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Thin position for knots, links, and graphs in $3\mkern-1.5mu$–manifolds

Scott A Taylor and Maggy Tomova

Algebraic & Geometric Topology 18 (2018) 1361–1409

We define a new notion of thin position for a graph in a 3–manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the ideas of thin position for knots first originated by Gabai. This thin position has the property that connect-summing annuli and pairs-of-pants show up as thin levels. In a forthcoming paper, this new thin position allows us to define two new families of invariants of knots, links, and graphs in 3–manifolds. The invariants in one family are similar to bridge number, and the invariants in the other family are similar to Gabai’s width for knots in the 3–sphere. The invariants in both families detect the unknot and are additive under connected sum and trivalent vertex sum.

3-manifold, knot, spatial graph, bridge number, width, bridge position, Heegaard splitting
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Secondary: 57M50
Received: 16 July 2016
Revised: 30 November 2017
Accepted: 15 January 2018
Published: 3 April 2018
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