Virtual knot cobordism and bounding the slice genus

TL;DR

This paper investigates the slice genus of low-crossing virtual knots, demonstrating the effectiveness of Turaev's graded genus as a concordance invariant and developing algorithms for its computation.

Contribution

It extends Turaev's graded genus to virtual knot concordance and provides an algorithm to compute the slice genus for small virtual knots.

Findings

1295 out of 92800 virtual knots with ≤6 crossings are slice

Most non-slice knots are identified using the graded genus invariant

The developed algorithm efficiently computes the slice genus for small virtual knots

Abstract

In this paper, we compute the slice genus for many low-crossing virtual knots. For instance, we show that 1295 out of 92800 virtual knots with 6 or fewer crossings are slice, and that all but 248 of the rest are not slice. Key to these results are computations of Turaev's graded genus, which we show extends to give an invariant of virtual knot concordance. The graded genus is remarkably effective as a slice obstruction, and we develop an algorithm that applies virtual unknotting operations to determine the slice genus of many virtual knots with 6 or fewer crossings.