How to read the length of a braid from its curve diagram

Wiest, Bert

doi:10.4171/ggd/143

Cited by 6 publications

(8 citation statements)

References 14 publications

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“…The following definition, which comes from [21], uses the notion of canonical length of a braid, which will be recalled in the next subsection. Definition 1.4.…”

Section: Corollary 13mentioning

confidence: 99%

Fast Algorithmic Nielsen–thurston Classification of Four-Strand Braids

Calvez

Wiest

2012

J. Knot Theory Ramifications

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Abstract. We give an algorithm which decides the Nielsen-Thurston type of a given four-strand braid. The complexity of our algorithm is quadratic with respect to word length. The proof of its validity is based on a result which states that for a reducible 4-braid which is as short as possible within its conjugacy class (short in the sense of Garside), reducing curves surrounding three punctures must be round or almost round. As an application, we give a polynomial time solution to the conjugacy problem for non-pseudo-Anosov four-strand braids.

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“…The following definition, which comes from [21], uses the notion of canonical length of a braid, which will be recalled in the next subsection. Definition 1.4.…”

Section: Corollary 13mentioning

confidence: 99%

Fast Algorithmic Nielsen–thurston Classification of Four-Strand Braids

Calvez

Wiest

2012

J. Knot Theory Ramifications

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“…The action by a positive permutation braid transforms an arc α with τ α ≡ 0 into an arc α ′ which, after reduction, has the following property: by an isotopy of D 2 that moves the n puncture points only in the vertical direction up or down, α ′ can be transformed into an arc whose imaginary coordinate is monotonically decreasing. Therefore, reduction curves C of complexity two can be characterized as follows: there exists a smooth arc α ′ disjoint from C but traversing horizontally all the punctures in the interior component of D 2 \C, such that τ α ′ only takes the values 0 and 1 (for a more detailed proof see [24]).…”

Section: Canonical Reduction Curves Of Reducible Positive Braids Witmentioning

confidence: 99%

Reducible braids and Garside Theory

González-Meneses

Wiest

2011

Algebr. Geom. Topol.

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We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its conjugacy class which we call the stabilized set of sliding circuits, and if it is reducible, then its reducibility is geometrically obvious: it has a round or almost round reducing curve. Moreover, for any given braid, an element of its stabilized set of sliding circuits can be found using the well-known cyclic sliding operation. This leads to a polynomial time algorithm for deciding the Nielsen-Thurston type of any braid, modulo one well-known conjecture on the speed of convergence of the cyclic sliding operation. 20F10, 20F36

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“…In the article [1], a certain labelling of the curve diagram of a braid by integer numbers was defined, called the winding number labelling. It was shown (Theorem 2.1) that the maximal and minimal label occurring in the curve diagram of a braidˇis equal to the supremum and infimum ofˇ, in the sense of Garside theory.…”

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confidence: 99%

“…Starting from the braidˇ1 we can even construct a counterexample to Proposition 3.1 of [1]: the curve diagram of the braidˇDˇ1ˇ2 2 B 6 , whereˇ2 D 3 2 3 4 2 1 2 3 2 5 , has six occurrences of the maximal label 3, as shown in Figure 1 (b). Yet, we believe that the slogan "maximally labelled arcs are rare" is philosophically true.…”

mentioning

confidence: 99%

“…parallel arcs labelled 1. (b) The maximally labelled arcs of the curve diagram of the braidˇDˇ1ˇ2, withˇ2 D 3 c) The error of the paper[1] occurred in the proof of Proposition 3.1 0 , where the arc shown in (c) was claimed not to exist. For any braidˇ2 B n and any i 2 f1; : : : ; n 1g, the arc E i ǒ f the curve diagram, equipped with the winding number labelling, contains no two parallel arc segments which both carry the maximal label.…”

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confidence: 99%

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