Narrow Results

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 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.

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 theorem of Friedl and Vidussi says that any 3-manifold $N$ and any non-fibered class in $H^1(N;\mathbb{Z})$ 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.

The derived group of a permutation representation, introduced by Crowell, unites many notions of knot theory. We survey Crowell's construction, and offer new applications.

The twisted Alexander group of a knot is defined. Using it, we obtain twisted Alexander polynomials. Also, we extend a well-known theorem of Neuwirth and Stallings giving necessary and sufficient conditions for a knot to be fibered.

Virtual Alexander polynomials provide obstructions for a virtual knot that must vanish if the knot has a diagram with an Alexander numbering. The extended group of a virtual knot is defined, and using it a more sensitive obstruction is obtained.

We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion, we derive a skein relation for a normalized form of this polynomial.

Let H(p) be the set of 2-bridge knots K(r), 0<r<1, such that there is a meridian-preserving epimorphism from G(K(r)), the knot group, to G(K(1/p)) with p odd. Then there is an algebraic integer s₀ such that for any K(r) in H(p), G(K(r)) has a parabolic representation ρ into SL(2, ℤ[s₀]) ⊂SL(2, ℂ). Let be the twisted Alexander polynomial associated to ρ. Then we prove that for any K(r) in H(p), and , where , μ ∈ ℤ[s₀]. The number μ can be recursively evaluated.

Let K be a prime knot in S³ and G(K) = π₁(S³ - K) the knot group. We write K₁ ≥ K₂ if there exists a surjective homomorphism from G(K₁) onto G(K₂). In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then 640, 800 should be considered. The existence of a surjective homomorphism can be proved by constructing it explicitly. On the other hand, the non-existence of a surjective homomorphism can be proved by the Alexander polynomial and the twisted Alexander polynomial.

Lin's definition of twisted Alexander knot polynomial is extended for finitely presented groups. Cha's fibering obstruction theorem is generalized. The group of the 2-twist-spun trefoil knot is seen to have a faithful representation that yields a trivial twisted Alexander polynomial.

The number of representations of a knot group is an invariant of knots. In this paper, we calculate these numbers associated to SL(2;ℤ/pℤ)-representations for all the knots in Rolfsen's knot table. Moreover, we show some properties of these numbers.

Let L = ℓ₁ ∪⋯∪ℓ_d+1 be an oriented link in 𝕊³, and let L(q) be the d-component link ℓ₁ ∪⋯∪ℓ_d regarded in the homology 3-sphere that results from performing 1/q-surgery on ℓ_d+1. Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtained. The behavior of the invariants as q increases without bound is described.

We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.

We study the twisted Alexander polynomial Δ_K,ρ of a knot K associated to a non-abelian representation ρ of the knot group into SL₂(ℂ). It is known for every knot K that if K is fibered, then for every non-abelian representation, Δ_K,ρ is monic and has degree 4g(K) – 2 where g(K) is the genus of K. Kim and Morifuji recently proved the converse for 2-bridge knots. In fact they proved a stronger result: if a 2-bridge knot K is non-fibered, then all but finitely many non-abelian representations on some component have Δ_K,ρ non-monic and degree 4g(K) – 2. In this paper, we consider two special families of non-fibered 2-bridge knots including twist knots. For these families, we calculate the number of non-abelian representations where Δ_K,ρ is monic and calculate the number of non-abelian representations where the degree of Δ_K,ρ is less than 4g(K) – 2.

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in [J. Dubois, Nonabelian twisted Reidemeister torsion for fibered knots, Canad. Math. Bull.49(1) (2006) 55–71] and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in [J. Dubois, V. Huynh and Y. Yamaguchi, Nonabelian Reidemeister torsion for twist knots, J. Knot Theory Ramifications18(3) (2009) 303–341].

When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume of the connected sum is the sum of each complex volumes modulo iπ² and the twisted Alexander polynomial of the connected sum is the product of each polynomials with normalization.

We give explicit formulas for the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of genus one two-bridge knots.

We fix the errors in the paper ‘Connected sum of representations of knot groups’ [J. Cho, Connected sum of representations of knot groups, J. Knot Theory Ramifications24(3) (2015) 18, 1550020].

Let $M$ be a non-abelian semi-direct product of a cyclic group $\mathbb{Z}/n$ and an elementary abelian $p$-group $A=\oplus k(\mathbb{Z}/p)$ of order $p^k$, $p$ being a prime and $gcd(n,p)=1$. Suppose that the knot group $G(K)$ of a knot $K$ in the $3$-sphere is represented on $M$. Then we conjectured (and later proved) that the twisted Alexander polynomial ${\mathrm{\Delta }}_{\gamma ,K}(t)$ associated to $\gamma :G(K)\to M\to GL(p^k,\mathbb{Z})$ is of the form: $\frac{{\mathrm{\Delta }}_K(t)}{1-t}F(t^n)$, where ${\mathrm{\Delta }}_K(t)$ is the Alexander polynomial of $K$ and $F(t^n)$ is an integer polynomial in $t^n$. In this paper, we present a proof of the following. For a $2$-bridge knot $K(r)$ in $H(p)$, if $n=2$ and $k=1$, then $F(t^2)$ is written as $f(t)f(t^{-1})$, where $H(p)$ is the set of $2$-bridge knots whose knot groups map on that of $K(1/p)$ with $p$ odd.

The adjoint twisted Alexander polynomial has been computed for twist knots [A. Tran, Twisted Alexander polynomials with the adjoint action for some classes of knots, J. Knot Theory Ramifications23(10) (2014) 1450051], genus one two-bridge knots [A. Tran, Adjoint twisted Alexander polynomials of genus one two-bridge knots, J. Knot Theory Ramifications25(10) (2016) 1650065] and the Whitehead link [J. Dubois and Y. Yamaguchi, Twisted Alexander invariant and nonabelian Reidemeister torsion for hyperbolic three dimensional manifolds with cusps, Preprint (2009), arXiv:0906.1500]. In this paper, we compute the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of twisted Whitehead links.

We consider classification of the oriented ribbon 2-knots presented by virtual arcs with up to four crossings. We show the difference by the 2-fold branched covering space, the Alexander polynomial, the number of representations of the knot group to SL$(2,𝔽)$, $𝔽$ a finite field, and the twisted Alexander polynomial.