Unknotting Number and Combinatorial Sutured Manifold Theory

Author (Your Name)

Alexander Rasmussen, Colby CollegeFollow

Date of Award

2013

Document Type

Honors Thesis (Colby Access Only)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Scott Taylor

Second Advisor

Andreas Malmendier

Abstract

The unknotting number u(K) of a knot K is an easily defined measure of its complexity. Yet little is known about unknotting number in general. Given two knots K₁ and K₂ we can form a third knot K₁#K₂ by “cutting” small segments from K₁ and K₂ and “pasting” them together. K₁#K₂ is called the connected sum of K₁ and K₂. An old conjecture states that unknotting number is additive under connected sums. That is, if K₁ and K₂ are knots in S³ and K₁#K₂ is their connected sum, then u(K₁#K₂) = u(K₁) + u(K₂). Martin Scharlemann proved that if K is a connected sum of two nontrivial knots then u(K) ≥ 2. With the goal of extending this result to connected sums of three nontrivial knots we study unknotting number for such connected sums using the techniques of Scharlemann's sutured manifold theory. We obtain some preliminary theorems in the direction of proving our result. We also present some lemmas that may be of use in continuing this project. Ultimately, this work may also prove useful in considering the results of performing more complicated rational tangle replacements on connected sums.

Keywords

unknotting number

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