Journal of Knot Theory and Its Ramifications

Milnor defined two kinds of link-homotopy invariants $\overline{\mu }$ and ${\mu }^{\ast }$. By definition it would seem that the ${\mu }^{\ast }$-invariant is weaker than the $\overline{\mu }$-invariant. In this paper, we show that this is indeed the case by giving an example of length greater than 4 where $\overline{\mu }=1$ and ${\mu }^{\ast }=0$. For non-repeated sequences of length not greater than 4, Milnor has shown that $\overline{\mu }={\mu }^{\ast }$.

We establish a new framework for diagramming both Legendrian rational tangles in the standard contact structure on ${ℝ}^3$ and the signed characteristic foliations of their associated compressing discs, as well as the technical means by which these diagrams can be used to study Legendrian isotopies of such tangles. We then establish a number of results that represent new progress in the ongoing effort to classify Legendrian rational tangles under a pair of operations known as Legendrian flypes. These operations, while topologically isotopies, are known to produce distinct Legendrian objects in many circumstances, a fact that has been of much interest throughout the study and classification of Legendrian knots.

Let $C(2n,3)$ be the family of two bridge knots of slope $(4n+1)/(6n+1)$. We calculate the volumes of the $C(2n,3)$ cone-manifolds using the Schläfli formula. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano and Montesinos-Amilibia and extend the Ham, Mednykh and Petrov’s methods. As an application, we give the volumes of the cyclic coverings over those knots. For the fundamental group of $C(2n,3)$, we take and tailor Hoste and Shanahan’s. As a byproduct, we give an affirmative answer for their question whether their presentation is actually derived from Schubert’s canonical two-bridge diagram or not.

We prove the following strengthened version of Dehn’s lemma: if a knot in a three-sphere possesses a compressing disc which intersects with the knot at only two points of opposite sign, then it is the unknot. This result was conjectured by Greene–Wiest.

Let $M$ be a $3$-manifold, $F$ be an essential planar surface which cuts $M$ into two 3-manifolds $M_1$ and $M_2$. Suppose $M_i=V_i{\bigcup }_{S_i}{W}_i$ $(i=1,2)$ is a Heegaard splitting of $M_i$, $M=V{\bigcup }_{S}W$ is the dual Heegaard splitting of $M_1=V_1{\bigcup }_{S_1}{W}_1$ and $M_2=V_2{\bigcup }_{S_2}{W}_2$ along $F$, where $V=V_1\cup (D\times [-1,2])\cup V_2$ and $W=cl(M-V)$. In this paper, we give a condition of unstabilized dual Heegaard splittings of $3$-manifolds by using Hempel’s distance and the method of proof of Gordon’s Conjecture. Also, we give a counterexample for stabilized dual Heegaard splittings of $3$-manifolds for any Hempel’s distance.

The space ${𝒞}$ of conservative vertex colorings (over a field $𝔽$) of a countable, locally finite graph $G$ is introduced. When $G$ is connected, the subspace ${{𝒞}}^0$ of based colorings is shown to be isomorphic to the bicycle space of the graph. For graphs $G$ with a cofinite free ${\mathbb{Z}}^d$-action by automorphisms, ${𝒞}$ is dual to a finitely generated module over the polynomial ring $𝔽[x_1^{\pm 1},\dots ,x_d^{\pm 1}]$. Polynomial invariants for this module, the Laplacian polynomials ${\mathrm{\Delta }}_k,k\ge 0$, are defined, and their properties are discussed. The logarithmic Mahler measure of ${\mathrm{\Delta }}_0$ is characterized in terms of the growth of spanning trees.

Generalizing the construction of the Kauffman bracket skein algebra, Roger and Yang defined the Kauffman bracket arc algebra of a punctured surface, so that it is a quantization of the decorated Teichmüller space. We provide an explicit, finite set of generators for their algebra.

We introduce a new basis of the Temperley–Lieb algebra. It is defined using a bijection between noncrossing partitions and fully commutative elements together with a basis introduced by Zinno, which is obtained by mapping the simple elements of the Birman–Ko–Lee braid monoid to the Temperley–Lieb algebra. The combinatorics of the new basis involve the Bruhat order restricted to noncrossing partitions. As an application we can derive properties of the coefficients of the base change matrix between Zinno’s basis and the well-known diagram or Kazhdan–Lusztig basis of the Temperley–Lieb algebra. In particular, we give closed formulas for some of the coefficients of the expansion of an element of the diagram basis in the Zinno basis.

A comprehensive study of geometric and topological properties of torus knots and unknots is presented. Torus knots/unknots are particularly symmetric, closed, space curves, that wrap the surface of a mathematical torus a number of times in the longitudinal and meridian direction. By using a standard parametrization, new results on local and global properties are found. In particular, we demonstrate the existence of inflection points for a given critical aspect ratio, determine the location and prescribe the regularization condition to remove the local singularity associated with torsion. Since to first approximation total length grows linearly with the number of coils, its nondimensional counterpart is proportional to the topological crossing number of the knot type. We analyze several global geometric quantities, such as total curvature, writhing number, total torsion, and geometric ‘energies’ given by total squared curvature and torsion, in relation to knot complexity measured by the winding number. We conclude with a brief presentation of research topics, where geometric and topological information on torus knots/unknots finds useful application.