Obstructing 4-torsion in the classical knot concordance group

1998/02/15 by Charles Livingston, Swatee Naik
Mathematics · #math.GT #msc:57M25

paper · pdf

published as J. Diff. Geom. 51 (1999), 1--12. · 7 pages

arxiv created 1998/02/15 · arxiv updated 2009/11/30

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 5t² - 11t + 5 is of infinite order in concordance; Levine proved that in higher dimensions all such knots are of order 4.

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