Gukov–Stošic conjecture on symmetric and antisymmetric coloured HOMFLY homology

Let KK be a knot, and let Hi,j,Λr(K)H^{\Lambda^r}_{i,j,*}(K) and Hi,j,Sr(K)H^{S^r}_{i,j,*}(K) denote, respectively, its antisymmetric- and symmetric-coloured HOMFLY homology groups, with indices ii, jj, and *. Gukov–Stošic conjecture. There is an isomorphism

Hi,j,Λr(K)Hi,j,Sr(K).H^{\Lambda^r}_{i,j,*}(K)\cong H^{S^r}_{i,-j,*}(K).

This conjecture predicts a relationship between knot homologies coloured by exterior and symmetric powers. The supplied text describes it as conjectural and gives no evidence of a resolution.

Sources & referencesView supporting material

Primary source

Jonathan Grant, “A categorification of the skew Howe action on a representation category of U_q(gl(m|n))”, arXiv:1504.07671 (2015).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.