EPFL Logo
fr|en
Login
Concept

Jones polynomial

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients. Suppose we have an oriented link , given as a knot diagram. We will define the Jones polynomial, , using Louis Kauffman's bracket polynomial, which we denote by . Here the bracket polynomial is a Laurent polynomial in the variable with integer coefficients. First, we define the auxiliary polynomial (also known as the normalized bracket polynomial) where denotes the writhe of in its given diagram. The writhe of a diagram is the number of positive crossings ( in the figure below) minus the number of negative crossings (). The writhe is not a knot invariant. is a knot invariant since it is invariant under changes of the diagram of by the three Reidemeister moves. Invariance under type II and III Reidemeister moves follows from invariance of the bracket under those moves. The bracket polynomial is known to change by a factor of under a type I Reidemeister move. The definition of the polynomial given above is designed to nullify this change, since the writhe changes appropriately by or under type I moves. Now make the substitution in to get the Jones polynomial . This results in a Laurent polynomial with integer coefficients in the variable . This construction of the Jones polynomial for tangles is a simple generalization of the Kauffman bracket of a link. The construction was developed by Vladimir Turaev and published in 1990. Let be a non-negative integer and denote the set of all isotopic types of tangle diagrams, with ends, having no crossing points and no closed components (smoothings). Turaev's construction makes use of the previous construction for the Kauffman bracket and associates to each -end oriented tangle an element of the free -module , where is the ring of Laurent polynomials with integer coefficients in the variable .

Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Mathematics
Related concepts (13)
HOMFLY polynomial
In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial, which is computed from a diagram of the knot and can be shown to be an invariant of the knot, i.e.
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984.
Knot polynomial
In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923. Other knot polynomials were not found until almost 60 years later. In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial.
Show more

Copyright © 2025 EPFL, all rights reserved

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.