Nonemptiness and minima for the smooth and locally flat slice genera

By [F1], choose a connected compact smooth oriented Seifert surface $\mathrm{\Sigma }\subset S^3$ with $\mathrm{\partial }\mathrm{\Sigma }=K$. By [F3], choose $0<\epsilon <1$ and identify a collar of $S^3=\mathrm{\partial }{B}^4$ with $S^3\times [0,\epsilon )$, where $(x,u)$ maps to $(1-u)x$. Apply the collar theorem in [F3] also to $\mathrm{\Sigma }$ and write $r\ge 0$ for its inward boundary-collar coordinate. Choose a smooth cutoff $\chi (r)$ equal to $1$ near $r=0$ and $0$ before the collar ends, and a constant $c\in (0,\epsilon )$; set $h=\chi (r)r+(1-\chi (r))c$ in the collar and $h=c$ beyond it. Since $0\le r,c<\epsilon <1$, $h$ takes values in $[0,1)$; it is smooth, vanishes exactly on $\mathrm{\partial }\mathrm{\Sigma }$, and is positive on the interior.

The graph $F=\{((1-h(x))x):x\in \mathrm{\Sigma }\}\subset B^4$ is embedded because the radial collar projection to $S^3$ recovers $x$. It is compact and connected, its boundary is exactly $K$, and all its nonboundary points lie in the interior of $B^4$. Its smooth charts give the local-flat pair charts in [F4]. The graph is homeomorphic to $\mathrm{\Sigma }$, so it has genus $g(\mathrm{\Sigma })$ from [F2]. Thus this one surface contributes an element to each of ${\mathcal{S}}_{\mathrm{s}\mathrm{m}}(K)$ and ${\mathcal{S}}_{\mathrm{t}\mathrm{o}\mathrm{p}}(K)$.

Each set is therefore a nonempty subset $A\subseteq {\mathbb{N}}_0$. Choose one $n\in A$. The finite set $A\cap \{0,1,\dots ,n\}$ is nonempty, so it has a least element $m$ by the ordinary finite ordering. If $a\in A$ and $a<m$, then $a\le n$ and hence $a\in A\cap \{0,1,\dots ,n\}$, contradicting the choice of $m$. Thus $m$ is least in $A$. Applying this argument to each of the two sets gives both minima; no well-ordering principle or additional choice axiom is used.