Takayuki Morifuji

Twisted Alexander polynomials of 2-bridge knots for parabolic representations

In this paper we show that the twisted Alexander polynomial associated to a parabolic representation determines fiberedness and genus of a wide class of 2-bridge knots. As a corollary we give an affirmative answer to a conjecture of Dunfield, Friedl and Jackson for infinitely many hyperbolic knots.

TWISTED ALEXANDER POLYNOMIALS AND CHARACTER VARIETIES OF 2-BRIDGE KNOT GROUPS

We study the twisted Alexander polynomial from the viewpoint of the SL(2,C)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL(2,C)-representations are all monic. In this paper, we show that the converse holds for 2-bridge knots. Furthermore we show that for a 2-bridge knot there exists a curve component in the SL(2,C)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexander polynomials are monic. We also show that for a 2-bridge knot of genus g, in the above curve component for all but finitely many characters the associated twisted Alexander polynomials have degree 4g-2.

The η-invariant of mapping tori with finite monodromies

Abstract The η-invariant of Riemannian 3-manifolds is defined by means of the spectrum of a certain elliptic operator. In this paper, we give a geometric interpretation of the deviation from the multiplicativity of the η-invariant for finite coverings. We then apply it to mapping tori with finite monodromies, and obtain a simple formula of the η-invariant for it.

Twisted Alexander polynomial for the braid group

In this paper, we study the twisted Alexander polynomial, due to Wada [ 11 ], for the Jones representations [ 6 ] of Artins braid group.

TWISTED ALEXANDER POLYNOMIALS ON CURVES IN CHARACTER VARIETIES OF KNOT GROUPS

For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2,C)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2,C)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.

A note on the reducibility of automorphisms of the Klein curve and the

We give a characterization for the reducibility of automorphisms of the genus 3 Klein curve in terms of the η-invariant of finite order mapping tori.

\eta

We give a characterization for the reducibility of elements of any finite subgroup of the mapping class group of genus 2 surface in terms of the η-invariant of finite order mapping tori.

-invariant of mapping tori

We give a characterisation for the vanishing of the η-invariant of prime order automorphisms of hyperelliptic Riemann surfaces through the mapping torus construction. To this end, we introduce a notion of s-symmetry for finite order surface automorphisms.

On the reducibility and the η-invariant of periodic automorphisms of genus 2 surface

In this paper, we discuss a relationship between the surface symmetry and the spectral asymmetry. More precisely we show that an automorphism of the Macbeath surface of genus 7, or one of the three Hurwitz surfaces of genus 14 is reducible if and only if the η-invariant of the corresponding mapping torus vanishes.

A remark on the η-invariant for automorphisms of hyperelliptic Riemann surfaces

In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2-bridge links. Moreover we consider a similar problem for parabolic representations of 2-bridge link groups.