TY - JOUR

T1 - Generalized Gaussian sums Chern-Simons-Witten-Jones invariants of len-spaces

AU - Li, Bang He

AU - Li, Tian Jun

PY - 1996/4

Y1 - 1996/4

N2 - Starting from evaluating all Gaussian sums, we calculate {τγ, γ ≥ 2} (in the 4r-th cyclotomic fields) and {ξγ, γ odd ≥ 3} (in the r-th cyclotomic fields) for all lens spaces L(p, q). We prove that they are all algebraic integers and show that ξγ determines the Dedekind sum s(q, p), and hence determines the generalized Casson invariant of lens space. We conjecture these two properties hold for more general 3-manifold, and some evidences are discussed. Though the formulae are not simple, we are able to give necessary and sufficient conditions for lens spaces to have the same CSWJ invariants. Examples of lens spaces with the same invariants but different topological types are given. Some applications in number theory are also included.

AB - Starting from evaluating all Gaussian sums, we calculate {τγ, γ ≥ 2} (in the 4r-th cyclotomic fields) and {ξγ, γ odd ≥ 3} (in the r-th cyclotomic fields) for all lens spaces L(p, q). We prove that they are all algebraic integers and show that ξγ determines the Dedekind sum s(q, p), and hence determines the generalized Casson invariant of lens space. We conjecture these two properties hold for more general 3-manifold, and some evidences are discussed. Though the formulae are not simple, we are able to give necessary and sufficient conditions for lens spaces to have the same CSWJ invariants. Examples of lens spaces with the same invariants but different topological types are given. Some applications in number theory are also included.

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U2 - 10.1142/S0218216596000151

DO - 10.1142/S0218216596000151

M3 - Article

AN - SCOPUS:0030557550

SN - 0218-2165

VL - 5

SP - 183

EP - 224

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

IS - 2

ER -