arXiv subjects

Adrián Jiménez Pascual

Publications and source records attributed to Adrián Jiménez Pascual.

2 recordsLinked to original sources

arXiv2017

On wrapping number, adequacy and the crossing number of satellite knots

In this work we establish the tightest lower bound up-to-date for the minimal crossing number of a satellite knot based on the minimal crossing number of the companion used to build the satellite. If $M$ is the wrapping number of the pattern knot, we essentially show that $c(Sat(P,C))>\frac{M^2}{2}c(C)$. The existence of this bound will be proven when the companion knot is adequate, and it will be further tuned in the case of the companion being alternating.

math.GT↗

arXiv2015

On lassos and the Jones polynomial of satellite knots

In this master thesis, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot $K$ with Alexander polynomial $Δ_K(t)$, I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial $Δ_K(t^d)$ where $d\in\mathbb{N}\cup \{0\}$. In particular, I prove that if $d\in\{0,1,2,3\}$ these satellite knots have different Jones polynomials. For this purpose, I give rise to a formula for calculating the Jones polynomial of a satellite knot in terms of the Jones polynomials of its pattern and companion.

math.GT↗