Homotopy classification of knotted defects in ordered media
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911292839690240 |
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| author | Nozaki, Yuta Kálmán, Tamás Teragaito, Masakazu Koda, Yuya |
| author_facet | Nozaki, Yuta Kálmán, Tamás Teragaito, Masakazu Koda, Yuya |
| contents | We give a homotopy classification of the global defects in ordered media, and explain it via the example of biaxial nematic liquid crystals, i.e., systems where the order parameter space is the quotient of the $3$-sphere $S^3$ by the quaternion group $Q$. As our mathematical model we consider continuous maps from complements of spatial graphs to the space $S^3/Q$ modulo a certain equivalence relation, and find that the equivalence classes are enumerated by the six subgroups of $Q$. Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of $Q$; once we pass to planar diagrams, these labels can be refined to elements of $Q$ associated to each arc. The same classification scheme applies not only in the case of $Q$ but also to arbitrary groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homotopy classification of knotted defects in ordered media Nozaki, Yuta Kálmán, Tamás Teragaito, Masakazu Koda, Yuya Soft Condensed Matter Geometric Topology 57Z05 (Primary) 57K10, 76A15 (Secondary) We give a homotopy classification of the global defects in ordered media, and explain it via the example of biaxial nematic liquid crystals, i.e., systems where the order parameter space is the quotient of the $3$-sphere $S^3$ by the quaternion group $Q$. As our mathematical model we consider continuous maps from complements of spatial graphs to the space $S^3/Q$ modulo a certain equivalence relation, and find that the equivalence classes are enumerated by the six subgroups of $Q$. Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of $Q$; once we pass to planar diagrams, these labels can be refined to elements of $Q$ associated to each arc. The same classification scheme applies not only in the case of $Q$ but also to arbitrary groups. |
| title | Homotopy classification of knotted defects in ordered media |
| topic | Soft Condensed Matter Geometric Topology 57Z05 (Primary) 57K10, 76A15 (Secondary) |
| url | https://arxiv.org/abs/2402.16079 |