Plabic links, quivers, and skein relations
Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 431-474.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.345
Classification: 13F60, 57K14, 14M15, 05E99
Keywords: plabic graph, quiver, link isotopy, Coxeter link, point count, HOMFLY polynomial, skein relation
Galashin, Pavel 1; Lam, Thomas 2

1 Department of Mathematics University of California Los Angeles 520 Portola Plaza Los Angeles CA 90025 USA
2 Department of Mathematics University of Michigan 2074 East Hall 530 Church Street Ann Arbor MI 48109-1043 USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2024__7_2_431_0,
     author = {Galashin, Pavel and Lam, Thomas},
     title = {Plabic links, quivers, and skein relations},
     journal = {Algebraic Combinatorics},
     pages = {431--474},
     publisher = {The Combinatorics Consortium},
     volume = {7},
     number = {2},
     year = {2024},
     doi = {10.5802/alco.345},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.345/}
}
TY  - JOUR
AU  - Galashin, Pavel
AU  - Lam, Thomas
TI  - Plabic links, quivers, and skein relations
JO  - Algebraic Combinatorics
PY  - 2024
SP  - 431
EP  - 474
VL  - 7
IS  - 2
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.345/
DO  - 10.5802/alco.345
LA  - en
ID  - ALCO_2024__7_2_431_0
ER  - 
%0 Journal Article
%A Galashin, Pavel
%A Lam, Thomas
%T Plabic links, quivers, and skein relations
%J Algebraic Combinatorics
%D 2024
%P 431-474
%V 7
%N 2
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.345/
%R 10.5802/alco.345
%G en
%F ALCO_2024__7_2_431_0
Galashin, Pavel; Lam, Thomas. Plabic links, quivers, and skein relations. Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 431-474. doi : 10.5802/alco.345. https://alco.centre-mersenne.org/articles/10.5802/alco.345/

[1] A’Campo, Norbert Real deformations and complex topology of plane curve singularities, Ann. Fac. Sci. Toulouse Math. (6), Volume 8 (1999) no. 1, pp. 5-23 | Numdam | MR | Zbl

[2] Arkani-Hamed, Nima; Lam, Thomas; Spradlin, Marcus Positive configuration space, Comm. Math. Phys., Volume 384 (2021) no. 2, pp. 909-954 | DOI | MR

[3] Arnol’d, V. I. The geometry of spherical curves and quaternion algebra, Uspekhi Mat. Nauk, Volume 50 (1995) no. 1(301), pp. 3-68 | DOI | MR | Zbl

[4] Bazier-Matte, Véronique; Schiffler, Ralf Knot theory and cluster algebras, Adv. Math., Volume 408 (2022), Paper no. 108609, 45 pages | DOI | MR | Zbl

[5] Blasiak, Jonah; Haiman, Mark; Morse, Jennifer; Pun, Anna; Seelinger, George H. A shuffle theorem for paths under any line, Forum Math. Pi, Volume 11 (2023), Paper no. e5, 38 pages | DOI | MR

[6] Burban, Igor; Schiffmann, Olivier On the Hall algebra of an elliptic curve, I, Duke Math. J., Volume 161 (2012) no. 7, pp. 1171-1231 | DOI | MR | Zbl

[7] Casals, Roger; Gao, Honghao Infinitely many Lagrangian fillings, Ann. of Math. (2), Volume 195 (2022) no. 1, pp. 207-249 | DOI | MR

[8] Casals, Roger; Gorsky, Eugene; Gorsky, Mikhail; Le, Ian; Shen, Linhui; Simental, José Cluster structures on braid varieties, 2022 | arXiv

[9] Casals, Roger; Gorsky, Eugene; Gorsky, Mikhail; Simental, José Positroid Links and Braid varieties, 2024 | arXiv

[10] Casals, Roger; Weng, Daping Microlocal theory of Legendrian links and cluster algebras, Geom. Topol., Volume 28 (2024) no. 2, pp. 901-1000 | DOI | MR | Zbl

[11] Deodhar, Vinay V. On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math., Volume 79 (1985) no. 3, pp. 499-511 | DOI | MR | Zbl

