Morsifications and mutations

Fomin, Sergey; Pylyavskyy, Pavlo; Shustin, Eugeniĭ; Thurston, Dylan P.

doi:10.1112/jlms.12566

Cited by 13 publications

(35 citation statements)

References 72 publications

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“…The following description of the plabic graph link L plab G , or plabic link for short, can be deduced from [1,54,12] by breaking the symmetry following [26]. (See Section 3.4 for a more invariant description.)…”

Section: Plabic Linkmentioning

confidence: 99%

“…The fact that reduced plabic graphs are leaf recurrent was shown in [43,Remark 4.7]. Another natural subclass of leaf recurrent plabic graphs are the plabic fences of [12,Section 12]. A plabic fence is a plabic graph obtained by drawing k horizontal strands and inserting an arbitrary number of black-white and white-black bridges between them; see Figure 10 it is mutation equivalent to an acyclic quiver), and thus the point count polynomial R(Q G ; q) can be easily checked to coincide with P top (L plab G ; q).…”

Section: Plabic Linkmentioning

confidence: 99%

“…Shende-Treumann-Williams-Zaslow [54] suggested the existence of cluster structures for certain moduli spaces of sheaves associated to a Legendrian link, leading to many further developments; see for example [53,7,10]. Fomin-Pylyavskyy-Shushtin-Thurston [12] studied the relation between quiver mutation and morsifications of algebraic links. Casals-Gorsky-Gorsky-Simental [9] and Mellit [40] studied braid varieties associated to positive braid words; cluster structures on these spaces are the subject of ongoing works [8,21,20].…”

Section: Introductionmentioning

confidence: 99%

“…Plabic graphs were introduced by Postnikov [46] who also speculated on the relation with cluster algebras. To an arbitrary plabic graph G one can associate a plabic graph link L plab G introduced in [12,54]. On the other hand, the set of equivalence classes of reduced plabic graphs is in bijection with open positroid varieties, to which we associated positroid links in [17].…”

Section: Introductionmentioning

confidence: 99%

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Plabic links, quivers, and skein relations

Galashin,

Lam

2024

Algebraic Combinatorics

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We study relations between cluster algebra invariants and link invariants. First, we show that several constructions of positroid links (permutation links, Richardson links, grid diagram links, plabic graph links) give rise to isotopic links. For a subclass of permutations arising from concave curves, we also provide isotopies with the corresponding Coxeter links.Second, we associate a point count polynomial to an arbitrary locally acyclic quiver. We conjecture an equality between the top a-degree coefficient of the HOMFLY polynomial of a plabic graph link and the point count polynomial of its planar dual quiver. We prove this conjecture for leaf recurrent plabic graphs, which includes reduced plabic graphs and plabic fences as special cases.

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Section: Plabic Linkmentioning

confidence: 99%

Section: Plabic Linkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Plabic links, quivers, and skein relations

Galashin,

Lam

2024

Algebraic Combinatorics

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“…Totally non-negative Grassmannians Gr TNN (k, n) are a special case of the generalization to reductive Lie groups by Lusztig [53,54] of the classical notion of total positivity [31,32,40,66]. As for classical total positivity, the Grassmannians Gr TNN (k, n) naturally arise in relevant problems in different areas of mathematics and physics [12,13,16,18,27,51,61,63,67]. In particular, the role of total positivity in the selection of real regular KP-II solutions was pointed out in [55].…”

Section: Introductionmentioning

confidence: 99%

Real regular KP divisors on $${\texttt {M}}$$-curves and totally non-negative Grassmannians

Abenda

Grinevich

2022

Lett Math Phys

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In this paper, we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell $${{\mathcal {S}}}_{{\mathcal {M}}}^{\text{ TNN }}$$ S M TNN in the totally non-negative Grassmannian $$Gr^{\text{ TNN }}(k,n)$$ G r TNN ( k , n ) to the spectral data for the relevant class of real regular Kadomtsev–Petviashvili II (KP-II) solutions, thus completing the search of real algebraic-geometric data for the KP-II equation started in Abenda and Grinevich (Commun Math Phys 361(3):1029–1081, 2018; Sel Math New Ser 25(3):43, 2019). The spectral curve is modeled on the Krichever construction for degenerate finite-gap solutions and is a rationally degenerate $${\texttt {M}}$$ M -curve, $$\Gamma $$ Γ , dual to the graph. The divisors are real regular KP-II divisors in the ovals of $$\Gamma $$ Γ , i.e. they fulfill the conditions for selecting real regular finite-gap KP-II solutions in Dubrovin and Natanzon (Izv Akad Nauk SSSR Ser Mat 52:267–286, 1988). Since the soliton data are described by points in $${{\mathcal {S}}}_{{\mathcal {M}}}^{\text{ TNN }}$$ S M TNN , we establish a bridge between real regular finite-gap KP-II solutions (Dubrovin and Natanzon, 1988) and real regular multi-line KP-II solitons which are known to be parameterized by points in $$Gr^{\text{ TNN }}(k,n)$$ G r TNN ( k , n ) (Chakravarty and Kodama in Stud Appl Math 123:83–151, 2009; Kodama and Williams in Invent Math 198:637–699, 2014). We use the geometric characterization of spaces of relations on plabic networks introduced in Abenda and Grinevich (Adv Math 406:108523, 2022; Int Math Res Not 2022:rnac162, 2022. https://doi.org/10.1093/imrn/rnac162) to prove the invariance of this construction with respect to the many gauge freedoms on the network. Such systems of relations were proposed in Lam (in: Current developments in mathematics, International Press, Somerville, 2014) for the computation of scattering amplitudes for on-shell diagrams $$N=4$$ N = 4 SYM (Arkani-Hamed et al. in Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge, 2016) and govern the totally non-negative amalgamation of the little positive Grassmannians, $$Gr^{\text{ TP }}(1,3)$$ G r TP ( 1 , 3 ) and $$Gr^{\text{ TP }}(2,3)$$ G r TP ( 2 , 3 ) , into any given positroid cell $${{\mathcal {S}}}_{{\mathcal {M}}}^{\text{ TNN }}\subset {Gr^{\text{ TNN }} (k,n)}$$ S M TNN ⊂ G r TNN ( k , n ) . In our setting they control the reality and regularity properties of the KP-II divisor. Finally, we explain the transformation of both the curve and the divisor both under Postnikov’s moves and reductions and under amalgamation of positroid cells, and apply our construction to some examples.

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Complexity of Ice Quiver Mutation Equivalence

Soukup

2023

Ann. Comb.

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Morsifications and mutations

Abstract: We describe and investigate a connection between the topology of isolated singularities of plane curves and the mutation equivalence, in the sense of cluster algebra theory, of the quivers associated with their morsifications.

Cited by 13 publications

References 72 publications

Plabic links, quivers, and skein relations

Plabic links, quivers, and skein relations

Real regular KP divisors on $${\texttt {M}}$$-curves and totally non-negative Grassmannians

Complexity of Ice Quiver Mutation Equivalence

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