Ribbon concordance

Let $F_0$ and $F_1$ be knotted surfaces of the same genus. We say that $F_1$ is ribbon concordant to $F_0$ if there is a concordance $C$ (a properly embedded orientable submanifold diffeomorphic to $F_0\times I$) in ${\mathbb{R}}^4\times [0,1]$ between $F_1\subset{\mathbb{R}}^4\times\{1\}$ and $F_0\subset{\mathbb{R}}^4\times\{0\}$ such that the restriction to $C$ of the projection ${\mathbb{R}}^4\times[0,1]\rightarrow[0,1]$ is a Morse function with critical points of index 0 and $1$ only. We write $F_1\geq F_0$.

Figure: Ribbon concordance

Note that if $F_1\geq F_0$, then there is a set of $n$ $1$-handles on a split union of $F_0$ and $n$ trivial sphere-knots, for some $n\geq 0$, such that $F_1$ is obtained by surgeries ($1$-handle additions) along these handles (Fig. ). Ribbon concordance was first defined in [Gor81]. It is defined in general for knots in any dimension. In [CSS03*], quandle cocycle invariants were used as obstructions to ribbon concordance for surfaces, and explicit examples of surfaces that are not related by ribbon concordance were given.

This obstruction is descrived as follows. For two multi-sets $A'$ and $A''$ of $A$, we use the notation $A'\stackrel{{\rm m}}{\subset}A''$ if for any $a\in A'$ it holds that $a\in A''$. In other words, $A'\stackrel{{\rm m}}{\subset}A''$ if and only if $\tilde{A'}\subset \tilde{A''}$ where $\tilde{A'}$ and $\tilde{A''}$ are the subsets of $A$ obtained from $A'$ and $A''$ by eliminating the multiplicity of elements, respectively.

If $F_1\geq F_0$, then with respect to a natural diagram of $F_1$ described as above, that is obtained from $F_0$ by adding small spheres and this tubes, any coloring of such a diagram of $F_1$ restricts to a coloring of a diagram of $F_0$, and since adding tubes do not introduce triple points, we obtain the condition $\Phi_{\theta}(F_1)\stackrel{{\rm m}}{\subset}\Phi_{\theta}(F_0)$.