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We prove a number of fundamental properties about instanton knot Floerhomology. Our arguments rely on general properties of sutured Floer theoriesand apply also in the Heegaard Floer and monopole Floer settings, where many ofour results were already known. Our main result is the connected sum formulafor instanton knot Floer homology. An extension of this result proves theoriented skein exact triangle for the minus version of instanton knot Floerhomology. Finally, we derive a new model of the minus version of instanton knotFloer homology, which takes the form of a free, finitely generated chaincomplex over a polynomial ring, as opposed to a direct limit. This constructionis new to all of the Floer theories. We explore these results also in thecontext of Heegaard Floer theory as well.

Connected sums and directed systems in knot Floer homologies