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.

Coloring link diagrams and Conway-type polynomial of braids

“…the variable a, evaluated at a = 1. Another interpretation of the later polynomial was recently given by the author in [4,5].…”

“…Let G be a Gauss diagram of L (for a precise definition see for example [13,22]). Another interpretation of the polynomial zP ′ a (L)| a=1 in terms of counting surfaces with two boundary components in G was given recently by the author in [4,5].…”

The three loop isotopy and framed isotopy invariants of virtual knots

“…The three loop invariant is an invariant which is given by a Gauss diagram formula [27] with labeled pairs of regions (see below for definition). Much work has been done on finding Gauss diagram formulae for classical and virtual knot invariants [8,7,9,14,4,13]. The aim of this section is to define the three loop invariant, prove that it is an invariant of virtual knots, and provide some examples of its computation.…”

A lattice of finite-type invariants of virtual knots

“…In addition, they found Gauss diagram formulae which compute the coefficients of the Conway polynomial up to any order. Related work for other knot polynomials has been done by Chmutov and Polyak [5] and Brandenbursky and Polyak [2]. The invariant ∇ asc may be defined as follows [4].…”