Algebraic & Geometric Topology

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

Scott Taylor and Maggy Tomova

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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.

Article information

Algebr. Geom. Topol., Volume 18, Number 3 (2018), 1361-1409.

Received: 16 July 2016
Revised: 30 November 2017
Accepted: 15 January 2018
First available in Project Euclid: 26 April 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds
Secondary: 57M50: Geometric structures on low-dimensional manifolds

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


Taylor, Scott; Tomova, Maggy. Thin position for knots, links, and graphs in $3$–manifolds. Algebr. Geom. Topol. 18 (2018), no. 3, 1361--1409. doi:10.2140/agt.2018.18.1361.

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