Free Differential Calculus. I: Derivation in the Free Group Ring

Fox, Ralph H.

doi:10.2307/1969736

Cited by 377 publications

(260 citation statements)

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“…is the twisted Reidemeister torsion of Φ with parameter t. For actual computation we use the Fox free differential calculus again [6]. Consider the 3(n − 1) × 3(n − 1) matrix Φ ∂ri ∂xj (i ∈ {1, .…”

Section: Theorem 34 ([46 Theorem 312])mentioning

confidence: 99%

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

Murakami

2014

Acta Math Vietnam

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Abstract. A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern-Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums and each summand is related to the Chern-Simons invariant and the Reidemeister torsion associated with a representation.Let J N (K; q) ∈ Z[q, q −1 ] be the colored Jones polynomial of a knot K in the three-sphere S 3 associated with the irreducible N -dimensional representation of the Lie algebra sl 2 (C) [14,19]. We normalize it so that J N (unknot; q) = 1. Note that J 2 (K; q) is the original Jones polynomial. R. Kashaev conjectured [16] that his knot invariant K N ∈ C introduced in [15] would grow exponentially with growth rate the volume of the knot complement S 3 \ K when the integer parameter N goes to the infinity if the knot K is hyperbolic. J. Murakami and the author [34] proved that Kashaev's invariant coincides with J N K; exp(2π √ −1/N ) and generalized Kashaev's conjecture to general knots.

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Section: Theorem 34 ([46 Theorem 312])mentioning

confidence: 99%

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

Murakami

2014

Acta Math Vietnam

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“…Fox showed that the partial derivations form a basis of the module of derivations [3]. That is, every derivation D : ZF s → ZF s is given by…”

Section: Tietze Transformationsmentioning

confidence: 99%

Twisted linking numbers via representations of fundamental groups

Kawagoe¹

2002

Pacific J. Math.

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“…We call them Fox derivatives in honor of R. Fox who considered them in a free group ring [5]. There is the augmentation homomorphism ε : A → K defined by ε(x i ) = 0, 1 ≤ i ≤ n. Its kernel ∆ is called the augmentation ideal of A; it is a free left A-module with a free basis X, so that every element u ∈ ∆ can be uniquely written in the form u = One can extend these derivations linearly to the whole A defining d i (1) = 0.…”

Section: Preliminariesmentioning

confidence: 99%