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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.10561 |
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Table of Contents:
- Gukov, Pei, Putrov, and Vafa developed a $q$-series invariant of negative definite plumbed $3$-manifolds with spin$^{c}$ structures, building on earlier work of Lawrence and Zagier. This was recently generalized to an an infinite family of two-variable $(t,q)$-series invariants by Akhmechet, Johnson, and Krushkal (AJK). We calculate one such series for all Seifert manifolds with $b_{1}=0.$ These results extend a previous theorem of Liles and McSpirit to any number of exceptional fibers and the Reduction Theorem of Gukov, Svoboda, and Katzarkov to the two-variable case. As a consequence, a previous result of Liles and McSpirit on modularity properties and radial limits is enhanced to a larger family of manifolds. We also calculate the infinite collection of $(t,q)$-series invariants for three infinite families of manifolds, finding mixed modularity properties for one such family.