Polynomial knots and their spaces

Abstract

The main focus of this thesis is to study the topology of some spaces associated with polynomial knots and determining the least polynomial degree in which a given knot can be represented A polynomial knot is an embedding of R in R 3 whose component functions are real polynomials The image of a polynomial knot is a long knot Polynomial knots were mainly studied by Vassiliev 1990 1996 Shastri 1992 and Mishra Prabhakar 1994 2009 Vassiliev looked at the topology of the space V d consisting of polynomial knots whose component functions are monic polynomials of degree d with no constant term whereas Shastri Mishra and Prabhakar focused on finding concrete polynomial representation of a given knot In this thesis we have studied polynomial knots from both the perspectives We have generalized the space V d giving rise to some interesting spaces and explored the topology path components and the homotopy type of those spaces Furthermore we have studied the homotopy type of the space of all polynomial knots with respect to some natural topology on it On the other side we have focused on the polynomial representations of the knots up to six crossings The knots 0 1 3 1 4 1 and 5 1 were known to have representations in their minimal degree We have found the polynomial representations of the knots 5 2 6 1 6 2 6 3 3 1 3 1 and 3 1 3 1 in degree 7 where the representations of the knots 3 1 3 1 and 3 1 3 1 are in their minimal degree We have shown that it is almost impossible to represent the knots 5 2 6 1 6 2 and 6 3 in degree less than 7 newline newline