Structure in the bipolar filtration of topologically slice knots

Cochran, Tim D; Horn, Peter D

doi:10.2140/agt.2015.15.415

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Structure in the bipolar filtration of topologically slice knots

Tim D Cochran and Peter D Horn

Algebraic & Geometric Topology 15 (2015) 415–428

DOI: 10.2140/agt.2015.15.415

Abstract

Let $T$ be the group of smooth concordance classes of topologically slice knots and suppose

\dots \subset T_{n + 1} \subset T_{n} \subset \dots \subset T_{2} \subset T_{1} \subset T_{0} \subset T

is the bipolar filtration of $T$ . We show that $T_{0} ∕ T_{1}$ has infinite rank, even modulo Alexander polynomial one knots. Recall that knots in $T_{0}$ (a topologically slice $0$ –bipolar knot) necessarily have zero $τ$ –, $s$ – and $ϵ$ –invariants. Our invariants are detected using certain $d$ –invariants associated to the $2$ –fold branched covers.

Keywords

knot, topologically slice, bipolar filtration

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57N70

References

Publication

Received: 3 March 2014

Revised: 4 August 2014

Accepted: 5 August 2014

Published: 23 March 2015

Authors

Tim D Cochran
Department of Mathematics MS-136 Rice University PO Box 1892 Houston, TX 77251-1892 USA
http://math.rice.edu/~cochran
Peter D Horn
Department of Mathematics Syracuse University 215 Carnegie Building Syracuse, NY 13244-1150 USA
http://pdhorn.expressions.syr.edu/

Volume 15, issue 1 (2015)