A characterization of shortest geodesics on surfaces

Neumann-Coto, Max

doi:10.2140/agt.2001.1.349

Cited by 19 publications

(17 citation statements)

References 4 publications

(19 reference statements)

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“…We will always assume that collections of arcs on a surface are in minimal position, in the sense that they cross each other transversaly, and the number of crossings is minimal. It is pointed out in [Thu08] that the results in [FHS82] and [NC01] imply that this assumption can always be satisfied (up to homotopy). Some examples of dual cellular dissections on different surfaces are represented in Figure 3.…”

Section: Accordion Complex Slalom Complex and Non-crossing Complexmentioning

confidence: 99%

Non-kissing and non-crossing complexes for locally gentle algebras

2019

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Starting from a locally gentle bound quiver, we define on the one hand a simplicial complex, called the non-kissing complex. On the other hand, we construct a punctured, marked, oriented surface with boundary, endowed with a pair of dual dissections. From those geometric data, we define two simplicial complexes: the accordion complex, and the slalom complex, generalizing work of A. Garver and T. McConville in the case of a disk. We show that all three simplicial complexes are isomorphic, and that they are pure and thin. In particular, there is a notion of mutation on their facets, akin to τ -tilting mutation. Along the way, we also construct inverse bijections between the set of isomorphism classes of locally gentle bound quivers and the set of homeomorphism classes of punctured, marked, oriented surfaces with boundary, endowed with a pair of dual dissections.YP, VP and PGP were partially supported by the French ANR grant SC3A (15 CE40 0004 01). VP was partially supported by the French ANR grant CAPPS (17 CE40 0018). 1 arXiv:1807.04730v2 [math.CO] 13 Jul 20182.1. Locally gentle bound quivers and their blossoming quivers. We consider a bound quiverQ = (Q, I), formed by a finite quiver Q and an ideal I of the path algebra kQ (the k-vector space generated by all paths in Q, including vertices as paths of length zero, with multiplication induced by concatenation of paths) such that I is generated by linear combinations of paths of length at least two. Note that we do not require that the quotient algebra kQ/I be finite dimensional. The following definition is adapted from [BR87].Definition 2.1. A locally gentle bound quiverQ := (Q, I) is a (finite) bound quiver where (i) each vertex a ∈ Q 0 has at most two incoming and two outgoing arrows, (ii) the ideal I is generated by paths of length exactly two, (iii) for any arrow β ∈ Q 1 , there is at most one arrow α ∈ Q 1 such that t(α) = s(β) and αβ / ∈ I (resp. αβ ∈ I) and at most one arrow γ ∈ Q 1 such that t(β) = s(γ) and βγ / ∈ I (resp. βγ ∈ I). The algebra kQ/I is called a locally gentle algebra. A gentle bound quiver is a locally gentle bound quiverQ such that the algebra kQ/I is finite-dimensional; the algebra is then called a gentle algebra.Definition 2.2. A locally gentle bound quiverQ is complete if any vertex a ∈ Q 0 is incident to either one (a is a leaf) or four arrows (a is an internal vertex). The pruned subquiver of a

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Section: Accordion Complex Slalom Complex and Non-crossing Complexmentioning

confidence: 99%

Non-kissing and non-crossing complexes for locally gentle algebras

2019

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“…For example, the left-hand side of Proof: This follows, for instance, from the fact that a diagram is taut if and only if it is length-minimizing with respect to some metric (17,18). □ Diagrams can be monotonically simplified.…”

Section: Preliminaries On Surfaces and Curvesmentioning

confidence: 99%

Positive basis for surface skein algebras

Thurston

2014

Proc. Natl. Acad. Sci. U.S.A.

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“…We will provide a sketch of Max Neumann-Coto's construction of such a metric. A more detailed argument can be found in Lemma 1.2 of his paper [14].…”

Section: Star Log Symplectic Surfacesmentioning

confidence: 99%