Date of Award
Honors Thesis (Colby Access Only)
Colby College. Mathematics and Statistics Dept.
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.
Recommended CitationRasmussen, Alexander, "Unknotting Number and Combinatorial Sutured Manifold Theory" (2013). Honors Theses. Paper 698.
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