The algebraic concordance order of a knot

TL;DR

This paper develops practical methods to determine the algebraic concordance order of knots, simplifying p-adic analysis and applying these techniques to classify all prime knots with up to 12 crossings.

Contribution

It introduces effective criteria to identify the order of algebraic concordance elements, reducing the complexity of p-adic analysis and enabling classification of a large knot set.

Findings

Simplified tests often avoid p-adic analysis

Determined algebraic orders for all prime knots up to 12 crossings

Provided background on p-adic numbers and Witt groups

Abstract

Levine defined the rational algebraic knot concordance group and proved that each nontrivial element is of order two, of order four, or of infinite order. The determination of the order of an element depends on a p-adic analysis for all primes p. Here we develop effective means to determine the order of any element that is in the image of the integral algebraic concordance group by restricting the set of primes that need to be considered and by finding simple tests that often avoid p-adic considerations. The paper includes an outline of how the results apply to give the determination of the algebraic orders of all 2,977 prime knots of 12 or fewer crossings. The paper also includes a short expository account of the necessary background in p-adic numbers and Witt groups of bilinear forms.