2021/07/20 by Chrisman, Micah, Mukherjee, Sujoy
#57K12 #57N70 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2107.09653
Torsion in the concordance group \mathscrC of knots in S3 can be studied with the algebraic concordance group \mathscrG^\mathbbF. Here \mathbbF is a field of characteristic χ(\mathbbF) ≠ 2. The group \mathscrG^\mathbbF was defined by J. Levine, who also obtained an algebraic classification when \mathbbF=ℚ. While the concordance group \mathscrC is abelian, it embeds into the non-abelian virtual knot concordance group \mathscrVC. It is unknown if \mathscrVC admits non-classical finite torsion. Here we define the virtual algebraic concordance group \mathscrVG^\mathbbF for almost classical knots . This is an analogue of \mathscrG^\mathbbF for homologically trivial knots in thickened surfaces Σ× [0,1], where Σ is closed and oriented. The main result is an algebraic classification of \mathscrVG^\mathbbF. A consequence of the classification is that \mathscrGℚ embeds into \mathscrVGℚ and \mathscrVGℚ contains many nontrivial finite-order elements that are not algebraically concordant to any classical Seifert matrix. For \mathbbF=ℤ/2ℤ, we give a generalization of the Arf invariant.