Spaces of polynomial knots in low degree

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.