## Yang-Baxter deformations of quandles and racks (2005)

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@MISC{Eisermann05yang-baxterdeformations,

author = {Michael Eisermann},

title = {Yang-Baxter deformations of quandles and racks},

year = {2005}

}

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### Citations

1516 | Exactly solved models in statistical mechanics - Baxter - 1982 |

452 |
mapping class groups
- Birman, Braids
- 1974
(Show Context)
Citation Context ...ls in statistical mechanics, and later in quantum field theory [9] in connection with the quantum inverse scattering method. It also has a very natural interpretation in terms of Artin’s braid groups =-=[1, 4]-=- and their tensor representations. These have received much attention over the last 20 years since V. Jones’s unexpected discovery of his knot polynomial [14, 15, 16]. Remark 2. Recall that the braid ... |

290 | Hecke algebra representations of braid groups and link - Jones - 1987 |

269 |
On the deformation of rings and algebras
- Gerstenhaber
- 1964
(Show Context)
Citation Context ... cQ is rigid. In the remaining cases, where infinitesimal deformations are possible, we show that higher-order obstructions are the same as in the quantum case. Introduction Following M. Gerstenhaber =-=[12]-=-, an algebraic deformation theory should • define the class of objects within which deformation takes place, • identify infinitesimal deformations as elements of a suitable cohomology, • identify the ... |

217 |
A polynomial invariant for knots via von Neumann algebras
- Jones
- 1985
(Show Context)
Citation Context ...tation in terms of Artin’s braid groups [1, 4] and their tensor representations. These have received much attention over the last 20 years since V. Jones’s unexpected discovery of his knot polynomial =-=[14, 15, 16]-=-. Remark 2. Recall that the braid group on n strands can be presented as 〈 〉 σiσj = σjσi for |i − j| ≥ 2 Bn = σ1, . . . , σn−1 ∣ , σiσjσi = σjσiσj for |i − j| = 1 where the braid σi performs a positiv... |

166 |
Quantum Groups, Graduate Texts
- Kassel
- 1995
(Show Context)
Citation Context ... y defines a Yang-Baxter operator (see §1 for definitions). For a trivial rack, where x y = x for all x, y ∈ Q, we see that cQ is the transposition operator. In this case the theory of quantum groups =-=[7, 18, 19]-=- produces a plethora of interesting deformations, which have received much attention over the last 20 years. It thus seems natural to study deformations of cQ in the general case, where Q is a non-tri... |

161 |
Theory of braids
- Artin
- 1947
(Show Context)
Citation Context ...ls in statistical mechanics, and later in quantum field theory [9] in connection with the quantum inverse scattering method. It also has a very natural interpretation in terms of Artin’s braid groups =-=[1, 4]-=- and their tensor representations. These have received much attention over the last 20 years since V. Jones’s unexpected discovery of his knot polynomial [14, 15, 16]. Remark 2. Recall that the braid ... |

157 |
A classifying invariant of knots, the knot quandle
- Joyce
- 1982
(Show Context)
Citation Context ...eful: Definition 7. Let Q be a set with a binary operation ∗. We call (Q, ∗) a quandle if it satisfies axioms (Q1–Q3), and a rack if satisfies axioms (Q2–Q3). The term “quandle” goes back to D. Joyce =-=[17]-=-. The same structure was called “distributive groupoid” by S.V. Matveev [21], and “crystal” by L.H. Kauffman [20]. Since quandles are close to groups, their applications in knot theory are in close re... |

126 |
Racks and links in codimension two
- Fenn, Rourke
- 1992
(Show Context)
Citation Context ...tion ̺(y): x ↦→ x ∗ y is an automorphism of (Q, ∗). This is why such a structure was called automorphic set by E. Brieskorn [5]. The somewhat shorter term rack was preferred by R. Fenn and C.P.Rourke =-=[10]-=-. Definition 8. Let Q be a rack. The subgroup of Aut(Q) generated by the family {̺(y) | y ∈ Q} is called the group of inner automorphisms, denoted Inn(Q). Two elements x, y ∈ Q are called behaviourall... |

118 |
Partition function of the eight-vertex lattice model
- Baxter
- 1972
(Show Context)
Citation Context ... I) = (I ⊗ c)(c ⊗ I)(I ⊗ c) in AutA(V ⊗3 ). Yang-Baxter operators first appeared in theoretical physics, in a paper by C.N. Yang [22] on the many-body problem in one dimension, in work of R.J. Baxter =-=[2, 3]-=- on exactly solvable models in statistical mechanics, and later in quantum field theory [9] in connection with the quantum inverse scattering method. It also has a very natural interpretation in terms... |

