## New points of view in knot theory (1993)

Venue: | Bull. Am. Math. Soc., New Ser |

Citations: | 92 - 0 self |

### BibTeX

@ARTICLE{Birman93newpoints,

author = {Joan S. Birman},

title = {New points of view in knot theory},

journal = {Bull. Am. Math. Soc., New Ser},

year = {1993},

pages = {253--287}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial [Jo3] in 1984. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the central role that braid

### Citations

612 |
Quantum field theory and the Jones polynomial
- Witten
- 1989
(Show Context)
Citation Context ...of space, one stands out: it concerns the generalizations of the quantum group invariants and Vassiliev invariants to knots and links in arbitrary 3manifolds, i.e., the program set forth by Witten in =-=[Wi]-=-. That very general program is inherently more difficult than the special case of knots and links or simply of knots in the 3-sphere. It is an active area of research, with new discoveries made every ... |

271 |
Invariants of 3-manifolds via link polynomials and quantum
- Reshetikhin, Turaev
- 1991
(Show Context)
Citation Context ...ly of knots in the 3-sphere. It is an active area of research, with new discoveries made every day. We thought, at first, to discuss, very briefly, the 3-manifold invariants of Reshetikhin and Turaev =-=[RT]-=- and the detailed working out of special cases of those invariants by Kirby and Melvin [KM]; however, we then realized that we could not include such a discussion and ignore Jeffrey’s formulas for the... |

241 |
Index for subfactors
- Jones
(Show Context)
Citation Context ... restricted to [4, ∞] ∪ [4 cos 2 (π/p)], where p ≥ 3 is a natural number. Moreover, each such real number occurs for some pair M0, M1. We now sketch the idea of Jones’s proof, which is to be found in =-=[Jo1]-=-. Jones begins with the type II1 factor M1 and a subfactor M0. There is also a tiny bit of additional structure: In this setting there exists a map e1 : M1 → M0, known as the conditional expectation o... |

235 |
A q–difference analogue of U(g) and the Yang–Baxter equation
- Jimbo
- 1985
(Show Context)
Citation Context ...sentations of simple Lie algebras. Explicit solutions to (7 ∗ ) which are associated to the fundamental representations of the nonexceptional classical Lie algebras may be found in the work of Jimbo (=-=[Ji1]-=- and especially [Ji2]). See [WX] for a discussion of the problem of finding “classical” solutions to the “quantum” equation (7 ∗ ). Now, representations of Bn do not always support a Markov trace. To ... |

210 | Dehn surgery on knots - Gordon, Shalen - 1987 |

205 |
links and mapping class groups
- Birman, Braids
- 1975
(Show Context)
Citation Context ...ˆ β ∗ which they define represent the12 JOAN S. BIRMAN Figure 8. Elementary braids, singular braids, and tangles. same link type. Markov’s theorem, announced in [Mar] and proved forty years later in =-=[Bi]-=-, asserts that Markov equivalence is generated by conjugacy in each Bn and the map Bn → Bn+1 which takes a word W(σ1, . . .σn−1) to W(σ1, . . .σn−1)σ ±1 n . We call the latter Markov’s second move. Th... |

146 | Cohomology of knot spaces// in: Theory of Singularities and its Applications - Vassiliev - 1990 |

137 |
An enumeration of knots and links, and some of their algebraic properties, from
- Conway
- 1967
(Show Context)
Citation Context ...)Aq(Kp0). This formula passed unnoticed for forty years. (We first learned about Alexander’s version of it in 1970 from Mark Kidwell.) Then, in 1968 it was rediscovered, independently, by John Conway =-=[C]-=-, who added a new observation: If you require, in addition to (1a), that: (1b) Aq(O) = 1,NEW POINTS OF VIEW IN KNOT THEORY 9 Figure 5. Related link diagrams. Figure 6. Three related diagrams for the ... |

