Eigenvalue varieties of Brunnian links
| Autor: | François Malabre |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Fundamental group
link Mathematics::General Topology Geometric Topology (math.GT) Torus A-polynomial Brunnian link Mathematics::Geometric Topology eigenvalue variety Combinatorics Mathematics - Geometric Topology Knot (unit) 57M27 knot 57M25 FOS: Mathematics Geometry and Topology Peripheral subgroup Locus (mathematics) Unknot Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Algebr. Geom. Topol. 17, no. 4 (2017), 2039-2050 |
| ISSN: | 1472-2739 1472-2747 |
| DOI: | 10.2140/agt.2017.17.2039 |
| Popis: | In this article, it is proved that the eigenvalue variety of t he exterior of a nontrivial, non-Hopf, Brunnian link in S 3 contains a nontrivial component of maximal dimension. The eigenvalue variety was first introduced in [ 12] to generalize the Apolynomial of knots in S 3 to manifolds with nonconnected toric boundary. The result presented here generalizes, for Brunnian links, the result s of [1] and [4], where it is proved that nontrivial knots in S 3 have a nontrivial A-polynomial. The A-polynomial of a knot in S 3 is a two-variable polynomial constructed from the SL2C-character variety of the knot exterior. Let K be a knot in S 3 and let π1K denote the fundamental group of the exterior of K; the peripheral subgroup Z 2 is generated by a meridian µ and a longitude λ and the zero-set of the A-polynomial AK is the locus of eigenvalues for a common eigenvector of ρ(µ) and ρ(λ) of representations ρ from π1K to SL2C. It was first introduced by Daryl Cooper, Marc Culler, Henri G illet, Darren Long and Peter Shalen in [2], where it is also proved that the A-polynomial of any knot contains the A-polynomial of the unknot as a factor. The A-polynomial of a knot is said to be nontrivial if it contains other factors and it was also proved, in the sam e [2], that hyperbolic knots and non-trivial torus knots always have a non-trivial A-polynomial. This was later established in full generality for all non-trivial knots by Nathan Dunfie ld and Stavros Garoufalidis in [4], and independently by Steve Boyer and Xingru Zhang in [1]; both proofs use a theorem by Peter Kronheimer and Tomasz Mrowka in [5] on Dehn-fillings on knots and representations in SU2. The notion of A-polynomial can be generalized to any 3-manifold M with connected toric boundary by specifying a peripheral system (generators of π1∂M → π1M ). Stimulated by the work of Alan Lash in [6], it was then extended to manifolds with nonconnected boundary by Stephan Tillmann. In his PhD thesis [11] and the subsequent article [12], Tillmann presented the eigenvalue varietyE(M) associated to a 3-manifold M with toric boundary. If the boundary of M consists in n tori, the associated eigenvalue |
| Databáze: | OpenAIRE |
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