Specialization formula for the adjoint HOMFLY invariant

About 22 years old · traced to

Let KK be a 00-framed knot, let U0U_0 denote the unknot, and let Had⁡H_{\operatorname{ad}} be the adjoint HOMFLY invariant. Let ff be the ring morphism used in the paper, and let V2(K)V_2(K) be the standardly normalized type 22 Vassiliev invariant. Set σ+=(q12+q22+q32)\sigma_+=(q_1^2+q_2^2+q_3^2) and σ−=(q1−2+q2−2+q3−2)\sigma_-=(q_1^{-2}+q_2^{-2}+q_3^{-2}).

Specialization conjecture.

f(Had⁡(K))f(Had⁡(U0))−1(v−1v)2∣v=1=−2V2(K)−1z2Q~(K)σ+−σ−∣σ+=σ−=z2+3.\left.\frac{\frac{f(H_{\operatorname{ad}}(K))}{f(H_{\operatorname{ad}}(U_0))}-1}{\left(v-\frac1v\right)^2}\right|_{v=1}=-2V_2(K)-\frac1{z^2}\left.\frac{\widetilde Q(K)}{\sigma_+-\sigma_-}\right|_{\sigma_+=\sigma_-=z^2+3}.

The source states that this conjecture follows from work of N. Geer, so it is presented as resolved rather than open.

References

Primary source

Bertrand Patureau-Mirand, “Quantum link invariant from the Lie superalgebra D(2,1,alpha)”, arXiv:math/0404548 (2009).

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