104 | Quantum groups. Graduate Texts in Mathematics, 155 - Kassel - 1995 |

97 | The Yang-Baxter equation and invariants of links - Turaev - 1988 |

96 |
Some exact results for the many-body problem in one dimension with repulsive delta-function interaction
- Yang
- 1967
(Show Context)
Citation Context ...he Yang-Baxter equation, also called braid relation, (c ⊗ I)(I ⊗ c)(c ⊗ I) = (I ⊗ c)(c ⊗ I)(I ⊗ c) in AutA(V ⊗3 ). Yang-Baxter operators first appeared in theoretical physics, in a paper by C.N. Yang =-=[22]-=- on the many-body problem in one dimension, in work of R.J. Baxter [2, 3] on exactly solvable models in statistical mechanics, and later in quantum field theory [9] in connection with the quantum inve... |

86 | On some unsolved problems in quantum group theory - Drinfel’d - 1992 |

82 | Quandle cohomology and state-sum invariants of knotted curves and surfaces
- Carter, Jelsovsky, et al.
(Show Context)
Citation Context ... and D.N. Yetter in [11]. The deformation of quandles and racks appeared as an example, but only diagonal deformations were taken into account. Diagonal deformations have been more fully developed in =-=[6]-=-, where quandle cohomology was used to construct state-sum invariants of knots. • In [23] Yetter considered deformations of braided monoidal categories in full generality; see also [24] and the biblio... |

79 |
Braided compact closed categories with applications to lowdimensional topology
- Freyd, Yetter
- 1989
(Show Context)
Citation Context ...obstructions to the integration of an infinitesimal deformation, • give criteria for rigidity, and possibly determine the rigid objects. In answer to a question initiated by P.J.Freyd and D.N. Yetter =-=[11]-=-, we carry out this programme for racks and their formal deformations in the space of Yang-Baxter operators. A rack is a set Q with a binary operation, denoted (x, y) ↦→ x y , such that cQ: x ⊗ y ↦→ y... |

67 |
Distributive groupoids in knot theory
- Matveev
- 1982
(Show Context)
Citation Context ... a quandle if it satisfies axioms (Q1–Q3), and a rack if satisfies axioms (Q2–Q3). The term “quandle” goes back to D. Joyce [17]. The same structure was called “distributive groupoid” by S.V. Matveev =-=[21]-=-, and “crystal” by L.H. Kauffman [20]. Since quandles are close to groups, their applications in knot theory are in close relationship to the knot group. We should point out, however, that there exist... |

64 |
Integrable models in (1 + 1)-dimensional quantum field theory, Recent advances in field theory and statistical mechanics (Les Houches
- Faddeev
- 1982
(Show Context)
Citation Context ...l physics, in a paper by C.N. Yang [22] on the many-body problem in one dimension, in work of R.J. Baxter [2, 3] on exactly solvable models in statistical mechanics, and later in quantum field theory =-=[9]-=- in connection with the quantum inverse scattering method. It also has a very natural interpretation in terms of Artin’s braid groups [1, 4] and their tensor representations. These have received much ... |

61 | Set-theoretical solutions of the quantum Yang–Baxter equation - Etingof, Schedler, et al. - 1999 |

54 | On knot invariants related to some statistical mechanics models - Jones - 1989 |

45 |
bialgebras, quantum groups, and algebraic deformations, from: “Deformation theory and quantum groups with applications to mathematical physics
- Gerstenhaber, Schack, et al.
- 1990
(Show Context)
Citation Context ...over Q[h] is equivalent to s · cQ with some constant factor s ∈ 1 + (h). In other words, cQ is rigid. □ Example 20. The smallest non-trivial example of a rigid operator is given by the set Q = {(12), =-=(13)-=-, (23)} of transpositions in the symmetric group S3, or equivalently the set of reflections in the dihedral group D3. Ordering the basis Q × QYANG-BAXTER DEFORMATIONS OF QUANDLES AND RACKS 7 lexicogr... |

33 | Quantum groups, volume 155 of Graduate Texts in Mathematics - Kassel - 1995 |

28 |
Drinfe l ’d, Quantum groups
- G
- 1986
(Show Context)
Citation Context ... y defines a Yang-Baxter operator (see §1 for definitions). For a trivial rack, where x y = x for all x, y ∈ Q, we see that cQ is the transposition operator. In this case the theory of quantum groups =-=[7, 18, 19]-=- produces a plethora of interesting deformations, which have received much attention over the last 20 years. It thus seems natural to study deformations of cQ in the general case, where Q is a non-tri... |

27 | On the set-theoretical Yang-Baxter equation - Lu, Yan, et al. |

23 |
Topological quantum field theories associated to finite groups and crossed G-sets
- Yetter
- 1992
(Show Context)
Citation Context ...but only diagonal deformations were taken into account. Diagonal deformations have been more fully developed in [6], where quandle cohomology was used to construct state-sum invariants of knots. • In =-=[23]-=- Yetter considered deformations of braided monoidal categories in full generality; see also [24] and the bibliographical references therein. He was thus led to define a cohomology theory, which is ess... |

