2019/11/29 by Jason Joseph, Joseph, Jason · 1 citation
Mathematics · Medicine · #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1911.13112
openalex publication_date 2019/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we provide a new obstruction to 0-concordance of knotted surfaces in S4 in terms of Alexander ideals. We use this to prove the existence of infinitely many linearly independent 0-concordance classes and to provide the first proof that the submonoid of 2-knots is not a group. The main result is that the Alexander ideal induces a homomorphism from the 0-concordance monoid \mathscrC0 of oriented surface knots in S4 to the ideal class monoid of ℤ[t±1]. Consequently, any surface knot with nonprincipal Alexander ideal is not 0-slice and in fact, not invertible in \mathscrC0. Many examples are given. We also characterize which ideals are the ideals of surface knots, generalizing a theorem of Kinoshita, and generalize the knot determinant to the case of nonprincipal ideals. Lastly, we show that under a mild condition on the knot group, the peripheral subgroup of a knotted surface is also a 0-concordance invariant.