Knot Diagrams of Treewidth Two

H.L. Bodlaender; Benjamin Burton; Fedor Fomin; Alexander Grigoriev

Knot Diagrams of Treewidth Two

H.L. Bodlaender, Benjamin Burton, Fedor Fomin, Alexander Grigoriev

Research output: Book/Report › Report

Abstract

In this paper, we study knot diagrams for which the underlying graph has treewidth two. We give a linear time algorithm for the following problem: given a knot diagram of treewidth two, does it represent the unknot? We also show that for a link diagram of treewidth two we can test in linear time if it represents the unlink. From the algorithm, it follows that a diagram of the unknot of treewidth 2 can always be reduced to the trivial diagram with at most n (un)twist and (un)poke Reidemeister moves.

Original language	English
Place of Publication	Cornell University Library, US
Publisher	arXiv.org at Cornell University Library
Number of pages	19
Volume	1904.03117v2
Publication status	Published - 8 Apr 2019

Publication series

Series	cs.DS

JEL classifications

c00 - Mathematical and Quantitative Methods: General

Access to Document

https://arxiv.org/abs/1904.03117

Cite this

Bodlaender, H. L., Burton, B., Fomin, F., & Grigoriev, A. (2019). Knot Diagrams of Treewidth Two. arXiv.org at Cornell University Library. cs.DS https://arxiv.org/abs/1904.03117

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N2 - In this paper, we study knot diagrams for which the underlying graph has treewidth two. We give a linear time algorithm for the following problem: given a knot diagram of treewidth two, does it represent the unknot? We also show that for a link diagram of treewidth two we can test in linear time if it represents the unlink. From the algorithm, it follows that a diagram of the unknot of treewidth 2 can always be reduced to the trivial diagram with at most n (un)twist and (un)poke Reidemeister moves.

AB - In this paper, we study knot diagrams for which the underlying graph has treewidth two. We give a linear time algorithm for the following problem: given a knot diagram of treewidth two, does it represent the unknot? We also show that for a link diagram of treewidth two we can test in linear time if it represents the unlink. From the algorithm, it follows that a diagram of the unknot of treewidth 2 can always be reduced to the trivial diagram with at most n (un)twist and (un)poke Reidemeister moves.

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