Spaces of polynomial knots in low degree

TL;DR

This paper investigates polynomial representations of knots up to six crossings, providing minimal degree representations, concrete examples, and analyzing the topology of polynomial knot spaces for degrees up to 7.

Contribution

It offers explicit polynomial representations for knots up to six crossings and studies the topological structure of polynomial knot spaces, partially addressing questions about degree constraints.

Findings

All knots up to 6 crossings can be represented by degree ≤7 polynomial knots.

Most knots are in their minimal degree representations, except for a few specific ones.

Lower bounds on the number of path components in polynomial knot spaces for degrees up to 7.

Abstract

We show that all knots up to $6$ crossings can be represented by polynomial knots of degree at most $7$, among which except for $5_2, 5_2^*, 6_1, 6_1^*, 6_2, 6_2^*$ and $6_3$ all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a question: Is there any $5$ crossing knot in degree $6$? In this paper we try to partially answer this question. For an integer $d\geq2$, we define a set $\mathcal{\tilde{P}}_d$ to be the set of all polynomial knots given by $t\mapsto\big(f(t),g(t),h(t)\big)$ such that $\text{deg}(f)=d-2$, $\text{deg}(g)=d-1$ and $\text{deg}(h)=d$. This set can be identified with a subset of $\mathbb{R}^{3d}$ and thus it is equipped with the natural topology which comes from the usual topology $\mathbb{R}^{3d}$. In this paper we determine a lower bound on the number of path components of $\mathcal{\tilde{P}}_d$ for $d\leq 7$. We define a path equivalence for polynomial knots in the space $\mathcal{\tilde{P}}_d$ and show that it is stronger than the topological equivalence.