Signature and concordance of virtual knots
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abstract
We introduce Tristram-Levine signatures of virtual knots and use them to
investigate virtual knot concordance. The signatures are defined first for
almost classical knots, which are virtual knots admitting homologically trivial
representations. The signatures and $\omega$-signatures are shown to give
bounds on the topological slice genus of almost classical knots, and they are
applied to address a recent question of Dye, Kaestner, and Kauffman on the
virtual slice genus of classical knots. A conjecture on the topological slice
genus is formulated and confirmed for all classical knots with up to 11
crossings and for 2150 out of 2175 of the 12 crossing knots.
The Seifert pairing is used to define directed Alexander polynomials, which
we show satisfy a Fox-Milnor criterion when the almost classical knot is slice.
We introduce virtual disk-band surfaces and use them to establish realization
theorems for Seifert matrices of almost classical knots. As a consequence, we
deduce that any integral polynomial $\Delta(t)$ satisfying $\Delta(1)=1$ occurs
as the Alexander polynomial of an almost classical knot.
In the last section, we use parity projection and Turaev's coverings of knots
to extend the Tristram-Levine signatures to all virtual knots. A key step is a
theorem saying that parity projection preserves concordance of virtual knots.
This theorem implies that the signatures, $\omega$-signatures, and Fox-Milnor
criterion can be lifted to give slice obstructions for all virtual knots. There
are 76 almost classical knots with up to six crossings, and we use our
invariants to determine the slice status for all of them and the slice genus
for all but four. Table 2 at the end summarizes our findings.
