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 $5_2,5_2^{\ast },6_1,6_1^{\ast },6_2,6_2^{\ast }$ 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 $d\ge 2$, we define a set ${\tilde{\mathcal{P}}}_d$ to be the set of all polynomial knots given by $t\mapsto (f(t),g(t),h(t))$ 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 ${\tilde{\mathcal{P}}}_d$ for $d\le 7$. We define a path equivalence for polynomial knots in the space ${\tilde{\mathcal{P}}}_d$ and show that it is stronger than the topological equivalence.

32 pages, 11 figures, 3 tables