[12] Fomin, Sergey; Pylyavskyy, Pavlo; Shustin, Eugenii; Thurston, Dylan Morsifications and mutations, J. Lond. Math. Soc. (2), Volume 105 (2022) no. 4, pp. 2478-2554 | DOI | MR

[13] Fomin, Sergey; Williams, Lauren; Zelevinsky, Andrei Introduction to Cluster Algebras. Chapters 1-3, 2021 | arXiv

[14] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | DOI | MR

[15] Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B. R.; Millett, K.; Ocneanu, A. A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.), Volume 12 (1985) no. 2, pp. 239-246 | DOI | MR

[16] Galashin, Pavel; Lam, Thomas Positroid Catalan numbers, 2021 | arXiv

[17] Galashin, Pavel; Lam, Thomas Positroids, knots, and q,t-Catalan numbers, Sém. Lothar. Combin., Volume 85B (2021), Paper no. 54, 12 pages | MR | Zbl

[18] Galashin, Pavel; Lam, Thomas Monotone links in DAHA and EHA, 2023 | arXiv

[19] Galashin, Pavel; Lam, Thomas Positroid varieties and cluster algebras, Ann. Sci. Éc. Norm. Supér. (4), Volume 56 (2023) no. 3, pp. 859-884 | DOI | MR

[20] Galashin, Pavel; Lam, Thomas; Sherman-Bennett, Melissa Braid variety cluster structures, II: general type, 2023 | arXiv

[21] Galashin, Pavel; Lam, Thomas; Sherman-Bennett, Melissa; Speyer, David Braid variety cluster structures, I: 3D plabic graphs, 2022 | arXiv

[22] Geck, Meinolf; Pfeiffer, Götz On the irreducible characters of Hecke algebras, Adv. Math., Volume 102 (1993) no. 1, pp. 79-94 | DOI | MR

[23] Gorsky, Eugene; Neguţ, Andrei Refined knot invariants and Hilbert schemes, J. Math. Pures Appl. (9), Volume 104 (2015) no. 3, pp. 403-435 | DOI | MR

[24] Gorsky, Eugene; Neguţ, Andrei; Rasmussen, Jacob Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology, Adv. Math., Volume 378 (2021), Paper no. 107542, 115 pages | DOI | MR | Zbl

[25] Hikami, Kazuhiro; Inoue, Rei Braids, complex volume and cluster algebras, Algebr. Geom. Topol., Volume 15 (2015) no. 4, pp. 2175-2194 | DOI | MR | Zbl

[26] Hirasawa, Mikami Visualization of A’Campo’s fibered links and unknotting operation, Topology Appl., Volume 121 (2002) no. 1, pp. 287-304 | DOI | MR | Zbl

[27] The Knot Atlas: The Thistlethwaite Link Table http://katlas.org/wiki/The_Thistlethwaite_Link_Table

[28] The Knot Atlas: The Rolfsen Knot Table http://katlas.org/wiki/The_Rolfsen_Knot_Table

[29] Khovanov, Mikhail Triply-graded link homology and Hochschild homology of Soergel bimodules, Internat. J. Math., Volume 18 (2007) no. 8, pp. 869-885 | DOI | MR

[30] Khovanov, Mikhail; Rozansky, Lev Matrix factorizations and link homology, Fund. Math., Volume 199 (2008) no. 1, pp. 1-91 | DOI | MR | Zbl

[31] Khovanov, Mikhail; Rozansky, Lev Matrix factorizations and link homology. II, Geom. Topol., Volume 12 (2008) no. 3, pp. 1387-1425 | DOI | MR | Zbl

[32] Knutson, Allen; Lam, Thomas; Speyer, David E Positroid varieties: juggling and geometry, Compos. Math., Volume 149 (2013) no. 10, pp. 1710-1752 | DOI | MR

[33] Lam, Thomas Totally nonnegative Grassmannian and Grassmann polytopes, Current developments in mathematics 2014, Int. Press, Somerville, MA, 2016, pp. 51-152 | MR | Zbl

[34] Lam, Thomas; Speyer, David E Cohomology of cluster varieties I: Locally acyclic case, Algebra Number Theory, Volume 16 (2022) no. 1, pp. 179-230 | DOI | MR

