#### Volume 18, issue 6 (2018)

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Symmetric chain complexes, twisted Blanchfield pairings and knot concordance

### Allison N Miller and Mark Powell

Algebraic & Geometric Topology 18 (2018) 3425–3476
##### Abstract

We give a formula for the duality structure of the $3$–manifold obtained by doing zero-framed surgery along a knot in the $3$–sphere, starting from a diagram of the knot. We then use this to give a combinatorial algorithm for computing the twisted Blanchfield pairing of such $3$–manifolds. With the twisting defined by Casson–Gordon-style representations, we use our computation of the twisted Blanchfield pairing to show that some subtle satellites of genus two ribbon knots yield nonslice knots. The construction is subtle in the sense that, once based, the infection curve lies in the second derived subgroup of the knot group.

##### Keywords
twisted Blanchfield pairing, symmetric Poincaré chain complex, knot concordance
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27, 57N70