Euler characteristic and genus of a Seifert disk-and-band surface

Given: ${\mathrm{A}\mathrm{C}}_{\omega }$, a connected disk-and-band surface made from $s$ disks and $c$ bands with one boundary component; for the final clause, $F$ is the output for an oriented knot diagram $D$ representing $K$.

The finite-cell Euler characteristic is the alternating sum of the numbers of cells. (Euler characteristic of a finite CW complex)

Under ${\mathrm{A}\mathrm{C}}_{\omega }$, a connected compact oriented surface with $b>0$ boundary circles has $\chi =2-2g-b$. (Finite surface normal forms, Jordan disks, and torsion control)

A connected compact oriented embedded surface with boundary $K$ is admissible in the set defining $g(K)$. (Seifert surfaces and smooth knot genus)

For a one-component regular diagram with $s$ smoothed circles and $c$ crossings, the algorithm uses $s$ disk records and one band record per crossing. (Seifert disk-and-band construction from an oriented knot diagram)

For such a knot diagram, the realized disk-and-band surface is connected, oriented, embedded, and has boundary the represented knot. (Seifert disks and bands form an embedded oriented spanning surface)

The Axiom of Countable Choice is the sequence principle stated in the item. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))