21 |
Quantum groups and knot invariants, Panoramas et Synthèses
- Kassel, Rosso, et al.
- 1997
(Show Context)
Citation Context ... y defines a Yang-Baxter operator (see §1 for definitions). For a trivial rack, where x y = x for all x, y ∈ Q, we see that cQ is the transposition operator. In this case the theory of quantum groups =-=[7, 18, 19]-=- produces a plethora of interesting deformations, which have received much attention over the last 20 years. It thus seems natural to study deformations of cQ in the general case, where Q is a non-tri... |

19 | On rack cohomology - Etingof, Graña |

15 |
algebra representations of Braid groups and link polynomials
- Hecke
- 1987
(Show Context)
Citation Context ...tation in terms of Artin’s braid groups [1, 4] and their tensor representations. These have received much attention over the last 20 years since V. Jones’s unexpected discovery of his knot polynomial =-=[14, 15, 16]-=-. Remark 2. Recall that the braid group on n strands can be presented as 〈 〉 σiσj = σjσi for |i − j| ≥ 2 Bn = σ1, . . . , σn−1 ∣ , σiσjσi = σjσiσj for |i − j| = 1 where the braid σi performs a positiv... |

15 | The rack space - Fenn, Rourke, et al. |

11 |
knot invariants related to some statistical mechanical models
- On
- 1989
(Show Context)
Citation Context ...tation in terms of Artin’s braid groups [1, 4] and their tensor representations. These have received much attention over the last 20 years since V. Jones’s unexpected discovery of his knot polynomial =-=[14, 15, 16]-=-. Remark 2. Recall that the braid group on n strands can be presented as 〈 〉 σiσj = σjσi for |i − j| ≥ 2 Bn = σ1, . . . , σn−1 ∣ , σiσjσi = σjσiσj for |i − j| = 1 where the braid σi performs a positiv... |

10 |
Automorphic sets and braids and singularities, Braids
- Brieskorn
- 1986
(Show Context)
Citation Context ...roup. Axioms (Q2) and (Q3) are equivalent to saying that every right translation ̺(y): x ↦→ x ∗ y is an automorphism of (Q, ∗). This is why such a structure was called automorphic set by E. Brieskorn =-=[5]-=-. The somewhat shorter term rack was preferred by R. Fenn and C.P.Rourke [10]. Definition 8. Let Q be a rack. The subgroup of Aut(Q) generated by the family {̺(y) | y ∈ Q} is called the group of inner... |

9 | Homological characterization of the unknot
- Eisermann
- 2003
(Show Context)
Citation Context ...o diagonal matrices, that is, to matrices f : Qn × Qn → m with f[ x1,...,xn y1,...,yn ] = 0 whenever xi ̸= yi for some i. In this case we obtain the cochain complex of quandle or rack cohomology (see =-=[6, 8]-=-). 3.2. Characterization of entropic maps. Recall from Definition 10 that a map f : AQ n → mQ n is entropic if and only if d0f = · · · = dnf = 0. The following lemma gives a useful reformulation: Lemm... |

7 | V Turaev, Quantum groups and knot invariants - Kassel, Rosso - 1997 |

7 | Knot colouring polynomials - Eisermann - 2005 |

6 | Functorial knot theory, Series on Knots and Everything - Yetter - 2001 |

3 | Knots and physics, volume 1 - Kauffman - 2001 |

3 | Quantum groups, from - Drinfel’d - 1986 |

2 | Automorphic sets and braids and singularities, from: “Braids - Brieskorn - 1986 |

2 | Drinfel’d ” Quantum groups”, from - G - 1986 |

2 | Drinfel ′ d. On some unsolved problems in quantum group theory - G - 1990 |

2 |
Knots and physics, Series on Knots and Everything
- Kauffman
(Show Context)
Citation Context ...–Q3), and a rack if satisfies axioms (Q2–Q3). The term “quandle” goes back to D. Joyce [17]. The same structure was called “distributive groupoid” by S.V. Matveev [21], and “crystal” by L.H. Kauffman =-=[20]-=-. Since quandles are close to groups, their applications in knot theory are in close relationship to the knot group. We should point out, however, that there exist many quandles that do not embed into... |

2 | Quantum groups and knot invariants, volume 5 - Kassel, Rosso, et al. - 1997 |

2 | Knots and physics, Series on Knots and Everything 1 - Kauffman - 2001 |

1 | Knot colouring polynomials (2005)Preprint - Eisermann |

1 | Functorial knot theory, volume 26 - Yetter |

1 |
knot theory, Series on Knots and Everything, vol. 26, World Scientific Publishing Co
- Functorial
(Show Context)
Citation Context ...ly developed in [6], where quandle cohomology was used to construct state-sum invariants of knots. • In [23] Yetter considered deformations of braided monoidal categories in full generality; see also =-=[24]-=- and the bibliographical references therein. He was thus led to define a cohomology theory, which is essentially equivalent to Yang-Baxter cohomology. He did not, however, calculate any examples. As f... |