Higher-rank surgery formula for Z^G\hat{Z}^G

From papers

Let KS3K\subset S^3 be a knot, let GG be a root system with positive roots Δ+\Delta^+, and let Lp/r(b)\mathcal{L}_{p/r}^{(b)} denote the surgery transform appearing in the formula. Higher-rank surgery formula. Whenever the right-hand side makes sense,

Z^bG(Sp/r3(K))Lp/r(b)[αΔ+(xα2rxα2r)FKG(x,q)].\hat{Z}_b^G\big(S_{p/r}^3(K)\big)\cong\mathcal{L}_{p/r}^{(b)}\left[\prod_{\alpha\in\Delta^+}\left(x^{\frac{\alpha}{2r}}-x^{-\frac{\alpha}{2r}}\right)F_K^G(\mathbf{x},q)\right].

The formula is stated as a higher-rank analogue of the SU(2){\rm SU}(2) surgery formula. It is proved for knots and 3-manifolds represented by negative-definite plumbings, while its general scope is left conjectural.

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Sources & referencesView supporting material

Primary source

Sunghyuk Park, “Higher rank Z and F_K”, arXiv:1909.13002 (2020).

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