Link invariants via counting surfaces

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4 Scopus citations

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.

Original languageEnglish
Pages (from-to)243-270
Number of pages28
JournalGeometriae Dedicata
Volume173
Issue number1
DOIs
StatePublished - 1 Dec 2014
Externally publishedYes

Keywords

  • Finite type invariants
  • Gauss diagram formulas
  • Knots
  • Links

ASJC Scopus subject areas

  • Geometry and Topology

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