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.