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

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Venue: | in: Proceedings of the CassonFest |

Citations: | 17 - 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}

}

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### 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... |

403 |
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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 ...al of a knot. Recall the definition of the deformation variery of a knot from Section 1.3. Since projection of affine algebraic varieties corresponds to elimination in their corresponding ideals (see =-=[CLO]-=-), it is clear that the deformation variety of a knot can in principle be computed via elimination. In fact, according to [CCGLS], the deformation variety D(K) of a knot K is essentially equal to a co... |

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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 ... algorithm. 1sof a knot K in 3-space is a sequence of Laurent polynomials, whose nth term JK(n) is the Jones polynomial of a knot colored with the n-dimensional irreducible representation of sl2; see =-=[Tu]-=-. We will normalize it by Junknot(n) = 1 for all n, and (for those who worry about framings), we will assume that K is zero-framed. The first two terms of the colored Jones function of a knot K are be... |

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Citation Context ...f the A-polynomial and the colored Jones polynomial 5s3. Proof of the conjecture for the Trefoil and Figure 8 knots 3.1. The colored Jones function and the A-polynomial of the 31 and 41 knots. Habiro =-=[H]-=- and Le give the following formula for the colored Jones function of the left handed trefoil (31) and Figure 8 (41) knots: (4) (5) J31(n) = J41(n) = ∞� k=0 ∞� k=0 (−1) k q k(k+3)/2 q nk (q −n−1 ; q −1... |

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Citation Context ...erms of powers of q n and q − 1, [G]. The physical meaning of this expansion is, according to Rozansky, a Feynman diagram expansion around a U(1)-connection in the knot complement with holonomy q n , =-=[R]-=-. 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... |

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Citation Context ...milar to the above conjecture and its polynomial version (Conjecture 2 below) were also raised by Frohman and Gelca who studied the colored Jones function of a knot via Kauffman bracket skein theory, =-=[Ge]-=-. Our approach to recursion relations in [GL] and here is via statistical mechanics sums and holonomic functions. A modest corollary of the above conjecture is the following: Corollary 1.3. If a knot ... |

15 |
The colored Jones function is q-holonomic, preprint math.GT/0309214
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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 ...boundary of the knot complement. Furthermore, the polynomial that defines the deformation variety can compute the variation of the volume function; see Cooper et all [CCGLS, Sec.4.5] and also Yoshida =-=[Y]-=- and Neumann-Zagier [NZ, eqn (47)]. Remark 1.8. Conjecture 1 reveals a close relation between the colored Jones function of a knot and its deformation variety. It does not explain though why we ought ... |

8 |
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(Show Context)
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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Citation Context ...mension of the vector space Fm/(Fm ∩ I) ⊗¡ [q ± ]s(q) (over the fields(q)) is a polynomial in m of degree equal (by definition) to d(M). Bernstein’s famous inequality (proved by Sabbah in the q-case, =-=[Sa]-=-) states that d(M) ≥ r, if M �= 0 and M has no monomial torsions, i.e., any non-trivial element of M cannot be annihilated by a monomial in Qi, Ei. Note that the left Ar module Mf := Ar · f ∼ = Ar/If ... |

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1 |
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Citation Context ...For a thorough discussion of this matter, see [PWZ, p.164]. In other words, P need not equal to Aq(K). The problem of computing right factors of an operator has been solved in theory by Petkovˇsek in =-=[BP]-=-. A computer implementation of this solution is not available at present. In case we are looking for right factors of degree 1 (this is equivalent to deciding whether a discrete function has closed fo... |

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