Panhandle polynomials of torus links and geometric applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911342628175872 |
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| author | Mironov, Andrei Sati, Hisham Singh, Vivek Kumar Stoimenov, Alexander |
| author_facet | Mironov, Andrei Sati, Hisham Singh, Vivek Kumar Stoimenov, Alexander |
| contents | We use a decomposition of the tensor of the fundamental representation of the quantum group $U_q(\mathfrak{sl}_N)$ and the Rosso-Jones formula to establish a peculiar ``panhandle'' shape of the HOMFLY-PT polynomial of the reverse parallel of torus knots and links. Due to their panhandle-like intrinsic properties, the HOMFLY-PT polynomial is referred to as a ``panhandle polynomial''. With the help of the $\ell$-invariant, this extends to links the Etnyre-Honda result about the arc index and maximal Thurston-Bennequin invariant of torus knots. It has further geometric consequences, related to the braid index, the existence of minimal string Bennequin surfaces for banded and Whitehead doubled links, the Bennequin sharpness problem, and the equivalence of their quasipositivity and strong quasipositivity. We extend these properties to torus links, which relate to the classification of their component-wise Thurston-Bennequin invariants. Finally, we discuss the definition of the $\ell$-invariant for general links. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22967 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Panhandle polynomials of torus links and geometric applications Mironov, Andrei Sati, Hisham Singh, Vivek Kumar Stoimenov, Alexander Geometric Topology High Energy Physics - Theory Mathematical Physics Quantum Algebra 57K14(Primary), 57K10, 57K33, 17B37, 20F36 (Secondary) We use a decomposition of the tensor of the fundamental representation of the quantum group $U_q(\mathfrak{sl}_N)$ and the Rosso-Jones formula to establish a peculiar ``panhandle'' shape of the HOMFLY-PT polynomial of the reverse parallel of torus knots and links. Due to their panhandle-like intrinsic properties, the HOMFLY-PT polynomial is referred to as a ``panhandle polynomial''. With the help of the $\ell$-invariant, this extends to links the Etnyre-Honda result about the arc index and maximal Thurston-Bennequin invariant of torus knots. It has further geometric consequences, related to the braid index, the existence of minimal string Bennequin surfaces for banded and Whitehead doubled links, the Bennequin sharpness problem, and the equivalence of their quasipositivity and strong quasipositivity. We extend these properties to torus links, which relate to the classification of their component-wise Thurston-Bennequin invariants. Finally, we discuss the definition of the $\ell$-invariant for general links. |
| title | Panhandle polynomials of torus links and geometric applications |
| topic | Geometric Topology High Energy Physics - Theory Mathematical Physics Quantum Algebra 57K14(Primary), 57K10, 57K33, 17B37, 20F36 (Secondary) |
| url | https://arxiv.org/abs/2512.22967 |