Existence of the augmentation polynomial for every knot

Let KK be a knot. The augmentation variety VKV_K is the set of augmentation values in (C)3(\mathbb{C}^*)^3, and the augmentation polynomial is defined when the maximal-dimensional part of the Zariski closure of VKV_K has codimension 11. Existence conjecture. The condition about this Zariski closure holds for every knot KK, so the augmentation polynomial is always defined. This asserts universal existence of the three-variable augmentation polynomial, beyond the cases where it has been explicitly computed.

Sources & referencesView supporting material

Primary source

Lenhard Ng, “A topological introduction to knot contact homology”, arXiv:1210.4803 (2014).

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