Hard Unknots and Collapsing Tangles

Details Hard Unknots and Collapsing Tangles Hard Unknots and Collapsing Tangles

2006-01-22

arxiv.org/abs/math/0601525v4

Mathematics

Geometric Topology

59 pages, 44 figures, LaTeX manuscript

Scientific paper

This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is an unknot. The key theorem shows that the numerator of the sum of two rational tangles [P/Q] and [R/S] is unknotted if and only if PS + QR has absolute value equal to 1. The paper uses these results in studying processive DNA recombination, finding minimal size unknot diagrams, generalizing to collapses to knots as well as to unknots, and in finding unknots with arbirarily high complexity. The paper is self-contained, with a review of the theory of rational tangles and a last section on relationships of the theme of the paper with other aspects of topology and number theory.

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