Year: 2012
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Brandenbursky, Michael
Institution: Max Planck Institute for Mathematics
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2952911255