Conference abstracts

Session A4 - Computational Geometry and Topology

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The lexicographic degree of the first two-bridge knots

Pierre-Vincent Koseleff

UPMC - Sorbonne Universités (Paris 6), France   -   pierre-vincent.koseleff@upmc.fr

We study the degree of polynomial representations of knots. We give here the lexicographic degree of all two-bridge knots with 11 or fewer crossings. The proof uses the braid theoretical method developed by Orevkov to study real plane curves [BKP2], isotopies on trigonal space curves and explicit parametrizations obtained by perturbing a triple point.

The lexicographic degree of a knot $K$ is the minimal multidegree, for the lexicographic order, of a polynomial knot whose closure in $S_3$ is isotopic to $K$.

We show that this degree is not necessarily obtained for curves that have the minimal number of real crossings. For example the knot $8_6$ has minimal degree $(3,10,14)$ that corresponds to a diagram with 9 crossings while the alternating diagram with 8 crossings has degree at least $(3, 11, 13)$.

Here we study two-bridge knots, namely those that admit a trigonal parametrization, that is to say a polynomial parametrization in degree $(3,b,c)$. We show the sharp lower bound $c \geq 3N - b$, where $N$ is the crossing number of $K$.

Lower bounds for $b$ are obtained by considering first Bézout-like conditions on the real crossings of the plane projection.

We also use the associated 3-strings braid associated to any trigonal algebraic plane curve that must be quasi-positive [Or2].

Upper bounds are given by previous constructions on Chebyshev diagrams [KP3] and slide isotopies on knot diagrams : isotopies for which the number of crossings never increases [BKP1].

In addition, we introduce a new reduction $R$ on trigonal plane curves. In many cases, the use of this reduction allows to subtract three to the number of crossings of the diagram and to the degree $b$, thanks to a result on real pseudoholomorphic curves deduced from [Or2]. On the other hand we show that we can explicitly give a polynomial curve of degree $(3, d+3)$ from a polynomial curve of degree $(3, d)$ by adding a triple point and perturbing the singularity.

We obtain the lexicographic degree for some infinite familes of two-bridge knots (the torus knots and generalized twist-knots) and for all two-bridge knots with 11 or fewer crossings. For every considered knot, we compute first an upper bound $b_C$ for $b$, using Chebyshev diagrams. We compute all diagrams with $b_C -1$ or fewer crossings that cannot be reduced by slide isotopies. Most of these diagrams may be reduced using $R$-reductions. For those that cannot be reduced, we consider all possible schemes corresponding to admissible rational algebraic curve of degree $(3,b)<(3,b_C)$ and compute their associated braids, that must be quasi-positive.

REFERENCES

[BKP1] E. Brugallé, P. -V. Koseleff, D. Pecker. Untangling trigonal diagrams, Journal Of Knot Theory And Its Ramificationsc (JKTR), 25(7) (2016), 10p.

[BKP2] E. Brugallé, P. -V. Koseleff, D. Pecker, On the lexicographic degree of two-bridge knots, Journal Of Knot Theory And Its Ramifications (JKTR), 25(7) (2016), 17p.

[KP3] P. -V. Koseleff, D. Pecker, Chebyshev diagrams for two-bridge knots, Geom. Dedicata 150 (2010), 405–425

[Or2] S. Yu. Orevkov. Classification of flexible $M$-curves of degree 8 up to isotopy, Geom. Funct. Anal., 12(4), 723–755, 2002.

Joint work with Erwan Brugallé (Centre de Mathématiques Laurent Schwartz, CMLA, École Polytechnique, France) and Daniel Pecker (UPMC - Sorbonne Universités, Paris 6, France).

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