We improve the upper bound on the superbridge index $sb[K]$ of a knot type $[K]$ in terms of the bridge index $b[K]$ from $sb[K] leq 5b -3$ to $sb[K]leq 3b[k] - 1$.
We show that every two-bridge knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_3(t), y = T_b(t), z =C(t)$ where $T_k(t)$ are the Chebyshev polynomials and $b+deg C = 3N$. If $C (t)= T_c(t)$ is a Chebyshev polynomial, we call s
uch a knot a harmonic knot. We give the classification of harmonic knots for $a le 3.$ Most results are derived from continued fractions and their matrix representations.
We provide a new proof of the following results of H. Schubert: If K is a satellite knot with companion J and pattern L that lies in a solid torus T in which it has index k, then the bridge numbers satisfy the following: 1) The bridge number of K is
greater than or equal to the product of k and the bridge number of J; 2) If K is a composite knot (this is the case k = 1), then the bridge number of K is one less than the sum of the bridge numbers of J and L.
We prove that the expected value of the ratio between the smooth four-genus and the Seifert genus of two-bridge knots tends to zero as the crossing number tends to infinity.
In this paper we show that the twisted Alexander polynomial associated to a parabolic representation determines fiberedness and genus of a wide class of 2-bridge knots. As a corollary we give an affirmative answer to a conjecture of Dunfield, Friedl and Jackson for infinitely many hyperbolic knots.
We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combin
ed with previous results from [KP10] and [BKP14]. We also give a sharp lower bound for the lexicographic degree of any knot, using real polynomial curves properties.