Published papers

In this paper we prove that every coefficient of twisted Alexander polynomials of torus knots associated with irreducible SLn(C)-representations is an A-valued locally constant function on the SLn(C)-character variety, where A is the ring of all algebraic integers over C. Moreover, as a generalization of a recent result of Kitano and Nozaki, we show that SLn(C)-Reidemeister torsions are algebraic integers for many Seifert fibered spaces. Also, we discuss the power sums of Reidemeister torsions of torus knots for low-dimensional irreducible representations that provide a mysterious relation to TQFT.

In this paper, we provide an explicit formula for the twisted Alexander polynomial of a once-punctured torus bundle over the circle with tunnel number one associated with a curve of irreducible representations into GL4(C). Consequently, we can derive the Reidemeister torsion on the curve by using this formula.

Mangum and Shanahan construct a complex curve of irreducible SL3 (C)-representations of the fundamental group of a once-punctured torus bundle over the circle. In this paper, we provide an explicit formula for the twisted Alexander polynomial of a once-punctured torus bundle with tunnel number one associated with the Mangum-Shanahan representations. As a corollary, we exhibit the interesting phenomenon that the Reidemeister torsion of the complement of the figure-eight knot is constant on the curve.

In this paper, we provide a volume presentation of a hyperbolic fibered knot using the Bell polynomials. More precisely, we show that the hyperbolic volume of a fibered knot can be expressed in terms of the traces of powers of a monodromy matrix. We also show that the hyperbolic volume of the figure-eight knot is obtained by the asymptotics of a positive integer sequence consisting of the special values of the twisted Alexander invariants.

Dunfield, Friedl and Jackson make a conjecture that the hyperbolic torsion polynomial determines the genus and fibering of hyperbolic knots. In this paper, we study a similar problem for the adjoint torsion polynomial, and show that it determines the genus and fibering of a large family of hyperbolic genus one two-bridge knots.

A theorem of Friedl and Vidussi says that any 3-manifold N and any non-fibered class in H1(N;Z) there exists a representation such that the corresponding twisted Alexander polynomial is zero. However, it seems that no concrete example of such a representation is known so far. In this paper, we provide several explicit examples of non-fibered knots and their representations with zero twisted Alexander polynomial.

A Hurwitz group is a conformal automorphism group of a compact Riemann surface with precisely 84(g − 1) automorphisms, where g is the genus of the surface. Our starting point is a result on the smallest Hurwitz group PSL(2, F7) which is the automorphism group of the Klein surface. In this paper, we generalize it to various classes of simple Hurwitz groups and discuss a relationship between the surface symmetry and spectral asymmetry for compact Riemann surfaces. To be more precise, we show that the reducibility of an element of a simple Hurwitz group is equivalent to the vanishing of the η-invariant of the corresponding mapping torus. Several wide classes of simple Hurwitz groups which include the alternating group, the Chevalley group and the Monster, which is the largest sporadic simple group, satisfy our main theorem.

In this paper, we discuss a relationship between the chirality of knots and higher-dimensional twisted Alexander polynomials associated with holonomy representations of hyperbolic 3-cone-manifolds. In particular, we provide a new necessary condition for a knot, that appears in a hyperbolic 3-cone-manifold of finite volume as a singular set, to be amphicheiral. Moreover, we can detect the chirality of hyperbolic twist knots, according to our criterion, using low-dimensional irreducible representations.

In this paper, we explicitly calculate the highest degree term of the hyperbolic torsion polynomial of an infinite family of pretzel knots. This gives supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. The verification of the genus part of the conjecture for this family of knots also follows from the work of Agol and Dunfield [1] or Porti [19].

© 2019 International Academic Printing Co. Ltd.. All rights reserved. In this short note, we calculate the highest degree term of the hyperbolic torsion polynomial of a pretzel knot with three tangles. It gives a supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot.