First Order Signatures and Knot Concordance

Abstract

Invariants of knots coming from twisted signatures have played a central role in the study of knot concordance. Unfortunately, except in the simplest of cases, these signature invariants have proven exceedingly difficult to compute. As a consequence, many knots which presumably can be detected by these invariants are not a well understood as they should be.

We study a family of signature invariants of knots and show that they provide concordance information. Significantly, we provide a tractable means for computing these signatures. Once armed with these tools we use them first to study the knot concordance group generated by the twist knots which are of order 2 in the algebraic concordance group. With our computational tools we can show that with only finitely many exceptions, they form a linearly independent set in the concordance group.

We go on to study a procedure given by Cochran-Harvey-Leidy which produces infinite rank subgroups of the knot concordance group which, in some sense are extremely subtle and difficult to detect. The construction they give has an inherent ambiguity due to the difficulty of computing some signature invariants. This ambiguity prevents their construction from yielding an actual linearly independent set. Using the tools we develop we make progress to removing this ambiguity from their procedure.