ScalingStacks

Example 7.2.13. In the table below, the first row depicts some chord diagrams. The second and third rows show the corresponding sutured surfaces with the $S^{+}$ part of the boundary in green and with the arcs $\omega_{z}$ in red; the second row applies the above construction directly, and the third row gives an alternate perspective. The fourth row shows the sutured surfaces as open-closed cobordisms (with empty source and with target colored in green); this interpretation is discussed in §7.2.5.

Under the strands algebra construction of §8.1, the first and second columns give rise to simple 2-representations of $\mathcal{U}$, categorifying the vector representation and its dual.

Tensor powers of the algebra of the first column give algebras very similar to the one considered by Tian [Ti]; in fact, Tian’s algebras were an important early clue in the development of the present work. Tensor powers of the algebra of the second column are studied from the Heegaard Floer perspective by the first-named author in [Man].

The algebra of the third column is the $n=3$ case of a family of algebras considered in [ManMarWi, LePo]. For general $n$, these are isomorphic to the algebras ${\mathcal{B}}(n)=\oplus_{k=0}^{n}{\mathcal{B}}(n,k)$ used by Ozsváth and Szabó in their theory of bordered knot Floer homology [OsSz4, OsSz5, OsSz6] (their notation is slightly different). The middle summand of the algebra of the fourth column is the undeformed version of a curved $A_{\infty}$-algebra used by Lipshitz-Ozsváth-Thurston [LiOzTh2, LiOzTh3] to define bordered $HF^{-}$ for $3$-manifolds with torus boundary. The middle summand of the algebra of the fifth column is the well-known “torus algebra” from bordered Floer homology. The fifth and sixth columns together illustrate our perspective on cornered Floer homology; following Zarev’s ideas, we view the cornered Floer gluing theorem as recovering the algebra of two matched intervals glued end-to-end, rather than as the invariants of two matched intervals with distinguished endpoints being glued to form a pointed matched circle.

The first, fifth, and sixth columns give algebras that are among Zarev’s strands algebras $\mathcal{A}(\mathcal{Z})$, although the first diagram is degenerate (equivalently, its sutured surface has closed circles in $S^{-}$). The second, third, and fourth columns do not satisfy the restrictions that Zarev imposes. As far as we are aware, our strands categories below give the first detailed description of strands algebras associated to general chord diagrams with circles as well as intervals; less formal descriptions have appeared previously, cf. [Au2, Proposition 11]. As indicated by Lipshitz-Ozsváth-Thurston’s work [LiOzTh2, LiOzTh3], curved $A_{\infty}$-deformations of the algebras appear necessary in the general setting when defining modules and bimodules for 3-manifolds with boundary, although in special cases like Ozsváth-Szabó’s bordered knot Floer homology (third column) this complication should be avoidable.