Obstructing 4-torsion in the classical knot concordance group

TL;DR

This paper establishes a new obstruction criterion based on homology and prime congruences that proves certain knots are of infinite order in the classical knot concordance group, especially those with specific Alexander polynomials.

Contribution

It introduces a novel obstruction method using 2-fold branched cover homology to determine infinite order in the knot concordance group, ruling out order 4 for many knots.

Findings

Certain knots with specific homology conditions are of infinite order in concordance.

All prime knots with ≤10 crossings are shown to be of infinite order, not order 4.

Knots with Alexander polynomial 5t^2 - 11t + 5 are of infinite order in concordance.

Abstract

We prove that if the order of the first homology of the 2-fold branched cover of a knot K in the 3-sphere is given by pm where p is a prime congruent to 3 mod 4 and gcd(p,m) =1, then K is of infinite order in the knot concordance group. This provides an obstruction to classical knots being of order 4. In particular, there are 11 prime knots with 10 or fewer crossings that are of order 4 in the algebraic concordance group; all are infinite order in concordance. Another corollary states that any knot with Alexander polynomial 5t^2 - 11t + 5 is of infinite order in concordance; Levine proved that in higher dimensions all such knots are of order 4.