On 3-super bridge knots

TL;DR

This paper proves that the 5_2 and 6_2 knots have super bridge index 3 by constructing degree 6 polynomial representations, expanding the known class of such knots beyond trefoil and figure-eight.

Contribution

It demonstrates that the 5_2 and 6_2 knots are 3-super bridge knots through explicit polynomial representations, addressing a conjecture and a question in polynomial knot theory.

Findings

5_2 and 6_2 knots are 3-super bridge knots

Constructed degree 6 polynomial representations for these knots

Answers the question about 5-crossing knots in degree 6

Abstract

It is known that there are only finitely many knots with super bridge index 3. Jin and Jeon have provided a list of possible such candidates. However, they conjectured that the only knots with super bridge index 3 are trefoil and the figure eight knot. In this paper, we prove that the $5_2$ knot and the $6_2$ knot are also 3-super bridge knots by providing a polynomial representation of these knots in degree $6.$ This also answers a question asked by Durfee and O'Shea in their paper on polynomial knots: is there any 5-crossing knot in degree 6?