Additional cases of positive twisted torus knots

Evan Amoranto; Brandy Doleshal; Matt Rathbun

doi:10.1142/s021821651750078x

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arXiv · 1701.03835

Additional cases of positive twisted torus knots

Evan Amoranto·Brandy Doleshal·Matt Rathbun

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Abstract

A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in $F$, the genus 2 Heegaard surface for $S^3$. Primitive/primitive and primitive/Seifert knots lie in $F$ in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean's conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with the same slope in $F$. In this paper, we extend Doleshal's results to show there is a four parameter family of positive twisted torus knots. Additionally, we provide new examples of twisted torus knots with distinct representatives with the same surface slope in $F$.

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Evan Amoranto, Brandy Doleshal, Matt Rathbun. 2017-10-30. Additional cases of positive twisted torus knots. https://doi.org/10.1142/s021821651750078x

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