Small intersection numbers in the curve graph

Aougab, Tarik; Taylor, Selwyn

doi:10.1112/blms/bdu057

Cited by 10 publications

(19 citation statements)

References 17 publications

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“…Input top identifications: 1,11,3,27,8,15,7,24,0,10,2,12,4,21,19,17,24,14,6,23,28,9,16,25,13,5,20,18,16 Input bottom identifications: 0,10,2,26,7,14,6,23,28, 9,1,11,3,22,20,18,25,13,5,22,27,8,15,26,12,4,21,19,17 In our genus 2 example, we saw that there were 6 curves representing vertices in the elementary circuit set Γ. In fact, MICC utilizes the set Γ ′ in its calculation.…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

“…We can specific a curve representing a vertex of Γ ′ by a sequence of boundary segment labels. For example, the sequence [0, 22, 5,17,24,3,20,8] corresponds to a curve in this figure that intersects in cyclic order the boundary segments in this list. The set of green arcs in the figure correspond to this label sequence and should be to understood as a curve, γ 1 , representingγ 1 ∈ Γ ′ .…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

“…Input top identifications: 1,6,11,4,3,2,7,0,5,9,8,7 Input bottom identifications: 0,5,10,3,2,1, 6,11,4,10,9,8 The reader should readily grasp the ladder metaphor by considering the representation in Figure 7-left of a regular neighborhood of (∂ + ∪ ∼ ∂ − )∪{all ω arcs} coming from Figure 5. In this illustration, our two boundary components which have been glued together to form α is represented by the horizontal segment that has its ends identified.…”

Section: Information For the User Of Miccmentioning

confidence: 99%

“…For our extended example, the output would be: Curve 1 Distance : 3 [2,7,12,5,9,8,7,1,6,11,4,3] [12, 5, 10, 3, 2, 1, 6,11,4,10,9,8] Curve 2 Distance : 3 [5,9,8,7,1,6,11,4,3,2,7,12] [12, 5, 10, 3, 2, 1, 6,11,4,10,9,8] Curve 3 Distance : 4+ [1,6,11,4,3,2,7,12,5,9,8,7] [12, 5, 10, 3, 2, 1, 6,11,4,…”

Section: Faces Commandmentioning

confidence: 99%

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MICC: A tool for computing short distances in the curve complex

Glenn

Menasco

Morrell

et al. 2017

Journal of Symbolic Computation

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Abstract. The complex of curves C(Sg) of a closed orientable surface of genus g ≥ 2 is the simplicial complex whose vertices, C 0 (Sg), are isotopy classes of essential simple closed curves in Sg. Two vertices co-bound an edge of the 1-skeleton, C 1 (Sg), if there are disjoint representatives in Sg. A metric is obtained on C 0 (Sg) by assigning unit length to each edge of C 1 (Sg). Thus, the distance between two vertices, d(v, w), corresponds to the length of a geodesic-a shortest edge-path between v and w in C 1 (Sg). In (4), Birman, Margalit and the second author introduced the concept of efficient geodesics in C 1 (Sg) and used them to give a new algorithm for computing the distance between vertices. In this note, we introduce the software package MICC (Metric in the Curve Complex), a partial implementation of the efficient geodesic algorithm. We discuss the mathematics underlying MICC and give applications. In particular, up to an action of an element of the mapping class group, we give a calculation which produces all distance 4 vertex pairs for g = 2 that intersect 12 times, the minimal number of intersections needed for this distance and genus.

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Section: Proofs Of Main Resultsmentioning

confidence: 99%

Section: Proofs Of Main Resultsmentioning

confidence: 99%

Section: Information For the User Of Miccmentioning

confidence: 99%

Section: Faces Commandmentioning

confidence: 99%

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MICC: A tool for computing short distances in the curve complex

Glenn

Menasco

Morrell

et al. 2017

Journal of Symbolic Computation

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“…For a fixed surface S g,p , we will let i g,p denote the minimal geometric intersection number of a filling pair on that surface. As discussed above, the values of i g,p were determined in almost all cases in the works of Aougab-Huang [1] and Aougab-Taylor [2]. However, in the case of S 2,p , p ≥ 3 odd, they showed only the bounds:…”

Section: Preliminariesmentioning

confidence: 99%

Minimally intersecting filling pairs on the punctured surface of genus two

Jeffreys

2019

Topology and its Applications

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Efficient geodesics and an effective algorithm for distance in the complex of curves

2016

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We give an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, our approach is new, simple, and more effective for all distances accessible by computer. Our method gives a new preferred finite set of geodesics between any two vertices of the complex, called efficient geodesics, which are different from the tight geodesics introduced by Masur and Minsky. FIGURE 1. Vertices of C (S 2 ) with distance 4 and intersection number 12; this is the smallest possible intersection for vertices with distance 4 2 JOAN BIRMAN, DAN MARGALIT, AND WILLIAM MENASCO One goal of this paper is to give an algorithm for distance-the efficient geodesic algorithm-that actually can be implemented, at least for small distances. The third author and Glenn, Morrell, and Morse [9] have in fact already developed an implementation of our algorithm, called Metric in the Curve Complex [8]. Their program is assembling a data bank of examples as we write.Known examples. Let S g denote a closed, connected, orientable surface of genus g and let i min (g, d) denote the minimal intersection number for vertices of C (S g ) with distance d. The Metric in the Curve Complex program has been used to show that:(1) i min (2, 4) = 12 and (2) i min (3, 4) ≤ 21. The highly symmetric example in Figure 1-which realizes i min (2, 4)-was discovered using the program. See Section 2 for a discussion of this example and a proof using the methods of this paper that the distance is actually 4.We are only aware of one other explicit picture in the literature of a pair of vertices of C (S 2 ) that have distance four, namely, the example of Hempel that appears in the notes of Saul Schleimer [16, Figure 2] (see [9, Example 1.6] for a proof that the distance is 4). This example has geometric intersection number 25.Using the bounded geodesic image theorem [12, Theorem 3.1] of Masur and Minsky (as quantified by Webb [21]) it is possible to explicitly construct examples of vertices with any given distance; see [17, Section 6]. We do not know how to keep the intersection numbers close to the minimum with this method, but Aougab and Taylor did in fact use this method to give examples of vertices of arbitrary distance whose intersection numbers are close to the minimum in an asymptotic sense; see their paper [3] for the precise statement.Local infinitude. One reason why computations with the complex of curves are so difficult is that it is locally infinite and moreover there are infinitely many geodesics (i.e. shortest paths) between most pairs of vertices. Masur and Minsky [12] addressed this issue by finding a preferred set of geodesics, called tight geodesics, and proving that between any two vertices there are finitely many tight geodesics; see Section 2.2 for the definition. Our first goal is to give a new class of geodesics that still has finitely many elements connecting any two vertices but is more amenable to certain computations.Efficient geodesics. Ou...

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Small intersection numbers in the curve graph

Cited by 10 publications

References 17 publications

MICC: A tool for computing short distances in the curve complex

MICC: A tool for computing short distances in the curve complex

Minimally intersecting filling pairs on the punctured surface of genus two

Efficient geodesics and an effective algorithm for distance in the complex of curves

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