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We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Oh-tsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev's quantum dilogarithm invariants for links. Therefore Kashaev's conjecture can be restated as follows: The… (More)

- HITOSHI MURAKAMI
- 2008

We clarify and refine the relation between the asymptotic behavior of the colored Jones polynomial and Chern-Simons gauge theory with complex gauge group SL(2, C). The precise comparison requires a careful understanding of some delicate issues, such as normalization of the colored Jones polynomial and the choice of polarization in Chern-Simons theory.… (More)

- Alan Edelman, H Murakami
- 1995

In classical linear algebra, the eigenvalues of a matrix are sometimes deened as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual deenition. We derive a rst order error analysis of this algorithm that sheds light on… (More)

- HITOSHI MURAKAMI
- 2004

We study the asymptotic behaviors of the colored Jones polyno-mials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern–Simons invariants of the three-manifolds obtained by Dehn surgeries. On the other hand it is proved that in some cases the limits give the inverse… (More)

We define A k-moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let A k-equivalence denote an equivalence relation generated by A k-moves and ambient isotopy. A k-equivalence implies A k−1-equivalence. Let F be an A k−1-equivalence class of the embeddings of a finite graph into the 3-sphere. Let G be the quotient set of F… (More)

BACKGROUND
Refusal of the oral polio vaccine (OPV) is a difficulty faced by the Polio Eradication Initiative (PEI) in multiple endemic areas, including the Khyber Pakhtunkhwa Province (KPP), Pakistan. In 2007, we investigated community perceptions of the OPV and estimated the prevalence of OPV refusal in three districts in Swat Valley, KPP, a polio-endemic… (More)

R.M. Kashaev conjectured that the asymptotic behavior of his link invariant, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots 6 3 , 8 9 and 8 20 and for the Whitehead link, the colored Jones polynomials are related to the… (More)

For a knot K in the 3-sphere, by using the linking form on the first homology group of the double branched cover of the 3-sphere, we investigate some numerical invariants, 4-genus g * (K), nonorientable 4-genus γ * (K) and 4-dimensional clasp number c * (K), defined from the four-dimensional viewpoint. T. Shibuya gave an inequality g * (K) ≤ c * (K), and… (More)

We calculate limits of the colored Jones polynomials of the figure-eight knot and conclude that in most cases they determine the volumes and the Chern–Simons invariants of the three-manifolds obtained by Dehn surgeries along it.

We define a filtration on the vector space spanned by Seifert matrices of knots related to Vassiliev's filtration on the space of knots. Further we show that the invariants of knots derived from the filtration can be expressed by coefficients of the Alexander polynomial. The theory of finite type invariants (Vassiliev invariants) for knots was first… (More)