Alphabeta MathBraid Groups
Braid groups are built from configuration spaces as well as presentations. The published pages compare ordered and unordered configurations, derive pure braid groups and Fadell-Neuwirth sequences, study punctured-disc mapping classes, and recover Artin presentations, braid combing, Garside normal forms, and the center. They also develop the Artin action on free groups, Burau representations, and Lawrence-Krammer-Bigelow linearity. Further pages connect braids to link closures, Markov equivalence, Hecke-Markov traces, and polynomial link invariants. A categorical branch treats braid actions through graded quiver algebras, derived tensor functors, Soergel bimodules, Rouquier complexes, matrix factorizations, Hochschild homology, and Yang-Baxter operators.
Topology and Algebraic Topology provide homotopy, covering-space, and configuration-space tools; Abstract Algebra and Category Theory support presentations and monoidal constructions. Categorification and Geometric Representation Theory draws on the Soergel material, braid actions, categorical braid complexes, and diagram constructions. Quantum Algebra and Representation Theory and Knot Theory also use the collection's representation and closure results.
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
4 pagesThis page develops the Garside structure of the Artin braid group from its positive part.
4 definitions, 12 lemmas, 1 proposition, 6 theorems, 1 corollaryExamples & counterexamples- Geometric Braids and Artin Generators10 results
This page sets up geometric braids on n strands as the level-preserving object: with h=1/4(n+1)∈(0,1/2) and the base configuration q j=((2j-n-1)h,0) on the horizontal…
3 definitions, 4 lemmas, 2 propositions, 1 theoremExamples & counterexamples Paths compose left to right, so the projective P i=A me i is spanned by the paths that end at the vertex i while iP=e iA m is spanned by those that begin there, and the…
10 definitions, 6 lemmas, 2 theoremsExamples & counterexamplesDeleted collision diagonals are what make the coordinate permutation action free, and the orbit quotient of that action is the unordered configuration space: the ordered…
5 definitions, 5 lemmas, 1 proposition, 3 theoremsExamples & counterexamples
Level 1
2 pagesThis page identifies geometric braids, defined as level-preserving motions of the fixed base tuple Q=(q 1,…,q n) in the open unit disk, with loops in the unordered configuration space.
1 definition, 5 lemmas, 1 proposition, 2 theorems, 1 corollaryExamples & counterexamplesThe type-A Soergel page builds the categorification of the Hecke algebra H S n in Soergel's normalization, where A=ℤ[v,v⁻¹], q=v⁻², (T i-q)(T i+1)=0 and H i=v(T i+1): the…
9 definitions, 14 lemmas, 9 theoremsExamples & counterexamples
Level 2
1 pageThis page identifies the braid group on n strands with the mapping class group of the disk with n marked points, and defines point pushing as the map that drags a puncture along a loop of the punctured surface.
4 definitions, 6 lemmas, 3 theorems, 1 corollaryExamples & counterexamples
Level 3
1 pageThis page develops the Fadell–Neuwirth account of the pure braid group: the coordinate-forgetting maps of the ordered planar configuration spaces are fibrations whose exact…
1 definition, 4 lemmas, 5 theorems, 2 corollariesExamples & counterexamples
Level 4
1 pageThis page proves that the Artin presentation is not merely a presentation that surjects onto the geometric braid group: the surjection φ n: B n^Artin→ B nᵍᵉᵒᵐ that sends each…
1 definition, 5 lemmas, 1 theorem, 1 corollaryExamples & counterexamples
Level 5
1 page- The Artin Action on a Free Group24 results
This page constructs the Artin representation ρ:B n toAut(F n), proves faithfulness under AC, and characterizes its image.
4 definitions, 14 lemmas, 1 proposition, 4 theorems, 1 corollaryExamples & counterexamples
Level 6
3 pagesThis page builds the Lawrence–Krammer–Bigelow (LKB) representation of the classical braid group and proves that it is faithful, so that every braid group is linear.
8 definitions, 16 lemmas, 2 theorems, 1 corollaryExamples & counterexamplesThis page builds the bridge between links in the three-sphere and braids.
10 definitions, 26 lemmas, 3 theoremsExamples & counterexamples- The Burau Representations21 results
This page constructs the reduced and unreduced Burau representations of the braid group topologically, identifies them with their classical matrices over the Laurent…
7 definitions, 10 lemmas, 2 propositions, 2 theoremsExamples & counterexamples
Level 7
3 pagesThis page builds the Khovanov–Seidel categorical braid action for the type A algebras A m and compares it with the unreduced Burau representation.
10 definitions, 16 lemmas, 1 proposition, 3 theoremsExamples & counterexamplesAssuming AC for the cited topological suppliers, this page builds the classical route from the braid group to polynomial link invariants.
9 definitions, 8 lemmas, 1 proposition, 5 theoremsExamples & counterexamplesThis page develops the categorical source of braid-group representations.
7 definitions, 4 lemmas, 1 proposition, 5 theorems, 1 corollaryExamples & counterexamples
Level 8
1 pageThis page builds Rouquier's 2-braid group for type A inside the homotopy category of bounded complexes of graded (R,R)-bimodules, starting from the type-A Soergel bimodules B i=R⊗ R^s iR(1) of the Soergel page.
4 definitions, 7 lemmas, 1 proposition, 3 theoremsExamples & counterexamples
Level 9
1 pageThis page develops the matrix-factorization construction of Khovanov–Rozansky link homology.
7 definitions, 3 lemmas, 5 theoremsExamples & counterexamples
Level 10
1 pageThis page develops Khovanov's realization of triply-graded link homology as Hochschild homology of Soergel bimodules.
5 definitions, 5 lemmas, 1 theorem, 2 corollariesExamples & counterexamples