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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Top images from around the web for Knot groups and fundamental groups The Power of Group Generators and Relations: An Examination of the Concept and Its Applications View original
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The Power of Group Generators and Relations: An Examination of the Concept and Its Applications View original
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The knot group of a knot K K K is defined as the fundamental group of the knot complement S 3 ∖ K S^3 \setminus K S 3 ∖ K
The knot complement is obtained by removing the knot K K K from the 3-dimensional sphere S 3 S^3 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 , … , x n x_1, x_2, \ldots, x_n x 1 , x 2 , … , x n
At each crossing , assign the relation x k = x i − 1 x j x i x_k = x_i^{-1} x_j x_i x k = x i − 1 x j x i or x k = x i x j x i − 1 x_k = x_i x_j x_i^{-1} x k = x i x j x i − 1 depending on the orientation and crossing type
x i x_i x i represents the generator for the arc passing under the crossing
x j x_j x j represents the generator for the arc passing over the crossing
x k x_k x k represents the generator for the outgoing arc
The Wirtinger presentation is written as ⟨ x 1 , x 2 , … , x n ∣ r 1 , r 2 , … , r m ⟩ \langle x_1, x_2, \ldots, x_n \mid r_1, r_2, \ldots, r_m \rangle ⟨ x 1 , x 2 , … , x n ∣ r 1 , r 2 , … , r m ⟩
x 1 , x 2 , … , x n x_1, x_2, \ldots, x_n x 1 , x 2 , … , x n are the generators, one for each arc in the diagram
r 1 , r 2 , … , r m r_1, r_2, \ldots, r_m r 1 , r 2 , … , 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 ): ⟨ x ∣ ⟩ ≅ Z \langle x \mid \rangle \cong \mathbb{Z} ⟨ x ∣ ⟩ ≅ Z
Trefoil knot : ⟨ x , y ∣ x y x = y x y ⟩ \langle x, y \mid xyx = yxy \rangle ⟨ x , y ∣ x y x = y x y ⟩
Figure-eight knot : ⟨ x , y ∣ x y − 1 x y − 1 = y − 1 x y x ⟩ \langle x, y \mid xy^{-1}xy^{-1} = y^{-1}xyx \rangle ⟨ x , y ∣ x y − 1 x y − 1 = y − 1 x y x ⟩
Hopf link : ⟨ x , y ∣ x y = y x ⟩ ≅ Z ⊕ Z \langle x, y \mid xy = yx \rangle \cong \mathbb{Z} \oplus \mathbb{Z} ⟨ x , y ∣ x y = y x ⟩ ≅ Z ⊕ Z
Whitehead link : ⟨ x , y ∣ x y x − 1 y x y − 1 = y − 1 x y x − 1 y x ⟩ \langle x, y \mid xyx^{-1}yxy^{-1} = y^{-1}xyx^{-1}yx \rangle ⟨ x , y ∣ x y x − 1 y x y − 1 = y − 1 x y x − 1 y x ⟩
Borromean rings : ⟨ x , y , z ∣ x y = y z , y z = z x , z x = x y ⟩ \langle x, y, z \mid xy = yz, yz = zx, zx = xy \rangle ⟨ x , y , z ∣ x y = yz , yz = z x , z x = x y ⟩