The theory of braid groups is one of the most fascinating chapters in low-dimensional topology. This includes background material in topology and algebra, often going beyond traditional presentations of the theory of braids.
Braids and Braid Groups
The Artin braid groups
- Basic definition
- Projection to the symmetric group
- Natural inclusions
- The group B 3
It is easy to check that simple transpositions satisfy the braid ratios. Prove that f induces an isomorphism Bn/[Bn, Bn]∼=Z, where [Bn, Bn] is a commutative subgroup of Bn. n−1 defines an involutional automorphism of Bn.
Braids and braid diagrams
- Geometric braids
- Braid diagrams
- Reidemeister moves on braid diagrams
- The group of braids
To specify a geometric braid, one can draw its projection to R×{0}×I along the second coordinate and indicate which string goes “under” the other at each intersection. The Abraid diagram onnstrands is a setD ⊂R×Isplit as a union ofntopological intervals, called the strands ofD, such that the following three conditions are met: i) The projection R×I → Maps each strand homeomorphically on I. iii) Each point of R×I belongs to a maximum of two strands.
Pure braid groups
- Pure braids
From the geometric description of the natural inclusionι:Pn−1→Pn it is clear thatfn◦ι= idPn−1. The braid Δn can be obtained from the trivial braid 1n by a half twist obtained by holding the top of the braid fixed and flipping over the row of bottom ends by an angle of π.
Configuration spaces
- Configuration spaces of ordered sets of points Let M be a topological space and let
- Configuration spaces of nonordered sets of points
- The space C n (R 2 ) as a space of polynomials
Consider again the configuration space Fm,n(M) associated with integers m ≥ 0, n ≥ 1 and a connected topological manifold M of dimension ≥ 2. These functions are invariant under the action of Sn on Fn(C) by permutation of coordinates and thus induces a map Cn(R2) = Cn(C) → Cn. Exercise 1.4.1.Prove the following generalization of Lemma 1.28.
Braid automorphisms of free groups
- Proof of Theorem 1.31
For a group defined by generators and relations, the word problem consists of finding an algorithm that makes it possible to decide whether a given word in the generators represents a neutral element of the group. To see this, define the length ϕ as the sum of the letter lengths of the words Akxμ(k)A−k1overk= 1,2,.
Braids and homeomorphisms
- Mapping class groups
- Half-twists
- The isomorphism B n ∼ = M (D, Q n )
During the deformation, the base point f(d) =d can move inD−Qn.) This implies that the homotopy classes of these two loops ρ(f)(xk) and xμ(k) are conjugate in π1(D−Qn, d) ) . Moving along ∂P in the direction given by the orientation of M, we encounter successive edges, e.g. α1, α2, .
Groups of homeomorphisms vs. configuration spaces
- Groups of homeomorphisms
- Parametrizing isotopies
- Proof of Theorem 1.33
- Applications
Moving in the direction given by the orientation of M, denote these arcs by α1, α2,. for each elementβ of the set generated byτα3, τα4,. Let {gt : D → D}t∈I be the isotopy of the identity map g0 = id : D → D to g1 = ταi obtained by rotating αi in D about its midpoint counterclockwise .
Notes
Prove that f is isotopic to the identity in the class of self-homeomorphisms of S2fixinga.
Braids, Knots, and Links
Knots and links in three-dimensional manifolds
- Basic definitions
- Link diagrams
- Ordered and oriented links
- The linking number
An orientation of a geometric link L in a 3-dimensional manifold M is an orientation of the underlying 1-dimensional manifold L. To display the orientation of the link presented by a link diagram on a surface , it is sufficient to orient all components of the diagram.
Closed braids in the solid torus
- Solid tori
- Closed braids
- Closure of braids
- Proof of Theorem 2.1
- Closed braid diagrams
Each braid β on n cords results in a closed n-braid in the full torus as follows. Any closed braid diagram in S1×I represents a closed braid in the full torus S1×I×I in an obvious way; cf.
