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Milnor's concordance invariants for knots on surfaces

Micah Chrisman

Algebraic & Geometric Topology 22 (2022) 2293–2353

Milnor’s μ¯–invariants of links in the 3–sphere S3 vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in S3. Here we consider knots in thickened surfaces Σ × [0,1], where Σ is closed and oriented. We construct new concordance invariants by adapting the Chen–Milnor theory of links in S3 to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan Zh map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver and Williams. Our extended μ¯–invariants obstruct concordance to homologically trivial knots in thickened surfaces. We use them to give new examples of nonslice virtual knots having trivial Rasmussen invariant, graded genus, affine index (or writhe) polynomial, and generalized Alexander polynomial. Furthermore, we complete the slice status classification of all virtual knots up to five classical crossings and reduce to 4 (out of 92800) the number of virtual knots up to six classical crossings having unknown slice status.

Our main application is to Turaev’s concordance group 𝒱𝒞 of long knots on surfaces. Boden and Nagel proved that the concordance group 𝒞 of classical knots in S3 embeds into the center of 𝒱𝒞. In contrast to the classical knot concordance group, we show 𝒱𝒞 is not abelian, answering a question posed by Turaev.

virtual knots, knot concordance, Milnor's concordance invariants
Mathematical Subject Classification
Primary: 57K10, 57K12, 57K45
Received: 6 July 2020
Revised: 13 January 2021
Accepted: 23 April 2021
Published: 25 October 2022
Micah Chrisman
Department of Mathematics
The Ohio State University at Marion
Marion, OH
United States