A relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion

Abstract

Abstract. We show a relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion for a λ-regular SU(2) or SL(2, C)-representation of a knot group. Then we give a method to calculate the non-acyclic Reidemeister torsion of a knot exterior. We calculate a new example and investigate the behavior of the non-acyclic Reidemeister torsion associated to a 2-bridge knot and SU(2)-representations of its knot group.

The quantum content of the gluing equations

“…In general there is some freedom in choosing them, but the torsion depends in a predictable way on the choice, cf. [Yam08].…”

“…Recall that although S 2,9 12 is defined modulo an integer multiple of 1/24, S 3,9 12 is defined without ambiguity and the numerator N 3 is a prime number of 103 digits. For a computation of the Reidemeister torsion τ R M of the discrete faithful representation of a cusped hyperbolic manifold M, we use a theorem of Yamaguchi [Yam08] to identify it with…”

Limit values of the non-acyclic Reidemeister torsion for knots

“…/ as the differential coefficient of the sign-determined Reidemeister torsion of C .M K I e sl 2 ‫/ރ.‬ / as follows (see Theorem 3.1.2 in Yamaguchi [22]):…”

“…Since we suppose that is -regular, we know that .t 1/ 2 divides det A 1 K ;Ad ı (see Section 3.3 in Yamaguchi [22]). …”

“…In fact, this function is expressed by using the Alexander polynomial of K and it has zeros at bifurcation points (see Milnor [15,17] and Turaev [20]). The other function on the non-abelian part is given by the non-acyclic Reidemeister torsion for non-abelian representations, Dubois [6,7], Porti [19] and Yamaguchi [23] deal with Reidemeister torsion in such a light. Though the function on the non-abelian part is partially defined and it is not defined on bifurcation points, we can consider limits of the non-acyclic Reidemeister torsion at bifurcation points.…”