A positive Hopf-plumbed surface for a torus knot

Use the solid-torus side with core the first-coordinate circle, so the second coordinate is the meridional disk coordinate. At longitudinal angle $u\in \mathbb{R}\mathrm{/}\mathbb{Z}$ in the first coordinate, the knot meets this disk at the $p$ points with second-coordinate angles $2\pi q(u+k)\mathrm{/}p$, $k=0,\dots ,p-1$. Place these points at the vertices of a regular $p$-gon. A counterclockwise rotation by $2\pi \mathrm{/}p$, tapered to fix the disk boundary, carries each point to the next and crosses consecutive points once in the order ${\sigma }_1,\dots ,{\sigma }_{p-1}$; each ${\sigma }_i$ is the positive half twist of adjacent points. Thus one such increment is the braid block $\delta ={\sigma }_1\cdots {\sigma }_{p-1}$, and the rotation through $2\pi q\mathrm{/}p$ gives ${\delta }^q$. The lifted curve has winding vector $(p,q)$ in the fixed Heegaard-torus coordinates, exactly the parameterization in [F1]. The braid block permutes the points by a $p$-cycle, so its $q$th power has one cycle because $\gcd (p,q)=1$.

At each positive crossing replace the crossing by the orientation-compatible smoothing and then attach one half-twisted rectangle in the original crossing ball; this is the oriented braid Seifert-surface construction. It gives $p$ disks joined by $q(p-1)$ positively half-twisted bands, one for each crossing. The first block has one band between each adjacent pair of disks, so its incidence graph is a path; attaching bands along a tree joins the disks to a disk. Each of the remaining $q-1$ blocks has $p-1$ bands.

Add the remaining blocks in order, and within each block add the bands in the order ${\sigma }_1,\dots ,{\sigma }_{p-1}$. For each added band, a small ball containing it and the corresponding band from the preceding block can be chosen disjoint from the other crossing balls after sliding their strand rectangles along the Seifert disks. The two bands are positive half twists on the same adjacent pair; their union with the two intervening disk strips is an annulus whose boundary components have linking number $+1$, hence a positive Hopf band. The old band meets the already-built surface in a square, the new annulus lies on the opposite side of that square, and the orientations agree because both bands come from positive crossings. Thus each added band is exactly a plumbing along a square as defined in the Statement. Repeating this for all $(q-1)(p-1)$ later bands gives the stated grid of positive Hopf plumbings.

A plumbing replaces the old square and the Hopf annulus by their union, so inclusion-exclusion for Euler characteristic gives $\chi (F_{p,q})=\chi (F_{p,q}^{\mathrm{o}\mathrm{l}\mathrm{d}})+0-1=\chi (F_{p,q}^{\mathrm{o}\mathrm{l}\mathrm{d}})-1$. Starting from the disk in step 2.1, after $(p-1)(q-1)$ plumbings the surface is connected and has $\chi (F_{p,q})=1-(p-1)(q-1)$.

The boundary of the constructed surface is the closure of the braid in step 1.1. Its permutation has one cycle, so the boundary has one component; the construction is connected and oriented at every band attachment. It is therefore a Seifert surface for $T(p,q)$ in the sense of [F2].

Since $(p-1)(q-1)$ is even, the one-boundary-component identity $\chi (F)=1-2g(F)$ and step 4.1 give $g(F_{p,q})=(p-1)(q-1)\mathrm{/}2$.