On braid groups

Birman, Joan S.

doi:10.1002/cpa.3160220104

Cited by 157 publications

(191 citation statements)

References 10 publications

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“…Also, let x, y and z be the number of δ i 's that map to (1, 2), (2, 3) and (1, 3), respectively. Note that the only elements ψ = 1 ∈ S 3 that satisfy ψ −1 (1, 2, 3)ψ = (1, 2, 3) are the 3-cycles (1, 2, 3) and (1,3,2). Since conjugation by 3-cycles is transitive on the set of 2-cycles, our desired inequalities follow from a simple counting argument.…”

Section: Remark 17mentioning

confidence: 95%

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Non-isotopic symplectic surfaces in product 4-manifolds

Hays

Park

2008

Trans. Amer. Math. Soc.

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Section: Remark 17mentioning

confidence: 95%

“…Let Br p (Σ h ) denote the braid group on p strings in the Riemann surface Σ h . We refer the reader to [1] or [2] for the definition of Br p (Σ h ). (Zariski seems to be the first to have studied this braid group.…”

Section: Generalization Of Smith's Constructionmentioning

confidence: 99%

Non-isotopic symplectic surfaces in product 4-manifolds

Hays

Park

2008

Trans. Amer. Math. Soc.

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“…For example, the figure-eight knot b(5/2), which has Conway parameters [2,2], is the knot K ψ with ψ = t β t α t β τ −1 m t ε (see Figure 10). …”

Section: Representation Of Two-bridge Knotsmentioning

confidence: 99%

(1, 1)-knots via the mapping class group of the twice punctured torus

2004

Advg

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We develop an algebraic representation for (1, 1)-knots using the mapping class group of the twice punctured torus M CG 2 (T ). We prove that every (1, 1)-knot in a lens space L(p, q) can be represented by the composition of an element of a certain rank two free subgroup of M CG 2 (T ) with a standard element only depending on the ambient space. As notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of type (k, ck + 2).

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“…Later Fadell and Van Buskirk computed B n (S 2 ) [19], Van Buskirk computed B n (RP 2 ) [20], Birman computed B n (T 2 ) [18], and finally Scott computed B n (X) for X any compact 2-manifold [21]. For example, B n (S 2 ) is the quotient of B n by the additional relation…”

Section: Spin Statistics and Framed Braidsmentioning

confidence: 99%

Topological aspects of spin and statistics in nonlinear sigma models

Baez

Ody

Richter

1995

Journal of Mathematical Physics

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We study the purely topological restrictions on allowed spin and statistics of topological solitons in nonlinear sigma models. Taking as space the connected d-manifold X, and considering nonlinear sigma models with the connected manifold M as target space, topological solitons are given by elements of π d (M ). Any topological soliton α ∈ π d (M ) determines a quotient Stat n (X, α) of the group of framed braids on X, such that choices of allowed statistics for solitons of type α are given by unitary representations of Stat n (X, α) when n solitons are present. In particular, when M = S 2 , as in the O(3) nonlinear sigma model with Hopf term, and α ∈ π 2 (S 2 ) is a generator, we compute that Stat n (R 2 , α) = Z, while Stat n (S 2 , α) = Z 2n . It follows that phase exp(iθ) for interchanging two solitons of type α on S 2 must satisfy the constraint θ = kπ/n, k ∈ Z, when n such solitons are present.

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On braid groups

Cited by 157 publications

References 10 publications

Non-isotopic symplectic surfaces in product 4-manifolds

Non-isotopic symplectic surfaces in product 4-manifolds

(1, 1)-knots via the mapping class group of the twice punctured torus

Topological aspects of spin and statistics in nonlinear sigma models

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