Colored HOMFLY-PT for hybrid weaving knot W ̂ 3 $$ {\hat{\mathrm{W}}}_3 $$ (m, n)
Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-06-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP06(2021)063 |
| _version_ | 1818440885019344896 |
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| author | Vivek Kumar Singh Rama Mishra P. Ramadevi |
| author_facet | Vivek Kumar Singh Rama Mishra P. Ramadevi |
| author_sort | Vivek Kumar Singh |
| collection | DOAJ |
| description | Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as W ̂ 3 $$ {\hat{W}}_3 $$ (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving ℛ $$ \mathrm{\mathcal{R}} $$ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional ℛ ̂ $$ \hat{\mathrm{\mathcal{R}}} $$ -matrices can be written in terms of infinite family of Laurent polynomials V n , t q $$ {\mathcal{V}}_{n,t}\left[q\right] $$ whose absolute coefficients has interesting relation to the Fibonacci numbers ℱ n $$ {\mathrm{\mathcal{F}}}_n $$ . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind. |
| first_indexed | 2024-12-14T18:19:27Z |
| format | Article |
| id | doaj.art-4edf1a9b4ff14a3980d876d1b48632c2 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-14T18:19:27Z |
| publishDate | 2021-06-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-4edf1a9b4ff14a3980d876d1b48632c22022-12-21T22:52:06ZengSpringerOpenJournal of High Energy Physics1029-84792021-06-012021613110.1007/JHEP06(2021)063Colored HOMFLY-PT for hybrid weaving knot W ̂ 3 $$ {\hat{\mathrm{W}}}_3 $$ (m, n)Vivek Kumar Singh0Rama Mishra1P. Ramadevi2Department of Mathematics, Indian Institute of Science Education and ResearchDepartment of Mathematics, Indian Institute of Science Education and ResearchDepartment of Physics, Indian Institute of Technology BombayAbstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as W ̂ 3 $$ {\hat{W}}_3 $$ (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving ℛ $$ \mathrm{\mathcal{R}} $$ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional ℛ ̂ $$ \hat{\mathrm{\mathcal{R}}} $$ -matrices can be written in terms of infinite family of Laurent polynomials V n , t q $$ {\mathcal{V}}_{n,t}\left[q\right] $$ whose absolute coefficients has interesting relation to the Fibonacci numbers ℱ n $$ {\mathrm{\mathcal{F}}}_n $$ . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.https://doi.org/10.1007/JHEP06(2021)063Quantum GroupsTopological StringsWilson, ’t Hooft and Polyakov loopsChern-Simons Theories |
| spellingShingle | Vivek Kumar Singh Rama Mishra P. Ramadevi Colored HOMFLY-PT for hybrid weaving knot W ̂ 3 $$ {\hat{\mathrm{W}}}_3 $$ (m, n) Journal of High Energy Physics Quantum Groups Topological Strings Wilson, ’t Hooft and Polyakov loops Chern-Simons Theories |
| title | Colored HOMFLY-PT for hybrid weaving knot W ̂ 3 $$ {\hat{\mathrm{W}}}_3 $$ (m, n) |
| title_full | Colored HOMFLY-PT for hybrid weaving knot W ̂ 3 $$ {\hat{\mathrm{W}}}_3 $$ (m, n) |
| title_fullStr | Colored HOMFLY-PT for hybrid weaving knot W ̂ 3 $$ {\hat{\mathrm{W}}}_3 $$ (m, n) |
| title_full_unstemmed | Colored HOMFLY-PT for hybrid weaving knot W ̂ 3 $$ {\hat{\mathrm{W}}}_3 $$ (m, n) |
| title_short | Colored HOMFLY-PT for hybrid weaving knot W ̂ 3 $$ {\hat{\mathrm{W}}}_3 $$ (m, n) |
| title_sort | colored homfly pt for hybrid weaving knot w ̂ 3 hat mathrm w 3 m n |
| topic | Quantum Groups Topological Strings Wilson, ’t Hooft and Polyakov loops Chern-Simons Theories |
| url | https://doi.org/10.1007/JHEP06(2021)063 |
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