Signature and concordance of virtual knots

Hans U. Boden, Micah Chrisman, Robin Gaudreau

Published 2017-08-27Version 1

We introduce signature invariants of virtual knots and use them to investigate virtual knot concordance. The signatures, which depend on a choice of Seifert surface, are defined first for almost classical knots and then extended to all virtual knots using parity projection and Turaev's coverings of knots. A key step is a result implying that parity projection preserves concordance of virtual knots. One obtains similar results for long virtual knots, and in this case the knot signature is seen to be independent of the choice of Seifert surface. In addition to the knot signatures, we use the Seifert pairing to define several other invariants for almost classical knots, including the Alexander-Conway polynomials, the nullity, the $\omega$-signatures, and the directed Alexander polynomials. These invariants are applied to the problem of determining the slice genus for almost classical knots. There are 76 almost classical knots with up to six crossings, and we determine the slice status for all of them and the slice genus for all but four. These results are summarized in Table 2.