A ribbon concordance from a knot \(K_0\) to a knot \(K_1\) is a smooth annulus \(C\subset [0,1]\times S^3\) with \(\partial C \cap \{0\} \times S^3 = -K_0\) and \(\partial C \cap \{1\} \times S^3 = K_1\), such that the projection \([0,1]\times S^3\to[0,1]\) restricts to a Morse function on \(C\) without critical points of Morse index 2. Let us write \(K_0 \leq K_1\) if such a \(C\) exists.
Note that a knot \(K\) is ribbon if and only if there is a ribbon concordance from the unknot to \(K\). Ribbon concordances were studied by C. McA. Gordon [Math. Ann. 257, 157–170 (1981; Zbl 0451.57001)], who conjectured that the relation \(\leq\) is a partial order, and found an obstruction coming from the knot group. Another previously known obstruction comes from the Blanchfield pairing/Seifert form [P. M. Gilmer, Topology Appl. 18, 313–324 (1984; Zbl 0568.57005)].
The main theorem of this paper states that a ribbon concordance \(C\) from \(K_0\) to \(K_1\) induces an injection \[ F_C\colon \widehat{HFK}(K_0) \to \widehat{HFK}(K_1) \] of knot Floer homologies. This provides a strong obstruction for \(K_0 \leq K_1\). Moreover, since knot Floer homology detects the three-genus \(g_3\), it follows as a corollary that for all knots \(K_0, K_1\) \[ K_0 \leq K_1 \quad\Longrightarrow\quad g_3(K_0) \leq g_3(K_1) \] and for \(J\) a band connected sum of knots \(J_1, \ldots, J_n\), \[ g_3(J_1) + \cdots + g_3(J_n) \leq g_3(J). \] Both of these statements were previously only known in certain special cases, e.g. if \(K_1\) is fibered in the first statement, and \(J\) is fibered or \(n = 2\) in the second statement.
To prove the main theorem, the author considers the composition \(\bar{C}C\) of \(C\) with \(\bar{C}\), the concordance from \(K_1\) to \(K_0\) obtained by turning \(C\) upside down. Although \(\bar{C}C\) is in general not isotopic to the identity concordance of \(K_0\), it does induce the identity on \(\widehat{HFK}(K_0)\), and so \(F_C\) has \(F_{\bar{C}}\) as a left inverse.
This paper has already inspired a number of similar results: injectivity also follows for Khovanov homology [A. S. Levine and I. Zemke, “Khovanov homology and ribbon concordance”, Preprint, arXiv:1903.01546], also for Khovanov-Rozansky and similar homologies [S. Kang, “Link homology theories and ribbon concordances”, Preprint, arXiv:1909.06969], and it follows for knot Floer homology under a weakened hypothesis [M. Miller and I. Zemke, “Knot Floer homology and strongly homotopy-ribbon concordances”, Preprint, arXiv:1903.05772].