Virtual knot cobordism and bounding the slice genus

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.

26 pages, many figures, supplementary dataset available online at https://micah46.wixsite.com/micahknots/slicegenus

References in corpus (3)

• Virtual knot groups and almost classical knots
• Alexander invariants for virtual knots
• Concordance group of virtual knots

Cited by in corpus (7)

• Signature and concordance of virtual knots
• Milnor's concordance invariants for knots on surfaces
• Virtual concordance and the generalized Alexander polynomial
• Computations of the slice genus of virtual knots
• A parity for 2-colourable links
• Chord index for knots in thickened surfaces
• The homological arrow polynomial for virtual links