Supersignature unknotting-number bound

Let u,vu,v be parameters and let LL be a link for which the supersignature σu,v(L)\sigma_{u,v}(L) exists and is finite. Let u(L)u(L) denote the unknotting number, and let TnT_n be the trivial link with nn components. The supersignature unknotting-number conjecture.

u(L){σu,v(L)σu,v(Tn)2,uv,σu,v(L)2,u=v.u(L) \geq \begin{cases}\dfrac{|\sigma_{u,v}(L)-\sigma_{u,v}(T_n)|}{2},&u\ne v,\\[6pt]\dfrac{|\sigma_{u,v}(L)|}{2},&u=v.\end{cases}

This proposes a supersignature analogue of the classical signature bound for unknotting number. The supplied status evidence identifies the associated supersignature existence problem as open, and the text gives no resolution of this proposed bound.

Sources & referencesView supporting material

Primary source

Jozef H. Przytycki, “Survey on recent invariants on classical knot theory”, arXiv:0810.4191 (2008).

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