Rational Witt classes of pretzel knots

TL;DR

This paper introduces rational Witt classes as computable invariants for algebraic concordance of knots, computes these for all pretzel knots, and applies them to obstruct sliceness, providing explicit formulas for determinants and signatures.

Contribution

It proposes rational Witt classes as a practical alternative to Levine's invariants and computes them explicitly for pretzel knots, aiding sliceness obstructions.

Findings

Computed rational Witt classes for all pretzel knots.

Provided explicit formulas for determinants and signatures.

Demonstrated applications to sliceness obstructions.

Abstract

In his pioneering work from 1969, Jerry Levine introduced a complete set of invariants of algebraic concordance of knots. The evaluation of these invariants requires a factorization of the Alexander polynomial of the knot, and is therefore in practice often hard to realize. We thus propose the study of an alternative set of invariants of algebraic concordance - the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the rational Witt classes of all pretzel knots. We give many examples and provide applications to obstructing sliceness for pretzel knots. We also obtain explicit formulae for the determinants and signatures of all pretzel knots. This article is dedicated to Jerry Levine and his lasting mathematical legacy; on the occasion of the conference "Fifty years since Milnor and Fox" held at Brandeis University on June 2-5, 2008.