Invariants of closed braids via counting surfaces

TL;DR

This paper introduces formulas for a family of link invariants based on counting surfaces in Gauss diagrams of closed braids, linking them to derivatives of the HOMFLY-PT polynomial.

Contribution

It provides new combinatorial formulas for invariants of closed braids using surface counting, connecting them to well-known polynomial derivatives.

Findings

Formulas for invariants via counting surfaces of specific genus and boundary components.

Identification of these invariants with partial derivatives of the HOMFLY-PT polynomial.

Simplification of invariant computation for closed braids.

Abstract

A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. 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 associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.