Abstract: Given a balanced sutured manifold (M, \gamma) and a properly embedded surface S inside M, we can construct a Z (integral) grading on the sutured monopole Floer homology (SHM) of (M,\gamma). This grading enables us to compute SHM in some cases. As an example, I will explain how to compute the SHM of a solid torus with any valid sutures on its boundary.

The grading also plays a crucial role in the construction of a minus version of the monopole knot Floer homology (KHM). Given a knot K in a closed oriented 3-manifold Y, the KHM is defined to be the direct limit of a direct system introduced by Etnyre, Vela-Vick and Zarev. This direct system is built on a sequence of balanced sutured manifolds, whose underlying manifolds are all the knot complement Y(K)=Y\N(K) but having different sutures on the boundary. In the talk I will also introduce how the grading is used to prove many interesting properties of this KHM.