117 |
Anew polynomial invariant of knots and
- Freyd, Hoste, et al.
- 1985
(Show Context)
Citation Context ...turns out that (3a)–(3b) determine an infinite sequence of onevariable polynomials which in turn extend uniquely to give a two-variable invariant which has since become known as the HOMFLY polynomial =-=[FHL]-=-. Later, (1a)–(3b) were replaced by a more complicated family of crossing-change formulas, yielding10 JOAN S. BIRMAN the Kauffman polynomial invariant of knots and links [Kau1]. A unifying principle ... |

109 |
On the 3-manifold invariants of Witten and Reshetikhin–Tureav for sl2(C), Invent
- Kirby, Melvin
- 1991
(Show Context)
Citation Context ...ery day. We thought, at first, to discuss, very briefly, the 3-manifold invariants of Reshetikhin and Turaev [RT] and the detailed working out of special cases of those invariants by Kirby and Melvin =-=[KM]-=-; however, we then realized that we could not include such a discussion and ignore Jeffrey’s formulas for the Witten invariants of the lens spaces [Je]. Reluctantly, we made the decision to restrict o... |

107 |
Knots are determined by their complements
- Gordon, Luecke
- 1989
(Show Context)
Citation Context ...which reduced repeatedly the number of distinct knot types which could have homeomorphic complements and/or isomorphic groups, until it was finally proved, very recently, that (i) X determines K (see =-=[GL]-=-) and (ii) if K is prime, then G determines K up to unoriented equivalence [Wh]. Thus there are at most four distinct oriented prime knot types which have the same knot group [Wh]. This fact will be i... |

98 |
Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct
- Belavin, Drinfeld
- 1982
(Show Context)
Citation Context ...3] + [r12, r13] = 0. The theory of the CYBE is well understood, its solutions having been essentially classified. Two good references on the subject, both with extensive bibliographies, are [Sem] and =-=[BD]-=-. The possibility of “quantization”, i.e., passage from solutions to the CYBE (7 ∗∗ ) to those for the QYBE (7 ∗ ) was proved by Drinfeld in a series of papers, starting with [Dr1]. See [Dr2] for a re... |

96 | An invariant of regular isotopy - Kauffman - 1990 |

90 |
The Yang-Baxter equation and invariants of links
- Turaev
- 1988
(Show Context)
Citation Context ...ck to the main road. We now describe a method, discovered by Jones (see the discussion of vertex models in [Jo5]) but first worked out in full detail by TuraevNEW POINTS OF VIEW IN KNOT THEORY 15 in =-=[Tu]-=-, which can be applied to give, in a unified setting, every generalized Jones invariant via a Markov trace on an appropriate matrix representation of Bn. As before, E is a ring with 1. Let V be a free... |

89 | polynomial and classical conjectures in knot theory, Topology 26 - Murasugi, Jones - 1987 |

87 | Plane curves associated to character varieties of 3-manifolds, Invent
- Cooper, Culler, et al.
- 1994
(Show Context)
Citation Context ...and link invariants. In addition to the Jones polynomial and its generalizations, we mention the knot group invariants of [Wa], the energy invariants of [FH], and the algebraic geometry invariants of =-=[CCG]-=-, all of which seemed to come from mutually unrelated directions! In addition, there were generalizations of the Jones invariants to knotted graphs [Y1] and, finally, the numerical knot invariants of ... |

86 | Vassiliev’s knot invariants, Adv - Kontsevich - 1993 |

85 | link polynomials and a new algebra - Birman, Wenzl - 1989 |

79 |
Topological invariants of knots and links
- Alexander
- 1928
(Show Context)
Citation Context ... for describing 3-manifolds via their fibered knots; however, he did it long before anyone had considered the concept of a fibered knot! Another example that is of direct interest to us now occurs in =-=[Al2]-=-, where he reports on the discovery of the Alexander polynomial. In equation (12.2) of that paper we find observations on the relationship between the Alexander polynomials of three links: Kp+, Kp−, a... |

