Torus knots and generalized Schröder paths
We relate invariants of torus knots to the counts of a class of lattice paths, which we call generalized Schröder paths. We determine generating functions of such paths, located in a region determined by a type of a torus knot under consideration, and show that they encode colored HOMFLY-PT polynomi...
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| Format: | Article |
| Language: | English |
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Elsevier
2025-03-01
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| Series: | Nuclear Physics B |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S0550321325000240 |
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| _version_ | 1823856928197967872 |
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| author | Marko Stošić Piotr Sułkowski |
| author_facet | Marko Stošić Piotr Sułkowski |
| author_sort | Marko Stošić |
| collection | DOAJ |
| description | We relate invariants of torus knots to the counts of a class of lattice paths, which we call generalized Schröder paths. We determine generating functions of such paths, located in a region determined by a type of a torus knot under consideration, and show that they encode colored HOMFLY-PT polynomials of this knot. The generators of uncolored HOMFLY-PT homology correspond to a basic set of such paths. Invoking the knots-quivers correspondence, we express generating functions of such paths as quiver generating series, and also relate them to quadruply-graded knot homology. Furthermore, we determine corresponding A-polynomials, which provide algebraic equations and recursion relations for generating functions of generalized Schröder paths. The lattice paths of our interest explicitly enumerate BPS states associated to knots via brane constructions, as well as encode 3-dimensional N=2 theories. |
| format | Article |
| id | doaj-art-b48a04cc98814d0b8eb38be0c4bd9d05 |
| institution | Kabale University |
| issn | 0550-3213 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Nuclear Physics B |
| spelling | doaj-art-b48a04cc98814d0b8eb38be0c4bd9d052025-02-12T05:30:29ZengElsevierNuclear Physics B0550-32132025-03-011012116814Torus knots and generalized Schröder pathsMarko Stošić0Piotr Sułkowski1CEMS.UL, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Edifício C6, Campo Grande, 1749-016 Lisbon, Portugal; Mathematical Institute SANU, Knez Mihajlova 36, 11000 Beograd, SerbiaFaculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland; Corresponding author.We relate invariants of torus knots to the counts of a class of lattice paths, which we call generalized Schröder paths. We determine generating functions of such paths, located in a region determined by a type of a torus knot under consideration, and show that they encode colored HOMFLY-PT polynomials of this knot. The generators of uncolored HOMFLY-PT homology correspond to a basic set of such paths. Invoking the knots-quivers correspondence, we express generating functions of such paths as quiver generating series, and also relate them to quadruply-graded knot homology. Furthermore, we determine corresponding A-polynomials, which provide algebraic equations and recursion relations for generating functions of generalized Schröder paths. The lattice paths of our interest explicitly enumerate BPS states associated to knots via brane constructions, as well as encode 3-dimensional N=2 theories.http://www.sciencedirect.com/science/article/pii/S0550321325000240 |
| spellingShingle | Marko Stošić Piotr Sułkowski Torus knots and generalized Schröder paths Nuclear Physics B |
| title | Torus knots and generalized Schröder paths |
| title_full | Torus knots and generalized Schröder paths |
| title_fullStr | Torus knots and generalized Schröder paths |
| title_full_unstemmed | Torus knots and generalized Schröder paths |
| title_short | Torus knots and generalized Schröder paths |
| title_sort | torus knots and generalized schroder paths |
| url | http://www.sciencedirect.com/science/article/pii/S0550321325000240 |
| work_keys_str_mv | AT markostosic torusknotsandgeneralizedschroderpaths AT piotrsułkowski torusknotsandgeneralizedschroderpaths |