ABSTRACT
For any Lie algebra, https://www.w3.org/1998/Math/MathML"> g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq2482.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and integral level, k, there is defined an invariant, https://www.w3.org/1998/Math/MathML"> Z k * ( M , L ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq2483.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , of embeddings of links L in 3-manifolds M, known as the Witten-Reshetikhin-Turaev invariant. When https://www.w3.org/1998/Math/MathML"> g = s l 2 , Z k * ( S 3 , L ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq2484.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a polynomial in https://www.w3.org/1998/Math/MathML"> q = exp 2 π i ( k + 2 ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq2485.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> namely the generalised Jones polynomial of the link L. This paper discusses the invariant https://www.w3.org/1998/Math/MathML"> Z r − 2 * ( M , ∅ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq2486.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> when g = sl2 for a simple family of rational homology 3-spheres, Mn,t , obtained by integer surgery around (2,n) type torus knots. In earlier work of the author it was shown that there is an associated holomorphic function Z∞(Mn,t ) of In g ∈ ‒\iℝ, related to Ohtsuki’s invariants, from which https://www.w3.org/1998/Math/MathML"> Z r − 2 * ( M n , t , ∅ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq2487.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> may be derived for all sufficiently large primes r. The current paper extends the results to prime powers and odd composite numbers coprime to |H1(Mn,t)|, showing how the invariant https://www.w3.org/1998/Math/MathML"> Z r − 2 * ( M n , t , ∅ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq2488.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> may be extracted from Z∞(Mn,t ).