Alexander’s theorem
- Closed braids in R 3
We only need to prove that if D,D represent isotopic closed braids in the solid torus, then D can be transformed into D by a finite series of isotopies and moves (Ωbr2 )±1,(Ωbr3 )±1. It is well known that any geometric link in R3 is isotopic to a polygonal link (cf. the proof of Theorem 1.6). We only need to prove that any oriented polygonal linkL ⊂R3 is isotopic to a closed braid.
Links as closures of braids: an algorithm
- Preliminaries
- Bending and tightening of link diagrams Consider an oriented link diagram D in R 2 . Let
- The algorithm
Note that the Seifert circles of DandD do not pass through the shaded areas in Figure 2.11. It suffices to verify that the number of pairs of incompatible Seifert rings involving S1 or S2 or both is equal to tod1+d2+ 2d+ 1. Since f1 (respectively f2) is not a defect face, it is adjoined by at most two Seifert rings .
Markov’s theorem
- Markov moves
- Markov functions
- A pivotal lemma
The following fundamental conclusion gives a description of the set of isotopy classes of R3-oriented bonds in terms of braiding. The standard orientation of the threads of a braiding diagram is from the inputs to the outputs. If is a braid diagram, then the inputs and outputs of a+, a− lie on the vertical sides of the square.
Deduction of Markov’s theorem from Lemma 2.11
- The move M 3
- Reduction of Theorem 2.8 to Claim 2.15
- Reduction to Lemma 2.17
- Proof of Lemma 2.17, part I
- Proof of Lemma 2.17, part III
Note that g preserves the set of Seifert circles of the diagram and therefore preserves its height. If necessary, by conjugation with the S2 isotopy, we can assume that the Seifert circles D are oriented counterclockwise, i.e. that D is a 0-diagram in R2. The same argument as in the proof of Lemma 2.6 shows that the Seifert circles D1 lying in D1 form a system of t≥1 concentric compatible circles, the outer circle being S1.
Proof of Lemma 2.11
- Ghost braids
- Proof of Lemma 2.11
An argument similar to the one above shows that this move and its inverse preserve the M-equivalence class of the braid. This completes the proof of (2.9) and of the lemma. 2.10) By Lemma 2.23 the M-equivalence class ofα+ is preserved under the transformation which replaces the term 1n in the factor 1m+r⊗1n by the braid. The first transformation in Figure 2.27 is a single motion M(μm,−). It would be more logical to write ++ in the box, but we simply write +.) The next two moves are isotopies in the class of closed braid diagrams (this amounts to conjugation of braids).
Homological Representations of the Braid Groups
The Burau representation
- Definition
- Unitarity
The study of the Burau representationψn : Bn → GLn(Λ) is to a large extent focused on its core and image. That ρ, ρ lies at the core of the Burau representation can in principle be verified by a direct calculation. However, these calculations shed no light on the geometric reasons that force ρ, ρ to lie in the core.
Nonfaithfulness of the Burau representation
- Homological representations
- Dehn twists
- Equivalence of representations
This sum does not depend on the choice of loops α,β in their homology classes and defines the bilinear form H × H → Z. Choose an arbitrary point d∈∂Σ lying above and consider the relative homology group H =H1(Σ, G d; Z), where G is the G-orbit d, i.e. the set of all points of Σ lying over d. In the annulus between these discs, both tc and τα2 act as a Dehn twist around the core circle of the annulus.
The reduced Burau representation
- The faithfulness of ψ 3
This is the geometric reason for the fact that the Burau representation can be reduced, but is not a direct sum of its reduced form with a one-dimensional representation. It is clear that this homomorphism is surjective and that its core is the normal subgroup generated by the braid (σ1σ2σ1)4. Use the homologous interpretation ofψnr; note that the element ofM(D, Q) corresponding to (σ1 · · ·σn−1)n is the Dehn rotation about a circle inD concentric to ∂D.) Note that a similar equality does not hold for ψn, because for example ψ2(σ12)=t2I2.
The Alexander–Conway polynomial of links
- An example of a Markov function
- The Alexander–Conway polynomial
A direct calculation shows that the product of the first three matrices (respectively the last three) on the right side of (3.12) is equal to. The Alexander-Conway polynomial (one variable) is a basis and historically the first polynomial invariant of R3-oriented connections. A directed connection diagram DonR2 is increasing if it satisfies the following two conditions: a) the components eD can be indexed by 1,.
