Quandles and Monodromy

@article{Yetter2002QuandlesAM,
  title={Quandles and Monodromy},
  author={David N. Yetter},
  journal={Journal of Knot Theory and Its Ramifications},
  year={2002},
  volume={12},
  pages={523-541}
}
  • D. Yetter
  • Published 14 May 2002
  • Mathematics
  • Journal of Knot Theory and Its Ramifications
We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometry are most conveniently described in terms of homomorphisms from a(n augmented) knot quandle associated with the base to a suitable (augmented) quandle associated with the fiber. We consider the cases of the monodromy of a branched covering, braid monodromy and the monodromy of a Lefschetz fibration. The present paper is an expanded and corrected version of [1]. 
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References

SHOWING 1-10 OF 29 REFERENCES

Quandles and Lefschetz Fibrations

We show that isotopy classes of simple closed curves in any oriented surface admit a quandle structure with operations induced by Dehn twists, the Dehn quandle of the surface. We further show that…

Quandle Homology Groups, Their Betti Numbers, and Virtual Knots

A classifying invariant of knots, the knot quandle

LOOPS AND DISKS IN SURFACES AND HANDLEBODIES

We consider a groupoid structure involving graphs and imbedded circles (resp. disks) in a closed orientable surface F (resp. the handleboby H it bounds). We look at generalized connected sums and…

A simple presentation for the mapping class group of an orientable surface

LetFn.k be an orientable compact surface of genusn withk boundary components. For a suitable choice of 2n + 1 simple closed curves onFn,1 the corresponding Dehn twists generate bothMn,o andMn,1. A…

The second homology group of the mapping class group of an orientable surface

In I-7] Mumford shows that the Picard group P ic (~ ' ) is isomorphic to H2(F; 2~) and conjectures the latter is rank one, g>3 . We prove this below for g>5 . Another interpretation of this theorem…

Automorphisms of Surfaces after Nielsen and Thurston

Preface Introduction 1. The hyperbolic plane H2 2. Hyberbolic structures on surfaces 3. Geodesic laminations 4. Structure of geodesic laminations 5. Surface automorphisms 6. Pseudo-Anosov…

Quandle cohomology and state-sum invariants of knotted curves and surfaces

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation…

RACKS AND LINKS IN CODIMENSION TWO

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a…

4-manifolds and Kirby calculus

4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic…