## Algebraic & Geometric Topology

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

#### 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.

#### Article information

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

Dates
Revised: 30 November 2017
Accepted: 15 January 2018
First available in Project Euclid: 26 April 2018

https://projecteuclid.org/euclid.agt/1524708095

Digital Object Identifier
doi:10.2140/agt.2018.18.1361

Mathematical Reviews number (MathSciNet)
MR3784008

Zentralblatt MATH identifier
06866402

#### Citation

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. https://projecteuclid.org/euclid.agt/1524708095

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