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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Additive invariants for knots, links and graphs in $3$–manifolds

Scott A Taylor and Maggy Tomova

Geometry & Topology 22 (2018) 3235–3286
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 ( 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