Spaces of Polynomial knots in low degree

Rama Mishra, Hitesh Raundal

Published 2014-10-21Version 1

{\it 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. We define the set $\pd$ to be the set of all polynomial knots given by $t\mapsto\fght$ where $f, g$ and $h$ are real polynomials with $\fghio$. This set can be identified with a subset of $\rtd$ and thus it is equipped with the natural topology which comes from the usual topology $\rtd$. In this paper we determine a lower bound on the number of path components of $\pd$ for $d\leq 7$. We define path equivalence between polynomial knots in the space $\pd$ and show that path equivalence is stronger than the topological equivalence.}