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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.10468 |
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| _version_ | 1866910672158195712 |
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| author | Singh, Vivek Kumar Chbili, Nafaa |
| author_facet | Singh, Vivek Kumar Chbili, Nafaa |
| contents | Marino's conjecture remains underexplored within the framework of $SO(N )$ string dualities. In this article, we investigated the reformulated invariants of a one-parameter family of knots $\left[ K\right]_p$ derived from tangle surgery on Manolescu's quasi-alternating knot diagrams. Within topological string dualities, we have verified Marino's integrality conjecture for these families of knots up to the Young diagram representation ${\bf R}$, with ${|\bf R|}\leq 2$. Furthermore, through our analysis, we have conjectured the closed structure of extremal refined BPS integers for the torus knots $ \left[{\bf 3_1}\right]_{2p+1}$ and $ \left[{\bf 8_{20}}\right]_{2p+1}$, $p \in \mathbb{Z}_{\geq 0}$. As the parameter $p$ of the knot diagram increases, the total crossing number of a knot exceeds $16$, which we describe as a complex knot. Interestingly, we discovered a maximum number of gaps in the BPS spectra associated with complex knot families. Moreover, our observations indicated that as $p$ increases, the size of these gaps also expands. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10468 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | BPS Spectra of complex knots Singh, Vivek Kumar Chbili, Nafaa High Energy Physics - Theory Marino's conjecture remains underexplored within the framework of $SO(N )$ string dualities. In this article, we investigated the reformulated invariants of a one-parameter family of knots $\left[ K\right]_p$ derived from tangle surgery on Manolescu's quasi-alternating knot diagrams. Within topological string dualities, we have verified Marino's integrality conjecture for these families of knots up to the Young diagram representation ${\bf R}$, with ${|\bf R|}\leq 2$. Furthermore, through our analysis, we have conjectured the closed structure of extremal refined BPS integers for the torus knots $ \left[{\bf 3_1}\right]_{2p+1}$ and $ \left[{\bf 8_{20}}\right]_{2p+1}$, $p \in \mathbb{Z}_{\geq 0}$. As the parameter $p$ of the knot diagram increases, the total crossing number of a knot exceeds $16$, which we describe as a complex knot. Interestingly, we discovered a maximum number of gaps in the BPS spectra associated with complex knot families. Moreover, our observations indicated that as $p$ increases, the size of these gaps also expands. |
| title | BPS Spectra of complex knots |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2410.10468 |