Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics

Józef Przytycki

Banach Center Publications (1995)

  • Volume: 34, Issue: 1, page 121-148
  • ISSN: 0137-6934

Abstract

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We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3-string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa.

How to cite

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Przytycki, Józef. "Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics." Banach Center Publications 34.1 (1995): 121-148. <http://eudml.org/doc/251307>.

@article{Przytycki1995,
abstract = {We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3-string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa.},
author = {Przytycki, Józef},
journal = {Banach Center Publications},
keywords = {knots; Jones polynomial; mutation; links},
language = {eng},
number = {1},
pages = {121-148},
title = {Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics},
url = {http://eudml.org/doc/251307},
volume = {34},
year = {1995},
}

TY - JOUR
AU - Przytycki, Józef
TI - Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics
JO - Banach Center Publications
PY - 1995
VL - 34
IS - 1
SP - 121
EP - 148
AB - We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3-string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa.
LA - eng
KW - knots; Jones polynomial; mutation; links
UR - http://eudml.org/doc/251307
ER -

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