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

TL;DR

This paper introduces a new concept of thin position for graphs in 3-manifolds, combining existing ideas for knots and manifolds, leading to the development of new invariants that detect the unknot and are additive under certain sums.

Contribution

It defines a novel thin position for graphs in 3-manifolds that unifies previous approaches and paves the way for new invariants of knots, links, and graphs.

Findings

Connect summing annuli and pairs-of-pants appear as thin levels.

New invariants similar to bridge number and Gabai's width are introduced.

Invariants detect the unknot and are additive under connected sum and vertex sum.

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 idea 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.