A Sequence of Degree One Vassiliev Invariants for Virtual Knots

TL;DR

This paper introduces a sequence of three degree one Vassiliev invariants for virtual knots, establishing the strongest as universal, and extends Turaev's based matrix invariant to singular virtual knots.

Contribution

It constructs a sequence of degree one invariants for virtual knots and proves the strongest is universal, extending Turaev's invariant to singular virtual knots.

Findings

The strongest invariant is universal for degree one Vassiliev invariants.

The space of degree one invariants for virtual knots is infinite dimensional.

Extension of Turaev's invariant to singular virtual knots.

Abstract

For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual strings to the class of singular virtual knots with one double-point.