A Bound on the Unknotting Number

Abstract In this paper we give a bound on the unknotting number of a knot whose quasitoric braid representation is of type (3, q). Mathematics Subject Classification: 57M2

Poncelet's theorem and Billiard knots

Let $D$ be any elliptic right cylinder. We prove that every type of knot can be realized as the trajectory of a ball in $D.$ This proves a conjecture of Lamm and gives a new proof of a conjecture of Jones and Przytycki. We use Jacobi's proof of Poncelet's theorem by means of elliptic functions

The canonical genus for Whitehead doubles of a family of alternating knots

For any given integer $r \geq 1$ and a quasitoric braid $\beta_r=(\sigma_r^{-\epsilon} \sigma_{r-1}^{\epsilon}...$ $\sigma_{1}^{(-1)^{r}\epsilon})^3$ with $\epsilon=\pm 1$, we prove that the maximum degree in $z$ of the HOMFLYPT polynomial $P_{W_2(\hat\beta_r)}(v,z)$ of the doubled link $W_2(\hat\beta_r)$ of the closure $\hat\beta_r$ is equal to $6r-1$. As an application, we give a family $\mathcal K^3$ of alternating knots, including $(2,n)$ torus knots, 2-bridge knots and alternating pretzel knots as its subfamilies, such that the minimal crossing number of any alternating knot in $\mathcal K^3$ coincides with the canonical genus of its Whitehead double. Consequently, we give a new family $\mathcal K^3$ of alternating knots for which Tripp's conjecture holds.Comment: 33 pages, 27 figure