Recognising conjugacy classes of Dehn twists on $\mathbb D_3$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910140019507200 |
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| author | Atalan, Ferihe Finashin, Sergey |
| author_facet | Atalan, Ferihe Finashin, Sergey |
| contents | We analyse the action of the basic Dehn twists on the essential curves, $γ$, in a disc with 3 marked points, $\mathbb D_3$. In particular, we interpret the induced dynamics on the Dynnikov plane in terms of the standard dynamics in homology $H_1({\rm T})=\mathbb Z^2$ of the branched covering torus with a hole, ${\rm T}\to \mathbb D_3$. Our explicit description of orbits of the action of the pure mapping class group ${\rm PMod}(\mathbb D_3)$ can be viewed as a solution of the conjugacy problem for the Dehn twists $t_γ$. We also present an ``untwisting algorithm'' for factorization of this problem into a minimal number of steps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14582 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Recognising conjugacy classes of Dehn twists on $\mathbb D_3$ Atalan, Ferihe Finashin, Sergey Geometric Topology 57K20, 20F10, 37E30, 20E45, 05C12 We analyse the action of the basic Dehn twists on the essential curves, $γ$, in a disc with 3 marked points, $\mathbb D_3$. In particular, we interpret the induced dynamics on the Dynnikov plane in terms of the standard dynamics in homology $H_1({\rm T})=\mathbb Z^2$ of the branched covering torus with a hole, ${\rm T}\to \mathbb D_3$. Our explicit description of orbits of the action of the pure mapping class group ${\rm PMod}(\mathbb D_3)$ can be viewed as a solution of the conjugacy problem for the Dehn twists $t_γ$. We also present an ``untwisting algorithm'' for factorization of this problem into a minimal number of steps. |
| title | Recognising conjugacy classes of Dehn twists on $\mathbb D_3$ |
| topic | Geometric Topology 57K20, 20F10, 37E30, 20E45, 05C12 |
| url | https://arxiv.org/abs/2603.14582 |