The Lawrence–Krammer–Bigelow representation
- Remarks
The function L →f(L) satisfies all conditions of the Alexander-Conway polynomial, except that a priori it takes values in the field of rational functions in s instead of in its subring of Laurent polynomials Z[s, s−1]. The definition of the numerical invariant under loops inC extends to loops in C2(D) word for word and gives a homotopy invariant of loops inC2(D). It can be shown that the fundamental class of torus represents a non-trivial homology class in H.
Noodles vs. spanning arcs
- Noodles
- Algebraic intersection of noodles and arcs
Let ηi− be the loop inΣ obtained as the product of the pathθ−βi− with the path going from z−i to d− along N. Indeed, being a compact subset of D•, the disc B can contain only a finite number of points of the (discrete) set p−1(Q)⊂ D•. Then B projects injectively to D•, the loopρ bounds a small disk containing zi−, and the union of this disk with p(B) is a digon forN, α.
Proof of Theorem 3.15
- Homology classes associated with spanning arcs
- Surfaces associated with noodles For a noodle N on D, the set
- End of the proof
It is clear that the α class is determined only by α up to addition of elements of the image of the homomorphismH2(U;Z) → Hinduced by the inclusionU →C and up to multiplication by monomials inq, t (the latter is due to the indeterminacy in the choice of Sα). We now show that the set of α-classes is independent of the choice of disks U1. The refinement of this result to the Alexander-Conway polynomial (Section 3.3 and the second claim of Theorem 3.13) is due to V. The Lawrence-Krammer-Bigelow representation is part of a family of representations introduced by Lawrence [Law 90].
Symmetric Groups and Iwahori–Hecke Algebras
The symmetric groups
- A presentation of S n by generators and relations
- Proof of Theorem 4.1
- Reduced expressions and length of a permutation
- Inversions and the exchange theorem
- Equivalence of reduced expressions
Proceeding as above, we show that to prove (4.13), it is enough to prove the implication. To prove the theorem, we only need to check that ρ is well defined, i.e. if si1· · ·sik etcj1· · ·sjk are reduced expressions forw∈Sn, then. Before proving the assertion, let us show that this implies the lemma foru and v=u−1w0=si1· · ·sir.
The Iwahori–Hecke algebras
- Presentation by generators and relations
- The one-parameter Iwahori–Hecke algebras
- Consequences of Theorem 4.17
Many authors consider the one-parameter Iwahori–Hecke algebra HnR(q), which by definition is HnR(q, z) with z = q− 1. We now show that Hn is a free R-module in the basis indexed by the elements of the symmetric group Sn. We have used Lemma 4.18 for the second and sixth equalities and Lemma 4.19 for the first, third, fifth and seventh equalities, while the fourth equality comes from the relations relacionisjsi=sjsisj in Sn. c) Equations LiLj = LjLi for |i−j|.
The Ocneanu traces
Show that the function δsi∗δsi is zero outside B∪BsiB, and for any∈B,. g) Conclude that the algebraC(B\G/B) is isomorphic to the Iwahori–. i). The relationτn+1(ωTi) =τn+1(Tiω) follows from the induction hypothesis. where the third equality follows from the induction hypothesis. There are four cases to consider. b1) Ifandbelong toHn−1, then they commute with Tn and the connection is clear.
The Jones–Conway polynomial
The uniqueness of PL(x, y) is proved in the same way as the uniqueness of the Alexander–Conway polynomial in Theorem 3.13. In the literature it is also called the HOMFLY polynomial, the HOMFLY-PT polynomial or the two-variable Jones polynomial.
Semisimple algebras and modules
- Semisimple modules
- Modules over a simple algebra
- The radical of a finite-dimensional algebra
- Semisimple algebras
- A structure theorem for semisimple algebras
- Modules over a semisimple algebra
If the algebras{Aλ}λ∈Λ are finite-dimensional and the indexing setΛ is finite, then the algebra#. Similarly, an algebra is semisimple if it is finite-dimensional and its trace form is nondegenerate. Clearly, A is finite-dimensional if and only if all Aλ are finite-dimensional.
Semisimplicity of the Iwahori–Hecke algebras
