Cokernel polynomial conjecture for quasi-alternating links

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Let κ=κβ\kappa=\kappa_\beta be the closure of a braid β∈Brm\beta\in Br_m with writhe ww, and let Pcoker(t)P_{\mathrm{coker}}(t) denote the associated cokernel polynomial. Cokernel polynomial conjecture. The polynomial

(−1)m+wt(m+w)/2Pcoker(t)(-1)^{m+w}t^{(m+w)/2}P_{\mathrm{coker}}(t)

is an oriented link invariant. For quasi-alternating links, it coincides with the normalised Jones polynomial:

Vκ(s)=(−1)m+wt(m+w)/2Pcoker(t)t1/2+t−1/2∣s=t−1.V_\kappa(s)=\left.\frac{(-1)^{m+w}t^{(m+w)/2}P_{\mathrm{coker}}(t)}{t^{1/2}+t^{-1/2}}\right|_{s=t^{-1}}.

This proposes a refined invariant arising from the localization map; the source does not provide a resolution status.

References

Primary source

Paul Seidel and Ivan Smith, “Localization for involutions in Floer cohomology”, arXiv:1002.2648 (2010).

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