[35] Lam, Thomas; Speyer, David E Cohomology of cluster varieties II: Acyclic case, J. Lond. Math. Soc. (2), Volume 108 (2023) no. 6, pp. 2377-2414 | MR | Zbl

[36] Leclerc, B. Cluster structures on strata of flag varieties, Adv. Math., Volume 300 (2016), pp. 190-228 | DOI | MR | Zbl

[37] Lee, Kyungyong; Schiffler, Ralf Cluster algebras and Jones polynomials, Selecta Math. (N.S.), Volume 25 (2019) no. 4, Paper no. 58, 41 pages | DOI | MR | Zbl

[38] Lickorish, W. B. Raymond An introduction to knot theory, Graduate Texts in Mathematics, 175, Springer-Verlag, New York, 1997, x+201 pages | DOI | MR

[39] Livingston, Charles; Moore, Allison H. KnotInfo: Table of Knot Invariants, 2022

[40] Mellit, Anton Cell decompositions of character varieties, 2019 | arXiv

[41] Muller, Greg Locally acyclic cluster algebras, Adv. Math., Volume 233 (2013), pp. 207-247 | DOI | MR | Zbl

[42] Muller, Greg Skein and cluster algebras of marked surfaces, Quantum Topol., Volume 7 (2016) no. 3, pp. 435-503 | DOI | MR

[43] Muller, Greg; Speyer, David E Cluster algebras of Grassmannians are locally acyclic, Proc. Amer. Math. Soc., Volume 144 (2016) no. 8, pp. 3267-3281 | DOI | MR

[44] Oblomkov, A.; Rozansky, L. Homfly-PT homology of Coxeter links, Transform. Groups, Volume 28 (2023) no. 3, pp. 1245-1275 | DOI | MR

[45] Ozsváth, Peter S.; Stipsicz, András I.; Szabó, Zoltán Grid homology for knots and links, Mathematical Surveys and Monographs, 208, American Mathematical Society, Providence, RI, 2015, x+410 pages | DOI | MR

[46] Postnikov, Alexander Total positivity, Grassmannians, and networks (2006) | arXiv

[47] Przytycki, Józef H.; Traczyk, Pawel Conway algebras and skein equivalence of links, Proc. Amer. Math. Soc., Volume 100 (1987) no. 4, pp. 744-748 | DOI | MR | Zbl

[48] Rolfsen, Dale Knots and links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Houston, TX, 1990, xiv+439 pages | MR

[49] Rutherford, Dan Thurston-Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: the Fuchs conjecture and beyond, Int. Math. Res. Not. (2006), Paper no. 78591, 15 pages | DOI | MR

[50] Schiffmann, Olivier; Vasserot, Eric The elliptic Hall algebra and the K-theory of the Hilbert scheme of 𝔸 2 , Duke Math. J., Volume 162 (2013) no. 2, pp. 279-366 | DOI | MR

[51] Scott, J. S. Grassmannians and cluster algebras, Proc. London Math. Soc. (3), Volume 92 (2006) no. 2, pp. 345-380 | DOI | MR | Zbl

[52] Serhiyenko, K.; Sherman-Bennett, M.; Williams, L. Cluster structures in Schubert varieties in the Grassmannian, Proc. Lond. Math. Soc. (3), Volume 119 (2019) no. 6, pp. 1694-1744 | DOI | MR

[53] Shen, Linhui; Weng, Daping Cluster structures on double Bott-Samelson cells, Forum Math. Sigma, Volume 9 (2021), Paper no. e66, 89 pages | DOI | MR | Zbl

[54] Shende, Vivek; Treumann, David; Williams, Harold; Zaslow, Eric Cluster varieties from Legendrian knots, Duke Math. J., Volume 168 (2019) no. 15, pp. 2801-2871 | DOI | MR

[55] Shende, Vivek; Treumann, David; Zaslow, Eric Legendrian knots and constructible sheaves, Invent. Math., Volume 207 (2017) no. 3, pp. 1031-1133 | DOI | MR

[56] Trinh, Minh-Tâm Quang From the Hecke Category to the Unipotent Locus, 2021 | arXiv

[57] Turaev, V. G. The Conway and Kauffman modules of a solid torus, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 167 (1988) no. 6, p. 79-89, 190 | DOI | MR

Cited by Sources: