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.