Braids, Links, and Mapping Class Groups
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A01=Joan S. Birman
Alexander polynomial
Algebraic structure
Author_Joan S. Birman
Automorphism
Braid group
Braid theory
Burau representation
Calculation
Category=PBF
Category=PBV
Characterization (mathematics)
Combinatorial group theory
Commutative property
Commutator subgroup
Configuration space
Conjugacy class
Corollary
Dehn twist
Determinant
Diagram (category theory)
Disjoint union
Double coset
Eigenvalues and eigenvectors
Enumeration
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Equation
Equivalence class
Exact sequence
Existential quantification
Free abelian group
Free group
Fundamental group
Group (mathematics)
Group ring
Groupoid
Heegaard splitting
Homeomorphism
Homomorphism
Homotopy
Homotopy group
Identity element
Identity matrix
Inclusion map
Knot polynomial
Knot theory
Line segment
Line-line intersection
Link group
Mapping class group
Mathematical induction
Matrix group
Notation
Orientability
Parity (mathematics)
Permutation
Polynomial
Prime knot
Projection (mathematics)
Proportionality (mathematics)
Quotient group
Riemann surface
Semigroup
Sequence
Special case
Subgroup
Submanifold
Subset
Symmetric group
Theorem
Topology
Trefoil knot
Unit vector
Variable (mathematics)
Word problem (mathematics)
Product details
- ISBN 9780691081496
- Weight: 312g
- Dimensions: 152 x 229mm
- Publication Date: 21 Feb 1975
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Paperback
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The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems. Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.
