Additive invariants for knots, links and graphs in 3–manifolds

Taylor, Scott; Tomova, Maggy

doi:10.2140/gt.2018.22.3235

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Additive invariants for knots, links and graphs in $3$–manifolds

Scott A Taylor and Maggy Tomova

Geometry & Topology 22 (2018) 3235–3286

DOI: 10.2140/gt.2018.22.3235

Abstract

We define two new families of invariants for ( $3$ –manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and ( $- \frac{1}{2}$ ) additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai’s width for knots in the $3$ –sphere. We give applications to the tunnel number and higher-genus bridge number of connected sums of knots.

Keywords

thin position, bridge number, tunnel number

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

References

Publication

Received: 16 July 2016

Revised: 6 October 2017

Accepted: 15 October 2017

Published: 23 September 2018

Proposed: Rob Kirby

Seconded: David Gabai, Cameron Gordon

Authors

Scott A Taylor
Department of Mathematics and Statistics Colby College Waterville, ME United States

Maggy Tomova
Department of Mathematics University of Iowa Iowa City, IA United States

Volume 22, issue 6 (2018)