Stabilization distance bounds from link Floer homology (Q6564517)

Let \(S\) be a connected and properly embedded surface in a 4-manifold \(W\) such that there is an embedded 4-ball \(B^4 \subset \mathrm{int}(W)\) satisfying that \(\partial B^4 \cap S\) is an unlink of \(m\) components \(U_1 \cup \cdots \cup U_m\) and \(S \cap B^4\) is a collection of \(m\) pairwise disjoint and properly embedded disks \(D_1 \cup \cdots \cup D_m\) that can simultaneously be smoothly isotoped into \(\partial B^4\) relative to their boundaries. Let \(S_0 \subset B^4\) be an oriented, connected, properly embedded surface such that \(\partial S_0=U_1 \cup \cdots \cup U_m\). The \((m,n)\)-stabilization of \(S\) along \((B^4, S_0)\) is the surface \(S'=(S \backslash \mathrm{int}(B^4)) \cup S_0\). We say that \(S\) is the \((m,n)\)-destabilization of \(S'\) if \(S'\) is the \((m,n)\)-stabilization of \(S\). The \((m,n)\)-stabilization is a generalization of a 1-handle stabilization. \N\NIn this paper, the authors construct invariants that provide lower bounds on the stabilization distance. Let \(K\) be a knot in \(S^3\). Let \(\mathrm{Surf}(K)\) be the set of isotopy classes of connected, properly embedded surfaces in \(B^4\) with boundary \(K\). The stabilization distance \(\mu_{\mathrm{st}}(S, S')\) of \(S, S' \in \mathrm{Surf}(K)\) is the minimum of \(\max \{g(S_1), \ldots, g(S_k)\}\) over sequences \(S_1, \ldots, S_k \in \mathrm{Surf}(K)\) such that \(S_1=S\), \(S_k=S'\) and \(S_{j-1}\) and \(S_j\) are related by a stabilization or a destabilization for each \(j\), where \(g(S_j)\) is the genus. For surfaces \(S\) and \(S'\), the stabilization distance between \(S\) and \(S'\) gives a lower bound of the number of 1-handle stabilizations required to make \(S\) and \(S'\) isotopic. In addition to the stabilization distance \(\mu_{\mathrm{st}}(S, S')\), the authors treat the double-point distance \(\mu_{\mathrm{Sing}}(S, S')\) and the cobordism distance \(\mu_{\mathrm{Cob}}(S, S')\). The double-point distance \(\mu_{\mathrm{Sing}}(S, S')\) is defined for \(S\) and \(S'\) with the same genus, using the maximal number of double points that appear during a regular homotopy from \(S\) to \(S'\). The cobordism distance \(\mu_{\mathrm{Cob}}(S, S')\) is defined for two slice surfaces \(S\) and \(S'\) of a knot \(K \subset S^3\), using the sum of the genera of the components of regular level sets of the cobordism between \(S\) and \(S'\). Using the link Floer TQFT, the authors give several invariants of pairs of surfaces \(S, S' \in \mathrm{Surf}(K)\), that give lower bounds on \(\mu_{\mathrm{st}}(S, S')\), \(\mu_{\mathrm{Cob}}(S, S')\) and \(\mu_{\mathrm{Sing}}(S, S')\). The authors compute these invariants for some examples. The authors show that for given \(n \geq 0\), there exists a knot \(K_n\) and a pair of slice disks \(D_1\) and \(D_2\) for \(K_n\) such that \(\mu_{\mathrm{st}}(D_1, D_2) \geq n\), \(\mu_{\mathrm{Sing}}(D_1, D_2) \geq n\) and \(\mu_{\mathrm{Cob}}(D_1, D_2) \geq n\).