Polynomial representation of strongly-invertible knots and strongly-negative-amphicheiral knots

Abstract

It is shown that the symmetric behaviour of certain class of knots can be realized by their polynomial representations. We prove that every strongly invertible knot (open) can be represented by a polynomial embedding $t\mapsto (f(t),g(t),h(t))$ of $\mathbb{R}$ in $\mathbb{R}^{3}$ where among the polynomials $f(t)$, $g(t)$ and $h(t)$ two of them are odd polynomials and one is an even polynomial. We also prove that a subclass of strongly negative amphicheiral knots can be represented by a polynomial embedding $t\mapsto(f(t),g(t),h(t))$ of $\mathbb{R}$ in $\mathbb{R}^{3}$ where all three polynomials $f(t)$, $g(t)$ and $h(t)$ are odd polynomials.