The chord index, its definitions, applications and generalizations

TL;DR

This paper introduces the chord index for virtual knots, extending chord parity, and develops new invariants like the indexed Jones polynomial and indexed quandle, with applications in virtual and twisted knot theories.

Contribution

It defines the chord index for virtual knots, generalizes classical invariants, and explores their applications and generalizations via biquandles.

Findings

Chord index can define finite type invariants for virtual knots.

Indexed Jones polynomial generalizes the classical Jones polynomial.

Applications in twisted knot theory are demonstrated.

Abstract

In this paper we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to define finite type invariants of virtual knots. The notions of indexed Jones polynomial and indexed quandle are introduced, which generalize the classical Jones polynomial and knot quandle respectively. Some applications of these new invariants are discussed. We also study how to define a generalized chord index via a fixed finite biquandle. Finally the chord index and its applications in twisted knot theory are discussed.