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Perverse sheaves and knot contact homology. (Faisceaux pervers et homologie de contact des noeuds.) (English. French summary) Zbl 1367.57007

As a category theoretical reformulation of the knot differential graded (DG) algebra \(\mathcal{A}_L\) for a link \(L\) in \(\mathbb{R}^3\) [L. Ng, Geom. Topol. 9, 247–297 (2005; Zbl 1111.57011); ibid. 9, 1603–1637 (2005; Zbl 1112.57001)], the knot DG category \(\tilde{\mathcal{A}}_L\) is defined and gives a new algebraic construction of knot contact homology [loc. cit.], which provides the full noncommutative knot DGA [T. Ekholm et al., Geom. Topol. 17, No. 2, 975–1112 (2013; Zbl 1267.53095)].
The outline of the paper is as follows: In Section 2, consider a category \(\mathcal{C}\) closed under finite colimits. Then it is stated that for any Reidemeister operator \(\sigma\), the isomorphism type of the normalized categorical braid closure (Definitions 2.16 and 2.5) is invariant under Markov moves, and hence gives a knot invariant (Theorem 2.17). Here, \(\sigma: A\sqcup A\to A\sqcup A\), with \(A\) an object of \(\mathcal{C}\), is said to be Reidemeister, if it is co-Cartesian Yang-Baxter (Definition 2.1), dualizable (Definition 2.11), and Wada (Definition 2.9, cf. [M. Wada, Topology 31, No. 2, 399–406 (1992; Zbl 0758.57008)]), with invertible torsion \(\chi(\sigma)\) (Definition 2.15). In Section 3, the construction of a categorical braid closure is extended to the homological setting and states the following. Let \(\sigma: A\sqcup A\to A\sqcup A\) be a Reidemeister operator on a pseudoflat object \(A\), and let \(\theta:A\to A\) be a \(\sigma\)-natural map (Definition 3.1). Then the isomorphism type of \(\mathrm{Ho}(\mathcal{C})\) (weak homotopy equivalence type in \(\mathcal{C}\)) of the \(\theta\)-colored normalized homotopy braid closure (Definition 3.2) is invariant under Markov moves, and hence gives a knot invariant (Theorem 3.4). Two examples are also given which illustrate how Theorem 3.4 allows to refine classical link invariants defined by categorical braid closure.
The input of the above construction is a natural action of the braid group \(B_n\) on the category of perverse sheaves on a two-dimensional disk with singularities at \(n\) marked points [S. Gelfand et al., Duke Math. J. 83, No. 3, 621–643 (1996; Zbl 0861.32022)]. This argument is formulated with a GMV operator in Theorem 4.1. The background ideas to this input are explained in the introduction. A slight modification of the original GMV action, which is used in Sections 5 and 6, is also explained. This section is concluded with a diagram which illustrates the relation between the knot DG category and the knot DG algebra. Then, the knot DG category of a knot \(K\) is defined in Definition 5.3. Adopting the Baues-Lemaire cylinder on a semi-free DG category [H. J. Baues and J. M. Lemaire, Math. Ann. 225, 219–245 (1977; Zbl 0322.55019)], the independence of the quasi-isomorphism type of the knot DG category of the choice of the knot DGA construction is proved (Theorem 5.5). By this theorem, an alternative proof is given of the fact that the underlying quasi-isomorphism type of the combinatorial knot DGA is a knot invariant (Theorem 5.6 and [R. Anno and T. Logvinenko, “Spherical DG functors”, Preprint, arXiv:1309.5035]). In Section 6, the knot \(k\)-category, the 0th homology of the knot DG category, is computed and a description of this category is given in terms of the knot group together with a peripheral pair (Theorem 6.2). In Section 7, the fully noncommutative link DG category is defined (Definition 7.1), and the main results of Section 5 and Section 6 are extended to this case (Theorems 7.4 and 7.7). The input for the fully noncommutative case is the original GMV action. This allows (Theorem 7.10) to relate the corresponding module category to perverse sheaves on \(\mathbb{R}^3\) that are singular along the link \(L\). In Section 8, the last section, two generalizations of the GMV operator are presented. They satisfy the Reidemeister condition, and hence the corresponding homotopy braid closure gives link invariants generalizing the link DG category associated with the original GMV action.
The definitions of this paper are clearly stated while most of the theorems lack proofs. The authors say that detailed proofs will appear in their forthcoming paper [“Homotopy braid closure and link invariants”, in preparation].

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
53D42 Symplectic field theory; contact homology
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
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References:

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