Two-torsion in the grope and solvable filtrations of knots

Abstract

The abelian monoid of knots under connected sum, modulo concordance relation, is called the knot concordance group. In the attempt to understand the group, Cochran, Orr and Teichner de ned three ltrations of the knot concordance group: the solvable filtration {Fh}, the grope fi ltration {Gh}, and the Whitney tower filtration {Hh}. We study new knots of order 2 in the knot concordance group in relation to these filtrations. We show that, for any integer n C 4, there are knots generating a Z2-infinite subgroup of Gn/Gn.5 and Hn/Hn.5 whose torsion elements have been unknown. Considering the solvable filtration, our knots generate a Z2-infinite subgroup of Fn/Fn.5 distinct from the subgroup generated by the previously known 2-torsion knots discovered by Cochran, Harvey, and Leidy. We also present a result on the 2-torsion part in the Cochran, Harvey, and Leidy's primary decomposition of the solvable fi ltration.