The AJ conjecture and connected sums of torus knots
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866915851817451520 |
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| author | Zhang, Xingru |
| author_facet | Zhang, Xingru |
| contents | The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $|p|>q\geq 2$. We verify the AJ conjecture for the connected sums $T(p,q)\# T(a,b)$ when $p$ and $a$ have the same sign. Notably, in cases where $pq=ab$ but $p\ne a$, the recurrence polynomial $α(t,M,L)$ of $T(p,q)\#T(a,b)$ has repeated factors involving the variable $L$ after evaluation at $t=-1$. These appear to be the first examples of knots exhibiting this phenomenon. Therefore, the AJ conjecture requires a slight modification to accommodate this possibility. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10235 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The AJ conjecture and connected sums of torus knots Zhang, Xingru Geometric Topology The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $|p|>q\geq 2$. We verify the AJ conjecture for the connected sums $T(p,q)\# T(a,b)$ when $p$ and $a$ have the same sign. Notably, in cases where $pq=ab$ but $p\ne a$, the recurrence polynomial $α(t,M,L)$ of $T(p,q)\#T(a,b)$ has repeated factors involving the variable $L$ after evaluation at $t=-1$. These appear to be the first examples of knots exhibiting this phenomenon. Therefore, the AJ conjecture requires a slight modification to accommodate this possibility. |
| title | The AJ conjecture and connected sums of torus knots |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2603.10235 |