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Therefore u = log(w(1 − z)) and v = 2 log(z(1 − z)) can be regarded as the logarithms of the actions by the meridian μ and the longitude λ, respectively. Since the meridian and the longitude commute in π1 (S 3 \ E), their images can be simultaneously triangularizable. 1). This is a geometric interpretation of u and v. Since u determines z and w, it deﬁnes a hyperbolic structure of S 3 \ E as the union of Δ(z) and Δ(w). When u = 0 this hyperbolic structure is incomplete. We can complete this incomplete structure by attaching either a point or a circle.

8. In the case of a hyperbolic knot, we can also propose a similar conjecture with the imaginary part as in the case of the ﬁgure-eight knot. For a general knot, a relation to the Chern–Simons invariant is also expected by using a combinatorial description of the Chern–Simons invariant by Zickert [44]. 9. 7 is known to be true for the ﬁgure-eight knot [30] and for torus knots [26]. See also [27] and [10] for a possible relation to the Chern–Simons invariant. 7. 10 (S. Garoufalidis and T. Lˆe [5]).

2) = labelings ⎛ ⎠ (R±1 )ij kl crossings ⎞ 2 (a power of ξ ) × N N ⎝ ⎠, + (ξN )i± (ξN )k ± (ξN )j ∓ (ξN )l∓ (ξ ) N m crossings 24 HITOSHI MURAKAMI where the summation is over all the labelings with {0, 1, . . , N − 1} corresponding to the basis {e0 , e1 , . . , eN −1 } and for a ﬁxed labeling the product is over all the ij (or (R−1 )i,j crossings, each of which corresponds to an entry Rkl kl , respectively), determined by the four labeled arcs around the vertex, of the R-matrix (or its inverse, respectively).