Narrow Results

In this paper we compute the trace field for the family of hyperbolic twist knots. We describe this field as a simple extension ℚ(z₀) where z₀ is a specified root of a particular irreducible polynemial Φ_n(z)∈ℤ[z]. As a consequence, we find that the degree of the trace field is precisely two less than the-minimal crossing number of a twist knot.

In this paper we prove that if M_K is the complement of a non-fibered twist knot K in , then M_K is not commensurable to a fibered knot complement in a ℤ/2ℤ-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by Calegari and Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.

This paper gives an explicit formula for the SL₂(ℂ)-non-abelian Reidemeister torsion as defined in [6] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot.

We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (possibly, with extra trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.

For a hyperbolic knot, an ideal triangulation of the knot complement corresponding to the colored Jones polynomial was introduced by Thurston. Considering this triangulation of a twist knot, we find a function which gives the hyperbolicity equations and the complex volume of the knot complement, using Zickert's theory of the extended Bloch group and the complex volume.

We also consider a formal approximation of the colored Jones polynomial. Following Ohnuki's theory of 2-bridge knots, we define another function which comes from the approximation. We show that this function is essentially the same as the previous function, and therefore it also gives the same hyperbolicity equations and the complex volume.

Finally we compare this result with our previous one which dealt with Yokota theory, and, as an application to Yokota theory, present a refined formula of the complex volumes for any twist knots.

If a knot has the Alexander polynomial not equivalent to 1, then it is linearly n-colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e. the minimal order of a quandle with which the knot is quandle colorable. For twist knots, we study the minimal quandle orders in detail.

We prove that any neighborhood N of a positive twist knot , m ≥ 3 and odd with extremal Euler number can be thickened to a standard neighborhood of a maximal Thurston–Bennequin number Legendrian representative of .

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].

We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in terms of the distance between the two axes fixed by two generators. In this way the calculation becomes easier than using the singular locus directly. The volumes of the hyperbolic twist knot cone-manifolds simpler than Stevedore's knot are known. As an application, we give the volumes of the cyclic coverings over the hyperbolic twist knots.

We investigate the nonorientable 4-genus ${\gamma }_4$ of a special family of 2-bridge knots, the double twist knots $C(m,n)$. Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that ${\gamma }_4(C(m,n))\le 3$. By using explicit constructions to obtain upper bounds on ${\gamma }_4$ and known obstructions derived from Donaldson’s diagonalization theorem to obtain lower bounds on ${\gamma }_4$, we produce infinite subfamilies of $C(m,n)$ where ${\gamma }_4=0,1,2,$ and $3$, respectively. However, there remain infinitely many double twist knots where our work only shows that ${\gamma }_4$ lies in one of the sets $\{1,2\},\{2,3\}$, or $\{1,2,3\}$. We tabulate our results for all $C(m,n)$ with $\left|m\right|$ and $\left|n\right|$ up to 50. We also provide an infinite number of examples which answer a conjecture of Murakami and Yasuhara.