Department of Mathematics
Polynomial Knot theory
Rama Mishra
Boise State University
Polynomials are the easiest functions to work with. If we have a space curve parametrized by polynomials which is an embedding of the real numbers in 3-space then its one point compactification will be a smooth embedding of the unit circle in the unit three sphere which is nothing but a knot in the classical sense. On the other hand it can esaily be seen that if we take an open knot K, then, up to equivalence, we can find a polynomial embedding from the real numbers to 3-space that can represent K. A knot represented by a polynomial embedding is referred to as a polynomial knot. Polynomial representations for equivalent knots are connected by a one parameter family of polynomial embeddings. Thus, there is a bijection between equivalence classes of knots and equivalence classes of polynomial knots.
In this talk we show some estimates on the degree of the polynomials to represent a given knot type. We also discuss that polynomial knot theory may be employed to compute some known knot invariants.
All interested persons are welcome.