5.2 Genus of a knot and its properties

Knot genus is a key measure of knot complexity. It's defined as the smallest genus among all surfaces bounded by the knot. This non-negative integer remains unchanged under ambient isotopy, making it useful for classifying knots.

Calculating knot genus involves Seifert surfaces. These are oriented surfaces with the knot as their boundary. By applying Seifert's algorithm to a knot diagram, we can construct these surfaces and use them to determine the knot's genus.

Genus of a Knot

Definition of knot genus

• Non-negative integer measuring complexity of knot denoted as $g(K)$
• Defined as minimal genus among all oriented surfaces bounded by knot
• Knot invariant remains unchanged under ambient isotopy
  • Knots with different genera not equivalent useful for knot classification
• Genus 0 knot is unknot bounds a disk (surface with genus 0)
• Higher genus knots more complex cannot be unknotted without cutting and reconnecting (trefoil knot, figure-eight knot)

Calculation using Seifert surfaces

• Seifert surface oriented surface whose boundary is given knot
  • Constructed by applying Seifert's algorithm to knot diagram
• Knot genus is minimal genus among all possible Seifert surfaces for knot
• Calculate genus using Seifert surface:
  1. Construct Seifert surface from knot diagram
  2. Count number of crossings $(c)$ in knot diagram
  3. Count number of Seifert circles $(s)$ in constructed surface
  4. Apply genus formula: $g = \frac{1}{2}(2 - s + c)$
• Genus obtained from specific Seifert surface may not always be minimal provides upper bound for knot's actual genus (trefoil knot, figure-eight knot)

Properties of Knot Genus

Additivity under connected sum

• Connected sum of two knots $K_1 \# K_2$ formed by removing small arc from each knot and joining resulting endpoints
• Theorem: For any two knots $K_1$ and $K_2$, genus of connected sum equals sum of individual genera
  • In other words, $g(K_1 \# K_2) = g(K_1) + g(K_2)$
• Proof:
  1. Let $S_1$ and $S_2$ be minimal genus Seifert surfaces for $K_1$ and $K_2$, respectively
  2. Remove small disk from each surface and connect surfaces along boundaries of removed disks
  3. Resulting surface $S$ is Seifert surface for $K_1 \# K_2$
  4. Genus of $S$ equals $g(S_1) + g(S_2)$ connecting surfaces does not change total genus
  5. Since $S_1$ and $S_2$ are minimal genus surfaces, $g(S)$ also minimal for $K_1 \# K_2$
  6. Therefore, $g(K_1 \# K_2) = g(S) = g(S_1) + g(S_2) = g(K_1) + g(K_2)$

Relationship to other invariants

• Crossing number $c(K)$ minimum number of crossings among all diagrams of knot
  • For any knot $K$, $g(K) \leq \frac{1}{2}(c(K) - 1)$
    • Inequality provides lower bound for crossing number in terms of genus (trefoil knot, figure-eight knot)
• Unknotting number $u(K)$ minimum number of crossing changes needed to transform knot into unknot
  • For any knot $K$, $g(K) \leq u(K)$
    • Changing crossing can be thought of as performing band surgery on Seifert surface increases genus by at most 1 (trefoil knot, figure-eight knot)
• Signature $\sigma(K)$ knot invariant derived from knot's Seifert matrix
  • For any knot $K$, $|\sigma(K)| \leq 2g(K)$
    • Inequality relates signature and genus of knot (trefoil knot, figure-eight knot)