Link invariants via counting surfaces
Abstract
A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by counting signs of crossings in a link diagram. Until recently, explicit formulas of this type were known only for few invariants of low degrees. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram. We then identify the resulting invariants with certain derivatives of the HOMFLYPT polynomial.
Keywords
Cite
@article{arxiv.1209.0420,
title = {Link invariants via counting surfaces},
author = {Michael Brandenbursky},
journal= {arXiv preprint arXiv:1209.0420},
year = {2016}
}
Comments
This is a revised version, to appear in Geom. Dedicata, 29 pages, many figures