Jablan–Sazdanović conjecture on unknotting numbers of positive odd pretzel knots

Let P(a1,,an)P(a_1,\dots,a_n) be the pretzel knot with nn odd and positive odd integers a1,,ana_1,\dots,a_n, ordered so that a1a2ana_1\leq a_2\leq\dots\leq a_n. Jablan–Sazdanović conjecture.

u(P(a1,,an))=i=1n1ai2.u(P(a_1,\dots,a_n))=\frac{\sum_{i=1}^{n-1}a_i}{2}.

This conjecture would determine the unknotting number for all pretzel knots with an odd number of positive odd parameters. The paper proves special cases, including families with parameters (3,1,,1,b)(3,1,\dots,1,b) and (3a,3b,3c)(3a,3b,3c), but the general assertion remains open.

Sources & referencesView supporting material

Primary source

Seph Shewell Brockway, “Computing the unknotting numbers of certain pretzel knots”, arXiv:1312.4502 (2013).

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