12.1 Introduction to homology theories in knot theory

Homology theories are powerful tools in knot theory, assigning algebraic structures to knots that remain unchanged under deformation. These theories provide a richer understanding of knots compared to polynomial invariants, offering more detailed information about their properties.

Chain complexes form the backbone of homology theories in knot theory. By assigning algebraic objects to knot diagram crossings and defining boundary maps, we create a structure whose homology groups serve as knot invariants, capturing essential topological information.

Homology Theories and Knot Invariants

Homology theories for knot invariants

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• Homology theories are algebraic tools that study topological spaces and their properties by assigning algebraic objects (groups or vector spaces) to capture essential information about the space
• In knot theory, homology theories construct knot invariants which are quantities or algebraic structures that remain unchanged under ambient isotopy (continuous deformations) of the knot
• Homology theories provide a systematic way to associate algebraic invariants to knots
• Examples of homology theories used in knot theory:
  • Khovanov homology
  • Knot Floer homology
  • Heegaard Floer homology

Homology vs polynomial knot invariants

• Polynomial knot invariants (Jones polynomial, HOMFLY-PT polynomial) can be derived from certain homology theories
  • Jones polynomial obtained from the Euler characteristic of Khovanov homology
  • HOMFLY-PT polynomial obtained from the Euler characteristic of Khovanov-Rozansky homology
• Homology theories provide a categorification (lifting algebraic structures to a higher categorical level) of polynomial invariants
• Homology theories can be seen as a richer and more refined version of polynomial invariants
• The relationship between homology theories and polynomial invariants allows for a deeper understanding of the properties and behavior of knots

Advantages of homology theories

• Homology theories are generally more powerful than polynomial invariants in distinguishing knots by providing more information about the knot than just a polynomial
• Homology groups associated with knots can have a richer algebraic structure (graded modules, bigraded complexes)
• Examples of knots that have the same polynomial invariants but different homology groups:
  • Khovanov homology can distinguish certain pairs of knots with the same Jones polynomial
  • Knot Floer homology can distinguish certain pairs of knots with the same Alexander polynomial
• Homology theories can provide additional geometric and topological information about knots by detecting certain properties (genus of a knot, existence of certain types of surfaces bounded by the knot)

Chain Complexes and Knot Diagrams

Chain complexes in knot diagrams

• Constructing a chain complex from a knot diagram:
  1. Assign an algebraic object (vector space or module) to each crossing in the knot diagram
  2. Define a boundary map between these algebraic objects based on the local structure of the knot diagram around each crossing
     • Boundary map encodes the relationship between the crossings and adjacent arcs in the knot diagram
     • Satisfies the property that the composition of two consecutive boundary maps is zero: $\partial \circ \partial = 0$
  3. The resulting algebraic structure is a chain complex associated with the knot diagram
• The homology of the chain complex is defined as the quotient of the kernel of the boundary map by its image: $H_*(C) = \ker(\partial) / \operatorname{im}(\partial)$
• The homology groups of the chain complex are knot invariants that remain unchanged under Reidemeister moves and ambient isotopy of the knot
• Different homology theories use different algebraic objects and boundary maps in the construction of the chain complex, determining the specific properties and behavior of the resulting homology theory