Seifert forms and slice Euler characteristic of links

TL;DR

This paper introduces the Witt coindex as a new concordance invariant derived from Seifert forms, providing bounds on the slice Euler characteristic of links and characterizing certain forms, with applications to link concordance and bounding properties.

Contribution

It defines the Witt coindex for links with non-trivial Alexander polynomial, extending previous work on knot genera and algebraic sliceness, and characterizes forms of coindex 1.

Findings

Witt coindex bounds the slice Euler characteristic of links.

Characterization of coindex 1 forms with non-degenerate symmetrized Seifert form.

Examples showing coindex detects non-sliceness of certain links.

Abstract

We define the Witt coindex of a link with non-trivial Alexander polynomial, as a concordance invariant from the Seifert form. We show that it provides an upper bound for the (locally flat) slice Euler characteristic of the link, extending the work of Levine on algebraically slice knots and Taylor on the genera of knots. Then we extend the techniques by Levine on isometric structures and characterize completely the forms of coindex $1$ under the condition that the symmetrized Seifert form is non-degenerate. We illustrate our results with examples where the coindex is used to show that a two-component link does not bound a locally flat cylinder in the four-ball, whereas any other known restriction does not show it.