76 | Knots and links, Publish or - Rolfsen |

67 |
Chern–Simons–Witten invariants of lens spaces and torus bundles, and the semiclassical approximation
- Jeffrey
- 1992
(Show Context)
Citation Context ...al cases of those invariants by Kirby and Melvin [KM]; however, we then realized that we could not include such a discussion and ignore Jeffrey’s formulas for the Witten invariants of the lens spaces =-=[Je]-=-. Reluctantly, we made the decision to restrict our attention to knots in 3-space, but still, we have given at best a restricted picture. For example, we could not do justice to the topological constr... |

66 |
The invariant theory of n × n matrices
- Procesi
- 1976
(Show Context)
Citation Context ... of this section, such identities are one source of the crossing-change formulas we mentioned earlier, in §2. Another source will be trace identities, which always exist in matrix groups. (See, e.g., =-=[Pr]-=-.) In general one needs many such identities (i.e., polynomial equations satisfied by the images of various special braid words) to obtain axioms which suffice to determine a link type invariant. In t... |

63 |
Exactly solvable models in statistical mechanics
- Baxter
- 1982
(Show Context)
Citation Context ...R23 = ˇ R23 ˇ R13 ˇ R12. In this form, it occurs in nature in many ways, for example, in the theory of exactly solvable models in statistical mechanics, where it appears as the star-triangle relation =-=[Bax]-=-. Not long after the discovery that the Jones polynomial generalized to the HOMFLY and Kauffman polynomials, workers in the area began to discover other, isolated cases of generalizations, all relatin... |

60 |
A lemma on systems of knotted curves
- Alexander
- 1924
(Show Context)
Citation Context ...ers end with an intriguing or puzzling comment or remark which, as it turned out with the wisdom of hindsight, hinted at future developments of the subject. For example, in his famous paper on braids =-=[Al1]-=-, which we will discuss in detail in §6, he proves that every knot or link may be represented as a closed braid. He then remarks (at the end of the paper) that this yields a construction for describin... |

60 | Quantum groups and subfactors of type - Wenzl - 1990 |

47 |
Algorithms for the complete decomposition of a closed 3
- Jaco, Tollefson
- 1995
(Show Context)
Citation Context ...ork of Haken [Ha] and the work of Hemion [He] show that there is an algorithm which distinguishes knots. In recent years efforts have been directed at making that algorithm workable (for example, see =-=[JT]-=- for a discussion of recent results), but much work remains to be done before it could be considered “practical”, even for the simplest examples. (vi) The work in [BM2] and related papers referenced t... |

46 |
Möbius energy of knots and unknots
- Freedman, He, et al.
- 1994
(Show Context)
Citation Context ...ossession of an overflowing cornucopia of knot and link invariants. In addition to the Jones polynomial and its generalizations, we mention the knot group invariants of [Wa], the energy invariants of =-=[FH]-=-, and the algebraic geometry invariants of [CCG], all of which seemed to come from mutually unrelated directions! In addition, there were generalizations of the Jones invariants to knotted graphs [Y1]... |

46 | Monodromy representations of braid groups and Yang-Baxter equations - Kohno - 1987 |

39 | Studying links via closed braids III. Classifying links which are closed 3-braids
- Birman, Menasco
- 1993
(Show Context)
Citation Context ...nd let Kβ, where β ∈ Bn, be any closed braid representative of K. Choose any α ∈ P k n . Then the Vassiliev invariants of order ≤ k of the knots Kβ and Kαβ coincide. Remarks. (i) Using the results in =-=[BM1]-=-, Stanford has constructed sequences α1, α2, . . . of 3-braids such that the knot types obtained via Theorem 3 are all distinct and prime. Intuition suggests that distinct αj’s will always give distin... |

39 |
Homological representations of the Hecke algebra
- Lawrence
- 1990
(Show Context)
Citation Context ...ly, we made the decision to restrict our attention to knots in 3-space, but still, we have given at best a restricted picture. For example, we could not do justice to the topological constructions in =-=[La]-=- and in [Koh1, Koh2] without making this review much longer than we wanted it to be, even though it seems very likely that those constructions are closely related to the central theme of this review. ... |

36 | representations of braid groups and classical Yang-Baxter equations - Kohno, Linear - 1988 |

