#### Volume 15, issue 1 (2015)

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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
##### Abstract

Let $\mathsc{T}$ be the group of smooth concordance classes of topologically slice knots and suppose

$\cdots \subset {\mathsc{T}}_{n+1}\subset {\mathsc{T}}_{n}\subset \cdots \subset {\mathsc{T}}_{2}\subset {\mathsc{T}}_{1}\subset {\mathsc{T}}_{0}\subset \mathsc{T}$

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

##### Keywords
knot, topologically slice, bipolar filtration
Primary: 57M25
Secondary: 57N70
##### 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/