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 92,800 virtual knots with six 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 six or fewer crossings.

KEYWORDS:

• virtual knots
• cobordism
• concordance
• slice knots
• slice genus
• graded genus.

2010 AMS SUBJECT CLASSIFICATION:

• Primary 57M25
• Secondary 57M27

Acknowledgments

We would like to thank J. Scott Carter and Andrew Nicas for useful discussions, as well as Louis Kauffman and William Rushworth for their input.

Notes

1 A precise definition of the slice genus g_s(K) for virtual knots is given below in Definition 2.1; it is the analogue for virtual knots of the 4-ball genus for classical knots.

2 This is a metaphor used in parts of North America to indicate a large quantity.

3 Note that a slightly different normalization is used here for W_K(t) than in [Cheng and Gao 13].

4 The correction term ϵ equals minus the linking number of {a, b} and {c, d}, viewed as two S⁰’s in S¹, in the core circle of $\bar{D}$.

5 Turaev uses the word cobordant.

Additional information

Funding

H. Boden was supported by a grant from the Natural Sciences and Engineering Research Council of Canada, M. Chrisman was supported by a Monmouth University Creativity and Research Grant, and R. Gaudreau was supported by a scholarship from the National Centre of Competence in Research SwissMAP.