordinates (0, -∞) then the condition that some two points have the same abscissa is exactly the same as the condition that these two points belong to a line passing through (0, ∞). Thus, having a braid-word, we get a word in a larger alphabet (a modification of the G_n+1³ presentation) with the initial word inside.

It is well known that the group G_n³ has lots of nice invariants of crossings valued in free products of cyclic groups.

The "pull-back" of this G_n³-approach allows one to construct various invariants of letters in a given Artin braid word in standard generators.

By looking at classical crossings, we can get various consequences on minimal crossing number and unknotting number for classical braids. This is a partial work with Seongjeong Kim.

If I have time, I will say how the higher groups G_n^k (with large k) are related to fundamental groups of configuration spaces of higher dimensions.

Where else can we read between letters to study group presentations better? Topology and geometry can lead us to some partial answers to this question.

Date received: December 7, 2016

Copyright © 2016 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbnq-41.