Knot group
Introduction
In the field of mathematics, particularly in topology, the concept of a knot group is a fundamental aspect of knot theory. A knot group is an invariant of a knot, which means it is a property that remains unchanged under certain transformations, specifically ambient isotopies. The study of knot groups provides deep insights into the structure and classification of knots, which are embeddings of a circle in three-dimensional space, \(\mathbb{R}^3\).
Definition and Basic Properties
The knot group of a knot \(K\) in \(\mathbb{R}^3\) is defined as the fundamental group of the complement of the knot in three-dimensional space. Formally, if \(K\) is a knot, then its knot group \(G(K)\) is given by:
\[ G(K) = \pi_1(\mathbb{R}^3 \setminus K) \]
where \(\pi_1\) denotes the fundamental group, and \(\mathbb{R}^3 \setminus K\) represents the space obtained by removing the knot from \(\mathbb{R}^3\). The knot group is a topological invariant, meaning that if two knots are equivalent (i.e., they can be transformed into each other through a series of deformations without cutting or passing through themselves), their knot groups are isomorphic.
Presentation of Knot Groups
Knot groups can often be described using a presentation with generators and relations. A common method to obtain such a presentation is through the use of a Wirtinger presentation, which involves labeling the arcs of a knot diagram and assigning generators to these arcs. The relations are derived from the crossings in the knot diagram. For a knot diagram with \(n\) crossings, the Wirtinger presentation typically involves \(n+1\) generators and \(n\) relations.
Examples of Knot Groups
Trefoil Knot
The trefoil knot, one of the simplest nontrivial knots, has a knot group that can be presented as:
\[ G(T) = \langle a, b \mid a^2 = b^3 \rangle \]
This group is isomorphic to the braid group \(B_3\) and is a non-abelian group, reflecting the nontrivial nature of the trefoil knot.
Figure-Eight Knot
The figure-eight knot, another fundamental knot, has a more complex group:
\[ G(F) = \langle a, b \mid aba^{-1}b^{-1}ab^{-1}a^{-1}b = 1 \rangle \]
The figure-eight knot group is known for being the simplest hyperbolic knot group, meaning it corresponds to a hyperbolic structure on the complement of the knot.
Properties and Applications
Knot groups are crucial in distinguishing between different knots. Two knots with non-isomorphic knot groups cannot be equivalent. However, the converse is not true; there exist distinct knots with isomorphic knot groups, known as mutant knots.
Knot groups also play a role in 3-manifold theory, as the complement of a knot in \(\mathbb{R}^3\) is a 3-manifold. The study of these groups can provide insights into the topology of 3-manifolds, particularly through the lens of Dehn surgery, which involves modifying the manifold by removing a tubular neighborhood of the knot and gluing it back in a different way.
Advanced Topics in Knot Groups
Alexander Polynomial
The Alexander polynomial is an important knot invariant that can be derived from the knot group. It is obtained from the first homology group of the infinite cyclic cover of the knot complement. The Alexander polynomial provides information about the topology of the knot and can be used to distinguish between knots that have the same knot group.
Representation Theory
The representation theory of knot groups involves studying homomorphisms from the knot group into other groups, such as the symmetric group or linear groups. These representations can yield additional invariants and insights into the structure of the knot.
Virtual Knot Groups
In the study of virtual knots, which are generalizations of classical knots, virtual knot groups extend the concept of knot groups to these more complex structures. Virtual knot groups are defined similarly but take into account the additional crossings and virtual crossings present in virtual knots.