Relative Spin-c structures and Chern classes for sutured knot exteriors

Let $(M,\gamma )$ be a balanced sutured manifold. Fix a nowhere-zero vector field $v_0$ along $\mathrm{\partial }M$ that points into $M$ on $R_{-}(\gamma )$, points out of $M$ on $R_{+}(\gamma )$, and on each annular suture region follows the gradient of a height coordinate whose level circles are the sutures. A sutured Spin-c structure is a homology class of nowhere-zero vector fields $v$ on $M$ whose boundary restriction is $v_0$. Two such fields are homologous when, in every connected component $M_i$ of $M$, their restrictions outside some open ball in $\mathrm{int}{M}_i$ are homotopic through nowhere-zero fields fixed on $\mathrm{\partial }{M}_i$; for $M=\mathrm{\varnothing }$ this condition is vacuous. The set of these classes is denoted ${\mathrm{Spin}}^c(M,\gamma )$.

For a balanced Heegaard diagram of $(M,\gamma )$ and a generator $\mathbf{x}$, choose a Morse function compatible with the diagram. The generator specifies the multi-trajectory joining its paired index-one and index-two critical points. Modify the gradient in a regular neighborhood of that trajectory to obtain a nowhere-zero vector field agreeing with $v_0$ on $\mathrm{\partial }M$; for this chosen Morse data its homology class is denoted $\mathfrak{s}(\mathbf{x})$. The theorem that the assignment is independent of auxiliary Morse and gradient choices is part of diagram invariance, not asserted by this definition. The chain subcomplex generated by generators assigned to a fixed class $\mathfrak{s}$ has homology denoted $SFH(M,\gamma ,\mathfrak{s};R)$ for coefficients $R$ whenever the diagram-level differential is defined. For a chosen trivialization $t$ of the oriented plane field $v_0^{\perp }$ along $\mathrm{\partial }M$, the relative Chern class $c_1(\mathfrak{s},t)\in H^2(M,\mathrm{\partial }M;\mathbb{Z})$ is the relative Euler obstruction to extending $t$ over $v^{\perp }$, where $v$ represents $\mathfrak{s}$ and the plane orientation is determined by the orientation of $M$ and the direction of $v$.

For an oriented null-homologous knot $K$ in a closed connected oriented $3$-manifold $Y$, let $M=Y\setminus \mathrm{int}N(K)$ and let ${\gamma }_{\mu }$ be the union of two disjoint annuli on $\mathrm{\partial }M$ whose core sutures are parallel meridians of $K$ with opposite orientations. A Seifert surface is regarded in $M$ after deleting a collar of its boundary, so it determines a relative class on the sutured knot exterior. The meridional direction supplies the boundary trivialization of $v_0^{\perp }$ used to normalize the relative Chern-class pairing.

Juhasz, Holomorphic discs and sutured manifolds, §4, connected homology relation applied componentwise

Balanced sutured manifolds and their regions

Balanced sutured Heegaard diagrams and generators

Juhasz, Floer homology and surface decompositions, §3

Ni, Knot Floer homology detects fibred knots, §§1–2