Satellites and telescopes: a concordance formula for bordered Floer homology

Abstract

Given a knot $K$ in $S^3$, its knot Floer completely determines the bordered Floer homology of its complement by a work of Lipshitz--Ozsv\'{a}th--Thurston. Furthermore, the determination is combinatorial: given a model for $CFK(S^3, K)$ there is a method for producing an explicit model for $\widehat{CFD}(S^3 \smallsetminus \nu(K))$. In this paper, we show that a similar formula holds between certain classes of chain endomorphisms of knot Floer chain complex and type D endomorphisms of bordered Floer homology, up to a 1-dimensional ambiguity in the type D side; both the formula and the ambiguity can be computed combinatorially. It follows that, for any concordance from a knot to itself and any satellite pattern, we can combinatorially compute the knot Floer cobordism map of the satellite concordance (up to conjugation) from the knot Floer cobordism map of the given concordance and the bordered Floer homology of the pattern complement.