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Knots and Representation Theory
May 11, 2020 18:30, Moscow

Milnor's concordance invariants for virtual knots and welded links (@knots-in-moscow)

Micah Chrisman

Abstract: Milnor's $\bar{\mu}$-invariants for links in the 3-sphere vanish on any link concordant to a boundary link. In particular, they are are trivial for any classical knot. Here we define an analogue of Milnor's concordance invariants for knots in thickened surfaces $\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented. The invariants are constructed from the nilpotent quotients of an extension of the group of a virtual knot due to Boden-Gaudreau-Harper-Nicas-White. These invariants, called $\overline{zh}$-invariants, vanish on any knot concordant to a homologically trivial knot in $\Sigma \times [0,1]$. Three applications are given. First, we use the $\overline{zh}$-invariants to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, writhe polynomial, and generalized Alexander polynomial. Moreover, we complete the slice status classification of the 2564 virtual knots having at most five classical crossings and reduces to four (of 90235) the number of virtual knots with six classical crossings having unknown slice status. As a second application, we prove that in contrast to the classical knot concordance group, the virtual knot concordance group is not abelian. Thirdly, as a byproduct of the construction of the $\overline{zh}$-invariants, we obtain a generalization of the $\bar{\mu}$-invariants of classical links to concordance invariants of welded links.