Knot groups are powerful tools for distinguishing knots. They capture the essential structure of a knot's topology, allowing mathematicians to prove that certain knots are distinct. This algebraic approach to knot classification provides a bridge between geometry and algebra.

However, knot groups have limitations. They can't always differentiate between non-equivalent knots or detect chirality. This has led mathematicians to develop additional invariants and techniques for a more comprehensive understanding of knot classification and properties.

Knot Groups and Knot Classification

Knot groups for distinction

• The knot group is the fundamental group of the knot complement
  • Knot complement is the 3-dimensional space obtained by removing the knot from $\mathbb{R}^3$ or $S^3$ ($\mathbb{R}^3$ is 3D Euclidean space, $S^3$ is the 3-sphere)
• Knot groups capture essential information about the knot's structure and entanglements
  • Provide an algebraic encoding of the knot's topology (crossings, twists, and loops)
• If two knots have different knot groups, they must be distinct knots
  • Knot group is a knot invariant, remains unchanged under ambient isotopy (continuous deformation without cutting or passing through itself)
• Examples of distinct knots with different knot groups:
  • Trefoil knot (simplest nontrivial knot) and unknot (simple loop with no crossings)
  • Figure-eight knot (4 crossings) and cinquefoil knot (5 crossings)

Application of knot group techniques

• To show that two knots are distinct using knot groups:
  1. Calculate the knot group for each knot using a knot diagram and the Wirtinger presentation (method for presenting the knot group based on the knot diagram)
  2. Simplify the presentations using Tietze transformations (operations that modify group presentations without changing the group) or other methods
  3. If the resulting groups are not isomorphic (have the same structure), the knots must be distinct
• Example: The trefoil knot and the unknot have different knot groups
  • Trefoil knot group has a presentation $\langle a, b | a^3 = b^2 \rangle$, while the unknot group is isomorphic to $\mathbb{Z}$ (the group of integers under addition)
• Knot groups can be used to prove the existence of non-equivalent knots with the same number of crossings
  • 5₁ and 5₂ knots (two distinct 5-crossing knots) have different knot groups

Limitations of knot groups

• Knot groups are not complete invariants, non-equivalent knots can have isomorphic knot groups
  • Square knot (two trefoil knots linked together) and granny knot (two trefoil knots linked with opposite handedness) have isomorphic knot groups but are distinct knots
• There exist infinitely many prime knots with isomorphic knot groups
  • Limits the effectiveness of using knot groups alone for classification
  • Prime knots cannot be decomposed into simpler knots (analogous to prime numbers)
• Knot groups cannot always detect the chirality (handedness) of a knot
  • A knot and its mirror image have isomorphic knot groups, but they may not be equivalent knots (cannot be deformed into each other)
  • Chirality is important in applications such as chemistry and biology

Knot groups vs other invariants

• The abelianization (making the group commutative) of the knot group yields the first homology group of the knot complement
  • Related to the linking number (measure of how two knots are linked) and the Alexander polynomial (polynomial invariant of knots)
• Knot group is related to the peripheral subgroup, which encodes information about the knot's meridian (loop around the knot) and longitude (loop along the knot)
  • This information is used in the construction of other invariants, such as the A-polynomial (polynomial invariant that captures the SL(2,C) representation space of the knot group)
• Knot group can be used to construct representations into other groups
  • Braid group (group of braids, which are intertwined strings) or symmetric group (group of permutations)
  • These representations give rise to additional knot invariants and provide connections to other areas of mathematics (algebra, topology, and representation theory)