4.2 Knot group and Wirtinger presentation

Knot groups are a powerful tool for understanding knots. They're defined as the fundamental group of the knot complement, capturing essential topological information. Even though equivalent knots have isomorphic knot groups, the reverse isn't always true.

Wirtinger presentations offer a way to compute knot groups from knot diagrams. By assigning generators to arcs and relations to crossings, we can create a group presentation. Simplifying these presentations helps us compare and analyze different knots more easily.

The Knot Group and Its Presentation

Knot groups and fundamental groups

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• The knot group of a knot $K$ is defined as the fundamental group of the knot complement $S^3 \setminus K$
  • The knot complement is obtained by removing the knot $K$ from the 3-dimensional sphere $S^3$
  • The fundamental group captures information about the loops and holes in a topological space (knot complement)
• The knot group encodes essential topological information about the knot
  • Equivalent knots (trefoil and its mirror image) have isomorphic knot groups
  • Non-equivalent knots (trefoil and figure-eight) may have isomorphic knot groups, but the converse does not hold

Wirtinger presentations from knot diagrams

• Assign an orientation to the knot and label the arcs of the diagram with generators $x_1, x_2, \ldots, x_n$
• At each crossing, assign the relation $x_k = x_i^{-1} x_j x_i$ or $x_k = x_i x_j x_i^{-1}$ depending on the orientation and crossing type
  • $x_i$ represents the generator for the arc passing under the crossing
  • $x_j$ represents the generator for the arc passing over the crossing
  • $x_k$ represents the generator for the outgoing arc
• The Wirtinger presentation is written as $\langle x_1, x_2, \ldots, x_n \mid r_1, r_2, \ldots, r_m \rangle$
  • $x_1, x_2, \ldots, x_n$ are the generators, one for each arc in the diagram
  • $r_1, r_2, \ldots, r_m$ are the relations, one for each crossing in the diagram

Simplification of Wirtinger presentations

• Apply Tietze transformations to modify the presentation without changing the group
  • Add or remove a generator that can be expressed using other generators
  • Add or remove a relation that follows from other relations
• Eliminate redundant generators and relations by substituting generators
• Identify patterns or symmetries in the presentation to further simplify it
• Use algebraic manipulations to rewrite relations in simpler forms

Computation of knot groups

• Unknot (trivial knot): $\langle x \mid \rangle \cong \mathbb{Z}$
• Trefoil knot: $\langle x, y \mid xyx = yxy \rangle$
• Figure-eight knot: $\langle x, y \mid xy^{-1}xy^{-1} = y^{-1}xyx \rangle$
• Hopf link: $\langle x, y \mid xy = yx \rangle \cong \mathbb{Z} \oplus \mathbb{Z}$
• Whitehead link: $\langle x, y \mid xyx^{-1}yxy^{-1} = y^{-1}xyx^{-1}yx \rangle$
• Borromean rings: $\langle x, y, z \mid xy = yz, yz = zx, zx = xy \rangle$