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Twisted signatures of fibered knots

Anthony Conway and Matthias Nagel

Algebraic & Geometric Topology 21 (2021) 1973–2036
Abstract

This paper concerns twisted signature invariants of knots and 3–manifolds. In the fibered case, we reduce the computation of these invariants to the study of the intersection form and monodromy on the twisted homology of the fiber surface. Along the way, we use rings of power series to obtain new interpretations of the twisted Milnor pairing introduced by Kirk and Livingston. This allows us to relate these pairings to twisted Blanchfield pairings. Finally, we study the resulting signature invariants, all of which are twisted generalizations of the Levine–Tristram signature.

Keywords
knot, fibered knot, twisted Alexander polynomial, Milnor pairing, Blanchfield pairing, Casson–Gordon invariants
Mathematical Subject Classification 2010
Primary: 57M25
References
Publication
Received: 23 January 2020
Revised: 30 May 2020
Accepted: 25 June 2020
Published: 18 August 2021
Authors
Anthony Conway
Max-Planck-Institut fur Mathematik
Bonn
Germany
Matthias Nagel
Department of Mathematics
ETH Zurich
Zürich
Switzerland