## Algebraic & Geometric Topology

### Structure in the bipolar filtration of topologically slice knots

#### Abstract

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

$⋯ ⊂Tn+1 ⊂Tn ⊂⋯ ⊂T2 ⊂T1 ⊂T0 ⊂T$

is the bipolar filtration of $T$. We show that $T0∕T1$ has infinite rank, even modulo Alexander polynomial one knots. Recall that knots in $T0$ (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.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 1 (2015), 415-428.

Dates
Revised: 4 August 2014
Accepted: 5 August 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510840917

Digital Object Identifier
doi:10.2140/agt.2015.15.415

Mathematical Reviews number (MathSciNet)
MR3325742

Zentralblatt MATH identifier
1318.57005

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57N70: Cobordism and concordance

#### Citation

Cochran, Tim D; Horn, Peter D. Structure in the bipolar filtration of topologically slice knots. Algebr. Geom. Topol. 15 (2015), no. 1, 415--428. doi:10.2140/agt.2015.15.415. https://projecteuclid.org/euclid.agt/1510840917

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