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The A ‐Polynomial has Ones in the Corners
Author(s) -
Cooper D.,
Long D. D.
Publication year - 1997
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609396002251
Subject(s) - mathematics , affine transformation , polynomial , algebraic number , combinatorics , eigenvalues and eigenvectors , mathematics subject classification , affine space , boundary (topology) , manifold (fluid mechanics) , pure mathematics , discrete mathematics , mathematical analysis , mechanical engineering , physics , quantum mechanics , engineering
1. Definition of the A‐polynomial The A ‐polynomial was introduced in [ 3 ] (see also [ 5 ]), and we present an alternative definition here. Let M be a compact 3‐manifold with boundary a torus T . Pick a basis λ, μ of π 1 T , which we shall refer to as the longitude and meridian. Consider the subset R U of the affine algebraic variety R = Hom (π 1 M , SL 2 C) having the property that ρ(λ) and ρ(μ) are upper triangular. This is an algebraic subset of R , since one just adds equations stating that the bottom‐left entries in certain matrices are zero. There is a well‐defined eigenvalue map ξ ≡ ( ξ λ × ξ μ ) : R U → ℂ 2given by taking the top‐left entries of ρ(λ) and ρ(μ). 1991 Mathematics Subject Classification 57M25, 57M50.