36 |
Classical solutions of the quantum Yang-Baxter equation
- Weinstein, Xu
- 1992
(Show Context)
Citation Context .... Explicit solutions to (7 ∗ ) which are associated to the fundamental representations of the nonexceptional classical Lie algebras may be found in the work of Jimbo ([Ji1] and especially [Ji2]). See =-=[WX]-=- for a discussion of the problem of finding “classical” solutions to the “quantum” equation (7 ∗ ). Now, representations of Bn do not always support a Markov trace. To obtain a Markov trace from the r... |

34 | Braids and the Jones polynomial - Franks, Williams - 1987 |

32 |
Quasitriangle Hopf algebras and invariants of tangles
- Reshetikhin
- 1990
(Show Context)
Citation Context ...ations of Bn do not always support a Markov trace. To obtain a Markov trace from the representation defined in (6), where R satisfies (7), one needs additional data in the form of an enhancement of R =-=[Re1]-=-. See Theorem 2.3.1 of [Tu]. The enhancement is a choice of invertible elements µ1, . . .,µm ∈ E which determine a matrix µ = diag(µ1, . . .,µm) such that (8a) µ ⊗ µ commutes with R = [R j1j2 i1i2 ], ... |

25 |
On the cl~cation of homeomorphisms of 2-manifolds and the classification of 3- manifolds
- Hemion
(Show Context)
Citation Context ...ply had given up too soon. However, this is probably the very best that one could hope for from the algebraic invariants. (v) By contrast to all of this, the work of Haken [Ha] and the work of Hemion =-=[He]-=- show that there is an algorithm which distinguishes knots. In recent years efforts have been directed at making that algorithm workable (for example, see [JT] for a discussion of recent results), but... |

24 |
polynomials and a graphical calculus
- Kauffman, Vogel
- 1992
(Show Context)
Citation Context ..., τj] = 0 if |i − j| ≥ 2, (11b) [σi, τi] = 0, (11c) σiσjσi = σjσiσj if |i − j| = 1, (11d) σiσjτi = τjσiσj if |i − j| = 1, where in all cases 1 ≤ i, j ≤ n − 1. The same set of relations also occurs in =-=[KV]-=- as generalized Reidemeister moves.The validity of these relations is easily established via pictures; for example, see Figure 14 for special cases of (11a)–(11d). To the best of our knowledge, howeve... |

22 |
invariants of finite type and perturbation
- Baez, Link
- 1991
(Show Context)
Citation Context ... distinguish between the σi’s and the τi’s by calling them crossing points and double points respectively in the singular braid diagram. Both determine double points in the projection. The manuscript =-=[Bae]-=- lists defining relations in SBn as: (11a) [σi, σj] = [σi, τj] = [τi, τj] = 0 if |i − j| ≥ 2, (11b) [σi, τi] = 0, (11c) σiσjσi = σjσiσj if |i − j| = 1, (11d) σiσjτi = τjσiσj if |i − j| = 1, where in a... |

22 |
Non-invertible knots exist. Topology 2
- TROTTER
- 1964
(Show Context)
Citation Context ...on on K (but not S 3 ). As noted earlier, Max Dehn proved in 1913 that nonamphicheiral knots exist [De], but remarkably, it took over forty years before it became known that noninvertible knots exist =-=[Tr]-=-. The relevance of this matter to our question is: While the quantum group invariants detect nonamphicheirality of knots, they do not detect noninvertibility of knots. So, if we could prove that Vassi... |

20 |
Theorie der Zopfe, Hamburg Abh
- Artin
- 1925
(Show Context)
Citation Context ...on of [Mu, We2]; the second is via the theory of R-matrices, as we shall do here.NEW POINTS OF VIEW IN KNOT THEORY 11 Figure 7. Braids. Our story begins with the by-now familiar notion of an n-braid =-=[Art]-=-. See Figure 7(a) for a picture, when n = 3. Our n-braid is to be regarded as living in a slab of 3-space R2 × I ⊂ R3 . It consists of n interwoven oriented strings which join n points, labeled 1, 2, ... |

