## On the characteristic and deformation varieties of a knot (2004)

Venue: | in: Proceedings of the CassonFest |

Citations: | 13 - 5 self |

### BibTeX

@INPROCEEDINGS{Garoufalidis04onthe,

author = {Stavros Garoufalidis},

title = {On the characteristic and deformation varieties of a knot},

booktitle = {in: Proceedings of the CassonFest},

year = {2004},

pages = {291--309}

}

### OpenURL

### Abstract

Abstract. The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose nth term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation of sl2. It was recently shown by TTQ Le and the author that the colored Jones function of a knot is q-holonomic, i.e., that it satisfies a nontrivial linear recursion relation with appropriate coefficients. Using holonomicity, we introduce a geometric invariant of a knot: the characteristic variety, an affine 1-dimensional variety in 2. We then compare it with the character variety of SL2 ( ) representations, viewed from the boundary. The comparison is stated as a conjecture which we verify (by a direct computation) in the case of the trefoil and figure eight knots. We also propose a geometric relation between the peripheral subgroup of the knot group, and basic operators that act on the colored Jones function. We also define a noncommutative version (the so-called noncommutative A-polynomial) of the characteristic variety of a knot. Holonomicity works well for higher rank groups and goes beyond hyperbolic geometry, as we explain in the last chapter. Contents

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Citation Context ... Thus, it is not surprising that M 2 = Q. It is more surprising that the longitude variable L corresponds to the shift operator E. This can be explained in the following way. According to Witten (see =-=[Wi]-=-), the Jones polynomial JK(n) of a knot K is the average over an infinite dimensional space of connections, of the trace of the holonomy around K, where the trace is computed in the n-dimensional repr... |

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Citation Context ...boundary torus. The deformation variety of a knot is of fundamental importance to hyperbolic geometry, and to geometrization, and was studied extensively by Cooper et al and Thurston; see [CCGLS] and =-=[Th]-=-. Given a knot K in S 3 , consider the complement M = S 3 − nbd(K) (a 3-manifold with torus boundary ∂M ∼ = T 2 ), and the set R(M) = Hom(π1(M), SL2(¡ )) of representations of π1(M) into SL2(¡ ). This... |

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Citation Context ...hat K is zero-framed. The first two terms of the colored Jones function of a knot K are better known. Indeed, JK(1) = 1, and JK(2) coincides with the Jones polynomial of a knot K, defined by Jones in =-=[J]-=-. Although we will not use it, note that the colored Jones function of a knot essentially encodes the Jones polynomial of a knot and its connected parallels. The starting point for our paper is the ke... |

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Citation Context ...r, and from this it follows that the Hilbert dimension of (Ar ⊗s(q))/I for generic q ∈ ¡ is r. Since dimension is upper semicontinuous and it coincides with the Hilbert dimension at the generic point =-=[S]-=-, the result follows. � Definition 4.4. If K is a knot in S 3 , and G as above, we define its G-characteristic variety VG(K) ⊂ ¡ 2r to be the characteristic variery of its g-colored Jones function. Si... |

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Citation Context ...tion to the boundary torus. The deformation variety of a knot is of fundamental importance to hyperbolic geometry, and to geometrization, and was studied extensively by Cooper et al and Thurston; see =-=[CCGLS]-=- and [Th]. Given a knot K in S 3 , consider the complement M = S 3 − nbd(K) (a 3-manifold with torus boundary ∂M ∼ = T 2 ), and the set R(M) = Hom(π1(M), SL2(¡ )) of representations of π1(M) into SL2(... |

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Citation Context ...ones polynomial of a knot and its connected parallels. The starting point for our paper is the key property that the colored Jones function is q-holonomic, as was shown in joint work with TTQ Le; see =-=[GL]-=-. Informally, a q-holonomic function is one that satisfies a nontrivial linear recursion relation, with appropriate coefficients. A convenient way to describe recursion relations is the operator point... |

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Citation Context ...r right factors of degree 1 (this is equivalent to deciding whether a discrete function has closed form), there is an algorithm qHyper of Petkovˇsek which decides about this problem in real time; see =-=[PWZ]-=-. In the special examples that we will consider, namely the colored Jones function of 31 and 41 knots, we can bypass the thorny issue of right factorization of an operator. 1 AJ are the initials of th... |

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