Full twists and stability of knots and quivers
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arXiv
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| Format: | Preprint |
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2025
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| author | Chauhan, Sachin Kucharski, Piotr Noshchenko, Dmitry Pichai, Ramadevi Singh, Vivek Kumar Stošić, Marko |
| author_facet | Chauhan, Sachin Kucharski, Piotr Noshchenko, Dmitry Pichai, Ramadevi Singh, Vivek Kumar Stošić, Marko |
| contents | We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of $Sym^r$-coloured HOMFLY-PT polynomials under the addition of a~full twist to the knot. On the other hand, we show that symmetric quivers exhibit analogous stable growth under unlinking or linking of the quiver augmented with the extra node; in some cases this augmented quiver captures the spectrum of motivic Donaldson-Thomas invariants of all quivers in the sequence. Combining these two versions of the stable growth, we conjecture that performing a~full twist on any knot corresponds to appropriate unlinking or linking of the corresponding augmented quiver -- this statement is an important step towards a~direct definition of the knot-quiver correspondence based on the knot diagram. We confirm the conjecture for all twist knots, $(2,2p+1)$ torus knots, and all pretzel knots up to 15 crossings with an~odd number of twists in each twist region. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Full twists and stability of knots and quivers Chauhan, Sachin Kucharski, Piotr Noshchenko, Dmitry Pichai, Ramadevi Singh, Vivek Kumar Stošić, Marko High Energy Physics - Theory Mathematical Physics Geometric Topology Quantum Algebra We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of $Sym^r$-coloured HOMFLY-PT polynomials under the addition of a~full twist to the knot. On the other hand, we show that symmetric quivers exhibit analogous stable growth under unlinking or linking of the quiver augmented with the extra node; in some cases this augmented quiver captures the spectrum of motivic Donaldson-Thomas invariants of all quivers in the sequence. Combining these two versions of the stable growth, we conjecture that performing a~full twist on any knot corresponds to appropriate unlinking or linking of the corresponding augmented quiver -- this statement is an important step towards a~direct definition of the knot-quiver correspondence based on the knot diagram. We confirm the conjecture for all twist knots, $(2,2p+1)$ torus knots, and all pretzel knots up to 15 crossings with an~odd number of twists in each twist region. |
| title | Full twists and stability of knots and quivers |
| topic | High Energy Physics - Theory Mathematical Physics Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2508.18417 |