Narrow Results

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.

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.

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.

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 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.