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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

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.

twisted Blanchfield pairing, symmetric Poincaré chain complex, knot concordance
Mathematical Subject Classification 2010
Primary: 57M25, 57M27, 57N70
Received: 29 October 2017
Revised: 30 April 2018
Accepted: 22 June 2018
Published: 18 October 2018
Allison N Miller
Department of Mathematics
University of Texas at Austin
Austin, TX
United States
Department of Mathematics
Rice University
Houston, TX
United States
Mark Powell
Department of Mathematical Sciences
Durham University
United Kingdom