Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance

Miller, Allison N.; Powell, Mark

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.

Citation

Miller, A. N., & Powell, M. (2018). Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance. Algebraic & geometric topology, 18(6), 3425-3476. https://doi.org/10.2140/agt.2018.18.3425

• Journal Article Type: Article
• Acceptance Date: Jun 20, 2018
• Online Publication Date: Oct 18, 2018
• Publication Date: Oct 18, 2018
• Deposit Date: Mar 29, 2018
• Journal: Algebraic and Geometric Topology
• Print ISSN: 1472-2747
• Electronic ISSN: 1472-2739
• Publisher: Mathematical Sciences Publishers (MSP)
• Peer Reviewed: Peer Reviewed
• Volume: 18
• Issue: 6
• Pages: 3425-3476
• DOI: https://doi.org/10.2140/agt.2018.18.3425
• Public URL: https://durham-repository.worktribe.com/output/1336815
• Related Public URLs: https://arxiv.org/abs/1709.08560