Key Knot Polynomials

Knot polynomials are essential tools in knot theory, providing ways to distinguish and analyze knots and links. Key examples include the Alexander, Jones, HOMFLY-PT, Kauffman, Conway polynomials, and Vassiliev invariants, each offering unique insights into knot properties.

1. Alexander polynomial

   • Introduced by James W. Alexander in 1928, it is a fundamental knot invariant.
   • It is defined using a presentation of the knot group and can be computed from a knot diagram.
   • The polynomial is a Laurent polynomial in one variable, typically denoted as ( A(K, t) ).
   • It can distinguish some knots but is not a complete invariant, meaning different knots can share the same Alexander polynomial.
   • The Alexander polynomial is related to the topology of the knot complement and can provide information about the knot's structure.

2. Jones polynomial

   • Discovered by Vaughan Jones in 1984, it revolutionized knot theory by introducing a new type of polynomial invariant.
   • The Jones polynomial is defined using a state model and is computed from a knot diagram using a recursive formula.
   • It is a Laurent polynomial in a variable ( t ) and is denoted as ( V(K, t) ).
   • The Jones polynomial can distinguish many knots that the Alexander polynomial cannot, making it a more powerful invariant.
   • It has connections to statistical mechanics and quantum groups, linking knot theory to other areas of mathematics.

3. HOMFLY-PT polynomial

   • The HOMFLY-PT polynomial generalizes both the Alexander and Jones polynomials, introduced by Habiro, Ohtsuki, Murakami, and others.
   • It is defined for oriented links and is a two-variable polynomial, denoted as ( P(K, m, l) ).
   • The polynomial can recover both the Alexander and Jones polynomials as special cases.
   • It is computed using a skein relation, similar to the Jones polynomial, and is useful for studying links with multiple components.
   • The HOMFLY-PT polynomial is a powerful tool for distinguishing links and understanding their properties.

4. Kauffman polynomial

   • Introduced by Louis Kauffman in 1987, it is another important polynomial invariant of knots and links.
   • The Kauffman polynomial is defined using a state-sum model and is a two-variable polynomial, denoted as ( F(K, a, z) ).
   • It can be related to the Jones polynomial and is computed using a skein relation.
   • The Kauffman polynomial can distinguish certain knots and links that other polynomials cannot.
   • It has applications in both knot theory and quantum field theory, highlighting its interdisciplinary relevance.

5. Conway polynomial

   • The Conway polynomial, introduced by John Horton Conway, is a polynomial invariant that is particularly useful for studying knots.
   • It is a Laurent polynomial in one variable, denoted as ( C(K, t) ), and is defined using a specific form of a knot diagram.
   • The Conway polynomial can be computed using a recursive method based on the knot's diagram.
   • It is known for its ability to distinguish certain classes of knots and has connections to the Alexander polynomial.
   • The Conway polynomial is also related to the concept of knot genus and can provide insights into the knot's topological properties.

6. Vassiliev invariants

   • Vassiliev invariants are a class of knot invariants that arise from the study of singularities in knot diagrams.
   • They are defined using a filtration of knot spaces and can be thought of as a sequence of invariants indexed by degree.
   • Vassiliev invariants can be used to construct more complex invariants, including the Jones polynomial and others.
   • They provide a systematic way to study the topology of knots and links through their singularities.
   • The study of Vassiliev invariants has led to significant developments in knot theory and its connections to algebraic topology.