On $q$-deformed Farey sum and a homological interpretation of $q$-deformed real quadratic irrational numbers
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909239797088256 |
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| author | Ren, Xin |
| author_facet | Ren, Xin |
| contents | The left and right $q$-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and they gave a homological interpretation for left and right $q$-deformed rational numbers. In the present paper, we focus on negative continued fractions and defined left $q$-deformed negative continued fractions. We give a formula for computing the $q$-deformed Farey sum of the left $q$-deformed rational numbers based on it. We use this formula to give a combinatorial proof of the relationship between the left $q$-deformed rational number and the Jones polynomial of the corresponding rational knot which was proved by Bapat, Becker and Licata using a homological technique. Finally, we combine their work and the $q$-deformed Farey sum, and give a homological interpretation of the $q$-deformed Farey sum. We also give an approach to finding a relationship between real quadratic irrational numbers and homological algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_06056 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On $q$-deformed Farey sum and a homological interpretation of $q$-deformed real quadratic irrational numbers Ren, Xin Representation Theory Combinatorics Number Theory 11A55, 05A30, 57K14, 18G80 The left and right $q$-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and they gave a homological interpretation for left and right $q$-deformed rational numbers. In the present paper, we focus on negative continued fractions and defined left $q$-deformed negative continued fractions. We give a formula for computing the $q$-deformed Farey sum of the left $q$-deformed rational numbers based on it. We use this formula to give a combinatorial proof of the relationship between the left $q$-deformed rational number and the Jones polynomial of the corresponding rational knot which was proved by Bapat, Becker and Licata using a homological technique. Finally, we combine their work and the $q$-deformed Farey sum, and give a homological interpretation of the $q$-deformed Farey sum. We also give an approach to finding a relationship between real quadratic irrational numbers and homological algebra. |
| title | On $q$-deformed Farey sum and a homological interpretation of $q$-deformed real quadratic irrational numbers |
| topic | Representation Theory Combinatorics Number Theory 11A55, 05A30, 57K14, 18G80 |
| url | https://arxiv.org/abs/2210.06056 |