19 |
Knot polynomials and Vassiliev’s invariants, Invent
- Birman, Lin
- 1993
(Show Context)
Citation Context ...wer series representation of the knot polynomials before one can understand the situation. This was first explained to the author and Lin by Bar Natan. Theorem 1 was first proved for special cases in =-=[BL]-=- and then generalized in [Li1]. In §§5–7 we describe a set of ideas which will be seen to lead to a new and very simple proof of Theorem 1. First, in §5, we review how Vassiliev’s invariants, like the... |

19 |
Vertex models, quantum groups and Vassiliev’s knot invariants, Columbia Univ
- lin
- 1991
(Show Context)
Citation Context ...the knot polynomials before one can understand the situation. This was first explained to the author and Lin by Bar Natan. Theorem 1 was first proved for special cases in [BL] and then generalized in =-=[Li1]-=-. In §§5–7 we describe a set of ideas which will be seen to lead to a new and very simple proof of Theorem 1. First, in §5, we review how Vassiliev’s invariants, like the Jones polynomial, the HOMFLY ... |

18 | On the Invariants of Torus Knots Derived from Quantum Groups - Rosso, Jones - 1993 |

15 | algebra representations of Braid groups and link polynomials - Hecke - 1987 |

15 | Threading knot diagrams - Morton - 1986 |

11 |
Constant quasiclassical solutions of the quantum Yang-Baxter equation
- Drinfeld
- 1983
(Show Context)
Citation Context ...phies, are [Sem] and [BD]. The possibility of “quantization”, i.e., passage from solutions to the CYBE (7 ∗∗ ) to those for the QYBE (7 ∗ ) was proved by Drinfeld in a series of papers, starting with =-=[Dr1]-=-. See [Dr2] for a review of the subject and another very extensive bibliography, including the definition of a quantum group and a development of the relevance of the theory of quantum groups to this ... |

11 |
knot invariants related to some statistical mechanical models
- On
- 1989
(Show Context)
Citation Context ...ubfactor in a type II1 factor! Its discovery opened a new chapter in knot and link theory. Back to the main road. We now describe a method, discovered by Jones (see the discussion of vertex models in =-=[Jo5]-=-) but first worked out in full detail by TuraevNEW POINTS OF VIEW IN KNOT THEORY 15 in [Tu], which can be applied to give, in a unified setting, every generalized Jones invariant via a Markov trace o... |

10 |
Über die freie Äquivalenz geschlossener Zöpfe
- Markov
- 1935
(Show Context)
Citation Context ...-equivalent if the closed braids ˆ β, ˆ β ∗ which they define represent the12 JOAN S. BIRMAN Figure 8. Elementary braids, singular braids, and tangles. same link type. Markov’s theorem, announced in =-=[Mar]-=- and proved forty years later in [Bi], asserts that Markov equivalence is generated by conjugacy in each Bn and the map Bn → Bn+1 which takes a word W(σ1, . . .σn−1) to W(σ1, . . .σn−1)σ ±1 n . We cal... |

9 |
Infinitely many knots with the same polynomial invariant
- Kanenobu
- 1986
(Show Context)
Citation Context ... G, a first wild guess would be that Jq(K), which does detect changes in the ambient space orientation (but not in knot orientation), classifies unoriented knot types; but this cannot be true because =-=[Kan]-=- constructs examples of infinitely many distinct prime knot types with the same Jones polynomial. Thus it seems interesting indeed to ask about the underlying topology behind the Jones polynomial. If ... |

8 | Groupes quantiques et modèles à vertex de V. Jones en théorie des noeuds - Rosso - 1988 |

8 |
Group invariants of links, Topology 31
- Wada
- 1992
(Show Context)
Citation Context ...rculated, topologists were in possession of an overflowing cornucopia of knot and link invariants. In addition to the Jones polynomial and its generalizations, we mention the knot group invariants of =-=[Wa]-=-, the energy invariants of [FH], and the algebraic geometry invariants of [CCG], all of which seemed to come from mutually unrelated directions! In addition, there were generalizations of the Jones in... |