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Inverted state sums, inverted Habiro series, and indefinite theta functions
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abstract
S. Gukov and C. Manolescu conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be re-summed into a two-variable series $F_K(x,q)$, which is the knot complement version of the 3-manifold invariant $\hat{Z}$ whose existence was predicted earlier by S. Gukov, P. Putrov and C. Vafa. In this paper we use an inverted version of the R-matrix state sum to prove this conjecture for a big class of links that includes all homogeneous braid links as well as all fibered knots up to 10 crossings. We also study an inverted version of Habiro's cyclotomic series that leads to a new perspective on $F_K$ and discovery of some regularized surgery formulas relating $F_K$ with $\hat{Z}$. These regularized surgery formulas are then used to deduce expressions of $\hat{Z}$ for some plumbed 3-manifolds in terms of indefinite theta functions.
Forward citations
Cited by 5 Pith papers
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3d-3d correspondence for knot complements with finite and large $N$
The SU(N) homological block in inverted-Habiro form equals a half-index of an explicit 3d N=2 theory for the figure-eight and the two trefoil knots, with a conjectural all-knot extension.
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3d-3d correspondence and abelian flat connection
The homological block of a knot complement is realized as a half-index of a 3d N=2 theory via a contour enclosing z=q^k poles, and the same integral at z=q^-k poles gives the colored Jones polynomial.
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Causal evidence for the primordiality of colours in trans-Neptunian objects
A causal-inference claim that TNO colours are primordial, asserted in the abstract, is unsupported because the submitted full text is an unrelated paper on ceff and bZ invariants.
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Quantum invariants of 3-manifolds and links: a review
This is a survey, not a new result: it reviews the q-series invariants Zhat, F_K, F_L and their supergroup analogues, collecting known conjectures, theorems, and examples.
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Knot-quiver correspondence: a brief review
A survey of the known knot-quiver correspondence, including quiver equivalences, diagonalization, and an extension to knot complements.
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