2023/01/14 by Hans Bodén, Boden, Hans U., Homayun Karimi +1
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2301.05946
A mock Seifert matrix is an integral square matrix representing the Gordon-Litherland form of a pair (K,F), where K is a knot in a thickened surface and F is an unoriented spanning surface for K. Using these matrices, we introduce a new notion of unoriented algebraic concordance, as well as a new group denoted m G\mathbb Z and called the unoriented algebraic concordance group. This group is abelian and infinitely generated. There is a surjection λ\colon v C → m G\mathbb Z, where v C denotes the virtual knot concordance group. Mock Seifert matrices can also be used to define new invariants, such as the mock Alexander polynomial and mock Levine-Tristram signatures. These invariants are applied to questions about virtual knot concordance, crosscap numbers, and Seifert genus for knots in thickened surfaces. For example, we show that m G\mathbb Z contains a copy of \mathbb Z^∞ ⊕ (\mathbb Z/2)^∞ ⊕(\mathbb Z/4)^∞.