Spaces of polynomial knots in low degree

Abstract

We show that all knots up to six crossings can be represented by polynomial knots of degree at most 7, among which except for 52,5∗2,61,6∗1,62,6∗2 and 63 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≥2, we define a set P˜d to be the set of all polynomial knots given by t↦(f(t),g(t),h(t)) such that deg(f)=d−2,deg(g)=d−1 and deg(h)=d. This set can be identified with a subset of R3d and thus it is equipped with the natural topology which comes from the usual topology R3d. In this paper we determine a lower bound on the number of path components of P˜d for d≤7. We define a path equivalence for polynomial knots in the space P˜d and show that it is stronger than the topological equivalence.