Closed 4-braids and the Jones unknot conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916212350386176 |
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| author | Korzun, Dmitriy Lanina, Elena Sleptsov, Alexey |
| author_facet | Korzun, Dmitriy Lanina, Elena Sleptsov, Alexey |
| contents | The Jones problem is a question whether there is a non-trivial knot with the trivial Jones polynomial in one variable $q$. The answer to this fundamental question is still unknown despite numerous attempts to explore it. In braid presentation the case of 4-strand braids is already open. S. Bigelow showed in 2000 that if the Burau representation for four-strand braids is unfaithful, then there is an infinite number of non-trivial knots with the trivial two-variable HOMFLY-PT polynomial and hence, with the trivial Jones polynomial, since it is obtained from the HOMFLY-PT polynomial by the specialisation of one of the variables $A=q^2$.
In this paper, we study four-strand braids and ask whether there are non-trivial knots with the trivial Jones polynomial but a non-trivial HOMFLY-PT polynomial. We have discovered that there is a whole 1-parameter family, parameterised by the writhe number, of 2-variable polynomials that can be HOMFLY-PT polynomials of some knots. We explore various properties of the obtained hypothetical HOMFLY-PT polynomials and suggest several checks to test these formulas. A generalisation is also proposed for the case of a large number of strands. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_02553 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Closed 4-braids and the Jones unknot conjecture Korzun, Dmitriy Lanina, Elena Sleptsov, Alexey Geometric Topology High Energy Physics - Theory Mathematical Physics Group Theory Representation Theory The Jones problem is a question whether there is a non-trivial knot with the trivial Jones polynomial in one variable $q$. The answer to this fundamental question is still unknown despite numerous attempts to explore it. In braid presentation the case of 4-strand braids is already open. S. Bigelow showed in 2000 that if the Burau representation for four-strand braids is unfaithful, then there is an infinite number of non-trivial knots with the trivial two-variable HOMFLY-PT polynomial and hence, with the trivial Jones polynomial, since it is obtained from the HOMFLY-PT polynomial by the specialisation of one of the variables $A=q^2$. In this paper, we study four-strand braids and ask whether there are non-trivial knots with the trivial Jones polynomial but a non-trivial HOMFLY-PT polynomial. We have discovered that there is a whole 1-parameter family, parameterised by the writhe number, of 2-variable polynomials that can be HOMFLY-PT polynomials of some knots. We explore various properties of the obtained hypothetical HOMFLY-PT polynomials and suggest several checks to test these formulas. A generalisation is also proposed for the case of a large number of strands. |
| title | Closed 4-braids and the Jones unknot conjecture |
| topic | Geometric Topology High Energy Physics - Theory Mathematical Physics Group Theory Representation Theory |
| url | https://arxiv.org/abs/2402.02553 |