The colored Jones polynomial and Kontsevich-Zagier series for double twist knots
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2017
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| _version_ | 1866910534114213888 |
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| author | Lovejoy, Jeremy Osburn, Robert |
| author_facet | Lovejoy, Jeremy Osburn, Robert |
| contents | Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots $K_{(-m,-p)}$ and $K_{(-m,p)}$ where $m$ and $p$ are positive integers. In the $(-m,-p)$ case, this leads to new families of $q$-hypergeometric series generalizing the Kontsevich-Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of $K_{(m,p)}$ gives a generalization of a duality at roots of unity between the Kontsevich-Zagier function and the generating function for strongly unimodal sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1710_04865 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | The colored Jones polynomial and Kontsevich-Zagier series for double twist knots Lovejoy, Jeremy Osburn, Robert Geometric Topology Combinatorics Number Theory 57M27 Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots $K_{(-m,-p)}$ and $K_{(-m,p)}$ where $m$ and $p$ are positive integers. In the $(-m,-p)$ case, this leads to new families of $q$-hypergeometric series generalizing the Kontsevich-Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of $K_{(m,p)}$ gives a generalization of a duality at roots of unity between the Kontsevich-Zagier function and the generating function for strongly unimodal sequences. |
| title | The colored Jones polynomial and Kontsevich-Zagier series for double twist knots |
| topic | Geometric Topology Combinatorics Number Theory 57M27 |
| url | https://arxiv.org/abs/1710.04865 |