Knot Concordance and Torsion

Details Knot Concordance and Torsion Knot Concordance and Torsion

1999-11-30

arxiv.org/abs/math/9911241v1

Asian Journal of Mathematics 5 (2001), 161--168

Mathematics

Geometric Topology

10 pages

Scientific paper

Let K be a knot in the 3-sphere with 2-fold branched covering space M. If for some prime p congruent to 3 mod 4 the p-torsion in the first homology of M is cyclic with odd exponent, then K is of infinite order in the knot concordance group. As one application, recall that the n-twisted double of an arbitrary knot has order 4 in Levine's algebraic concordance group if and only if n is positive and some prime congruent to 3 mod 4 has odd exponent in 4n+1; we show that all such knots are of infinite order in the knot concordance group. As a second application, the 2-bridge knot K(r,s) has infinite order in the knot concordance group if some prime congruent to 3 mod 4 has odd exponent in r.

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