Title : Polynomial Representation of Strongly Invertible Knots and Strongly negative Amphicheiral Knots
Speaker : Rama Mishra (Indian Institute of Technology)
Abstract :

It has been proved that every smooth knot in $S^3$ is isotopy equivalent to the closure of the image of an embedding $\phi:\mbR\mapsto \mbR^3$ defined by $\phi(t)=(f(t),g(t),h(t))$ where $f(t),$ $g(t)$ and $h(t)$ are polynomials over the field of real numbers $\mbR.$ In fact any two such polynomial embeddings representing the same knot-type can be joined by a polynomial isotopy. In 1992, Shastri constructed polynomial embeddings which represents the trefoil knot and the figure eight knot respectively. Later, we could find a general procedure to construct a polynomial embedding representing any torus knot of type $(p,q).$ After observing the pattern of these polynomial embeddings Kawauchi made the following two conjectures: \begin{enumerate} \item Every strongly invertible knot can be represented by a polynomial embedding $t\mapsto (f(t),g(t),h(t))$ where among $f(t),$ $g(t)$ and $h(t)$, two of them must be odd polynomials and one must be an even polynomial.\\ \item Every Strongly negative amphicheiral knot can be represented by a polynomial embedding $t\mapsto (f(t),g(t),h(t))$ where all three polynomials $f(t),$ $g(t)$ and $h(t)$ must be odd polynomials. \end{enumerate} In this talk we will prove that both the conjectures are true and present some examples of such embeddings.

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