Constructing Seifert Surface for $(p,q)$-Torus Knot: The torus knot $T(p,q)$ lives on the standard torus $T^2 \subset S^3$. A canonical Seifert surface is:

1. Fiber surface: The torus $T^2$ minus a small neighborhood of the knot
2. Compress: Surgically modify to reduce genus using handle decomposition
3. Result: Surface with genus $g = \frac{(p-1)(q-1)}{2}$

For $T(2,3)$ (trefoil): $g = \frac{1 \cdot 2}{2} = 1$ (once-punctured torus).

Alternatively, view $T(p,q)$ as the closure of the $(p,q)$-braid. Seifert's algorithm on the standard diagram produces a genus-$g$ surface directly.

Minimal genus computation for pretzel knots: The pretzel knot $P(-2,3,7)$ has:

Using Seifert's algorithm on standard diagram:

• Crossing number: $c = |-2| + 3 + 7 = 12$
• Seifert circles: $s = 3$ (one per tangle strand)
• Upper bound: $g \leq \frac{12 - 3 + 1}{2} = 5$

Using Alexander polynomial:

• $\deg(\Delta) = 10$ (can be computed)
• Lower bound: $g \geq 5$

Therefore $g(P(-2,3,7)) = 5$ exactly. The Seifert algorithm yields minimal genus in this case!

Fiber surface for figure-eight knot: The figure-eight $4_1$ is fibered, so its minimal Seifert surface is a fiber $F$ of the fibration $S^3 \setminus 4_1 \to S^1$.

Explicitly:

• $F$ is a punctured torus (genus 1 surface with one boundary component)
• Monodromy $h: F \to F$ is a Dehn twist
• The fiber is incompressible and minimal genus

The fibration structure makes $4_1$ particularly tractable: many invariants can be computed from the monodromy alone.

Non-minimal Seifert surfaces: The unknot has minimal genus 0, but non-minimal Seifert surfaces exist with any genus $g \geq 0$.

Explicitly construct genus $g$ Seifert surface for unknot:

1. Start with a standard disk (genus 0)
2. Attach $g$ handles arbitrarily
3. Result: genus $g$ surface with boundary the unknot

These "stabilized" surfaces demonstrate that higher-genus Seifert surfaces always exist, making genus minimization nontrivial.