Links in orthoplicial Apollonian packings
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929370264764416 |
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| author | Alfonsín, Jorge L. Ramírez Rasskin, Iván |
| author_facet | Alfonsín, Jorge L. Ramírez Rasskin, Iván |
| contents | In this paper, we establish a connection between Apollonian packings and knot theory. We introduce new representations of links realized in the tangency graph of the regular crystallographic sphere packings. Particularly, we prove that any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing. We use these representations to improve the upper bound on the ball number of an infinite family of alternating algebraic links. Furthermore, the later allow us to reinterpret the correspondence of rational tangles and rational numbers and to reveal geometrically primitive solutions for the Diophantine equation $x^4 + y^4 + z^4 = 2t^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_03089 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Links in orthoplicial Apollonian packings Alfonsín, Jorge L. Ramírez Rasskin, Iván Geometric Topology Combinatorics 52C26, 57K10, 11D72 In this paper, we establish a connection between Apollonian packings and knot theory. We introduce new representations of links realized in the tangency graph of the regular crystallographic sphere packings. Particularly, we prove that any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing. We use these representations to improve the upper bound on the ball number of an infinite family of alternating algebraic links. Furthermore, the later allow us to reinterpret the correspondence of rational tangles and rational numbers and to reveal geometrically primitive solutions for the Diophantine equation $x^4 + y^4 + z^4 = 2t^2$. |
| title | Links in orthoplicial Apollonian packings |
| topic | Geometric Topology Combinatorics 52C26, 57K10, 11D72 |
| url | https://arxiv.org/abs/2301.03089 |