Knot contact homology is a modern topological link invariant. Given a link \(L\subset \mathbb R^3\), its conormal lift \(\Lambda_{L}\) is a Legendrian submanifold in the unit cotangent bundle \(ST^{\ast}(\mathbb R^3)\) with contact structure equal to the kernel of the Liouville form. It is defined as the Legendrian contact homology of the differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization of \(ST^{\ast}(\mathbb R^3)\) with Lagrangian boundary condition \(\mathbb R\times \Lambda_{L}\). It also admits a combinatorial interpretation.
In the paper under review, the authors prove that knot contact homology detects each of the torus knots as well as being a cable or composite. More precisely, the authors use the fully non-commutative degree zero knot contact homology with \(U=1\), denoted by \(\widetilde{HC_0}\), for these purposes and prove the following:
(1) Given relatively prime integers \(p\) and \(q\), an oriented knot \(K\subset \mathbb R^3\) and the \((p,q)\)-torus knot \(T_{p,q}\), if \(\widetilde{HC_0}(K)\simeq \widetilde{HC_0}(T_{p,q})\), then \(K\) is isotopic to \(T_{p,q}\).
(2) If \(\Lambda_K\) is Legendrian isotopic to \(\Lambda_{T_{p,q}}\), then \(K\) is smoothly isotopic to \(T_{p,q}\) or its mirror. In addition, if the Legendrain isotopy sends meridian to meridian and longitude to longitude, then \(K\) is isotopic to \(T_{p,q}\).
(3) Given an oriented knot \(K\subset \mathbb R^3\) and the \((p,q)\)-cable of a knot \(J\) denoted by \(C_{p,q}(J)\), if \(\widetilde{HC_0}(K)\simeq \widetilde{HC_0}(C_{p,q}(J))\), then \(K\) is isotopic to a \((p',q')\)-cable of a knot \(J'\) where \(pq=p'q'\). On the other hand, if \(\widetilde{HC_0}(K)\simeq \widetilde{HC_0}(J)\) and \(J\) is composite, then so is \(K\).
The key component of the proof is a theorem of Cielebak, Ekholm, Latschev, and Ng which relates \(\widetilde{HC_0}\) to the knot group, see [K. Cieliebak et al., J. Éc. Polytech., Math. 4, 661–780 (2017; Zbl 1380.53101)].