Colored Jones Polynomials and the Volume Conjecture
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912777894887424 |
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| author | Hughes, Mark Jejjala, Vishnu Ramadevi, P. Roy, Pratik Singh, Vivek Kumar |
| author_facet | Hughes, Mark Jejjala, Vishnu Ramadevi, P. Roy, Pratik Singh, Vivek Kumar |
| contents | Using the vertex model approach for braid representations, we compute polynomials for spin-1 placed on hyperbolic knots up to 15 crossings. These polynomials are referred to as 3-colored Jones polynomials or adjoint Jones polynomials. Training a subset of the data using a fully connected feedforward neural network, we predict the volume of the knot complement of hyperbolic knots from the adjoint Jones polynomial or its evaluations with 99.34% accuracy. A function of the adjoint Jones polynomial evaluated at the phase $q=e^{ 8 πi / 15 }$ predicts the volume with nearly the same accuracy as the neural network. From an analysis of 2-colored and 3-colored Jones polynomials, we conjecture the best phase for $n$-colored Jones polynomials, and use this hypothesis to motivate an improved statement of the volume conjecture. This is tested for knots for which closed form expressions for the $n$-colored Jones polynomial are known, and we show improved convergence to the volume. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Colored Jones Polynomials and the Volume Conjecture Hughes, Mark Jejjala, Vishnu Ramadevi, P. Roy, Pratik Singh, Vivek Kumar Geometric Topology Machine Learning High Energy Physics - Theory Using the vertex model approach for braid representations, we compute polynomials for spin-1 placed on hyperbolic knots up to 15 crossings. These polynomials are referred to as 3-colored Jones polynomials or adjoint Jones polynomials. Training a subset of the data using a fully connected feedforward neural network, we predict the volume of the knot complement of hyperbolic knots from the adjoint Jones polynomial or its evaluations with 99.34% accuracy. A function of the adjoint Jones polynomial evaluated at the phase $q=e^{ 8 πi / 15 }$ predicts the volume with nearly the same accuracy as the neural network. From an analysis of 2-colored and 3-colored Jones polynomials, we conjecture the best phase for $n$-colored Jones polynomials, and use this hypothesis to motivate an improved statement of the volume conjecture. This is tested for knots for which closed form expressions for the $n$-colored Jones polynomial are known, and we show improved convergence to the volume. |
| title | Colored Jones Polynomials and the Volume Conjecture |
| topic | Geometric Topology Machine Learning High Energy Physics - Theory |
| url | https://arxiv.org/abs/2502.18575 |