Torus knot filtered embedded contact homology of the tight contact 3-sphere (Q6564510)

Embedded contact homology (ECH) is a Floer-type invariant of contact three-manifolds introduced by \textit{M. Hutchings} [J. Eur. Math. Soc. (JEMS) 4, No. 4, 313--361 (2002; Zbl 1017.58005)] that is related to Seiberg-Witten Floer homology and Heegard Floer homology. The paper under review is concerned with the knot filtered ECH introduced by \textit{M. Hutchings} [J. Mod. Dyn. 10, 511--539 (2016; Zbl 1402.37066)]. The input to define knot filtered ECH is a fixed embedded Reeb orbit realizing a transverse knot.\N\NThe paper under review computes knot filtered ECH for the right handed torus knot \(T(2,q)\) for \(q \geq 1\) odd, denoted by \(b_0\). Namely, given \(p,q\in \mathbb R\), let \(N(p,q) = (pm+qn)_{m,n\in \mathbb Z_{\geq 0}}\) be written in increasing order with multiplicity, the \(k\)th element of which is denoted by \(N_k(p,q)\). Then\N\[\NECH_{\ast}^{\mathcal F_b \leq K}(S^3,\xi_{\text{std}},b_0,2q) = \begin{cases} \mathbb Z/2 & K \geq N_k(2,q) \text{ and } \ast = 2k, \\\N0 & \text{otherwise.} \end{cases}\N\]\NComputation of the knot filtration relies on computations of the ECH index and the action of every homologically essential generator. This is done via open book decompositions. The computation is complicated (and more subtle compared to earlier computations for the unknot and Hopf link) by the fact that some fibers project to orbifold points, making the use of the constant trivialization for those fibers impossible.