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arXiv · 2304.03761

The homotopy cardinality of the representation category for a Legendrian knot

Justin Murray
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Abstract

Given a Legendrian knot in $(\mathbb{R}^3, \ker(dz-ydx))$ one can assign a combinatorial invariants called ruling polynomials. These invariants have been shown to recover not only a (normalized) count of augmentations but are also closely related to a categorical count of augmentations in the form of the homotopy cardinality of the augmentation category. In this article, we prove that that the homotopy cardinality of the $n$-dimensional representation category is a multiple of the $n$-colored ruling polynomial. Along the way, we establish that two $n$-dimensional representations are equivalent in the representation category if they are conjugate DGA homotopic. We also provide some applications to Lagrangian concordance.

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Justin Murray. 2023-04-07. The homotopy cardinality of the representation category for a Legendrian knot. https://arxiv.org/abs/2304.03761

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