Thin position for knots, links, and graphs in 3–manifolds

Taylor, Scott; Tomova, Maggy

doi:10.2140/agt.2018.18.1361

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

To Appear

Other MSP Journals

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

DOI: 10.2140/agt.2018.18.1361

Abstract

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.

Keywords

3-manifold, knot, spatial graph, bridge number, width, bridge position, Heegaard splitting

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

Secondary: 57M50

References

Publication

Received: 16 July 2016

Revised: 30 November 2017

Accepted: 15 January 2018

Published: 3 April 2018

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 18, issue 3 (2018)