On Arf invariants of colored links

TL;DR

This paper investigates extensions of the Arf invariant from classical knots to colored links, using generalized Seifert forms, and finds that these invariants are mostly determined by linking numbers.

Contribution

It introduces a method to extend the Arf invariant to colored links via generalized Seifert forms and characterizes when these extensions are well-defined.

Findings

New Arf invariants are determined by linking numbers for colored links.

Extensions of the Arf invariant are possible using quadratic forms from Seifert forms.

Most new invariants coincide with known invariants except in the case of oriented links.

Abstract

Several classical knot invariants, such as the Alexander polynomial, the Levine-Tristram signature and the Blanchfield pairing, admit natural extensions from knots to links, and more generally, from oriented links to so-called colored links. In this note, we explore such extensions of the Arf invariant. Inspired by the three examples stated above, we use generalized Seifert forms to construct quadratic forms, and determine when the Arf invariant of such a form yields a well-defined invariant of colored links. However, apart from the known case of oriented links, these new Arf invariants turn out to be determined by the linking numbers.