Unknotting with a single twist

Samantha Allen (Dartmouth)

Ohyama showed that any knot can be unknotted by performing two full twists, each on a set of parallel strands. We consider the question of whether or not a given knot can be unknotted with a single full twist, and if so, what are the possible linking numbers associated to such a twist. It is observed that if a knot can be unknotted with a single twist, then some surgery on the knot bounds a rational homology ball. Using tools such as classical invariants and invariants arising from Heegaard Floer theory, we give obstructions for a knot to be unknotted with a single twist of a given linking number. In this talk, I will discuss some of these obstructions, their implications (especially for alternating knots), many examples, and some unanswered questions. This talk is based on joint work with Charles Livingston.

2 thoughts on “Unknotting with a single twist

  1. Just in case, I would like to let you know the following article,
    that you might be interested in.

    Knots that cannot be obtained from the trivial knot by a twisting
    Katsura Miyazaki, Akira Yasuhara
    Contemporary Mathematics 164 139 – 150 1994

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