0-concordance of knotted surfaces and Alexander ideals

Abstract

In this paper we provide a new obstruction to 0-concordance of knotted surfaces in $S^4$ 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 ${\mathcal{C}}_0$ of oriented surface knots in $S^4$ to the ideal class monoid of $\mathbb{Z}[t^{\pm 1}]$. Consequently, any surface knot with nonprincipal Alexander ideal is not 0-slice and in fact, not invertible in ${\mathcal{C}}_0$. 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.

25 pages, 1 figure. Comments welcome