What is a skein relation?

It is a recursive identity relating the polynomial of a link to those of the links obtained by changing or smoothing a single crossing; iterating it reduces any diagram to simple unknotted pieces whose values are known.

Does the Jones polynomial detect the unknot?

It is unknown whether the Jones polynomial distinguishes every nontrivial knot from the unknot; this remains a notable open problem, illustrating that even powerful polynomial invariants may not be complete.

Knot Polynomials

Knot polynomials assign a polynomial to each knot or link that is unchanged by deformation, packaging deep topological information into computable algebra and connecting knots to operator algebras and quantum physics.

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Definition

A knot polynomial is a polynomial-valued knot invariant, typically defined recursively through a skein relation that links the polynomials of knots differing at a single crossing, making it computable from any diagram.

Scope

This topic develops the principal polynomial invariants: the Alexander polynomial from the knot's homology or Seifert matrix, the bracket and Jones polynomial defined by a skein relation arising from the Temperley-Lieb algebra, and the two-variable HOMFLY-PT and Kauffman polynomials that generalize them. It treats skein relations as the computational mechanism, the strengths and known limitations of each polynomial in distinguishing knots, and the categorification of the Jones polynomial by Khovanov homology.

Core questions

How does a skein relation determine a polynomial invariant from its values on simple links?
What different topological information do the Alexander and Jones polynomials encode?
Why did the Jones polynomial, arising from von Neumann algebras, reveal unexpected connections to physics?
What are the limits of polynomial invariants, and how does categorification strengthen them?

Key concepts

Alexander polynomial from the Seifert matrix
Kauffman bracket and the Jones polynomial
Skein relations as a computational rule
HOMFLY-PT and Kauffman two-variable polynomials
Khovanov homology and categorification

Clinical relevance

The Jones polynomial linked knot theory to statistical mechanics, the Yang-Baxter equation, and topological quantum field theory, and knot polynomials provide invariants relevant to quantum computing and to distinguishing the entanglement of biological and physical filaments.

History

Alexander introduced the first knot polynomial in 1928; Jones's 1985 polynomial, discovered through the study of von Neumann algebras, was rapidly generalized to the HOMFLY-PT and Kauffman polynomials and later categorified by Khovanov, reshaping the field around quantum invariants.

Key figures

James W. Alexander
Vaughan Jones
Mikhail Khovanov

Seminal works

lickorish1997
jones1985

Frequently asked questions

What is a skein relation?: It is a recursive identity relating the polynomial of a link to those of the links obtained by changing or smoothing a single crossing; iterating it reduces any diagram to simple unknotted pieces whose values are known.
Does the Jones polynomial detect the unknot?: It is unknown whether the Jones polynomial distinguishes every nontrivial knot from the unknot; this remains a notable open problem, illustrating that even powerful polynomial invariants may not be complete.