Narrow Results

We prove that for any odd $n\ge 3$, the $n$-palette number of any effectively $n$-colorable $2$-bridge knot is equal to $2+\lfloor {log}_{2}n\rfloor$. Namely, there is an effectively $n$-colored diagram of the $2$-bridge knot such that the number of distinct colors that appeared in the diagram is exactly equal to $2+\lfloor {log}_{2}n\rfloor$.

An $n$-string tangle is a pair $(B,A)$ such that $A$ is a disjoint union of properly embedded $n$ arcs in a topological $3$-ball $B$. And an $n$-string tangle is said to be trivial (or rational)${}^{\mathrm{\text{a}}}$, if it is homeomorphic to $(D\times I,\{x_1,\dots ,x_n\}\times I)$ as a pair, where $D$ is a 2-disk, $I$ is the unit interval and each $x_i$ is a point in the interior of $D$. A stick tangle is a tangle each of whose arcs consists of finitely many line segments, called sticks. For an $n$-string stick tangle its stick-order is defined to be a nonincreasing sequence $(s_1,s_2,\dots ,s_n)$ of natural numbers such that, under an ordering of the arcs of the tangle, each $s_i$ denotes the number of sticks constituting the $i$th arc of the tangle. And a stick-order $S$ is said to be trivial, if every stick tangle of the order $S$ is trivial.

In this paper, restricting the $3$-ball $B$ to be the standard 3-ball, we give the complete list of trivial stick-orders.

A result about spanning forests for graphs yields a short proof of Krebes’s theorem concerning embedded tangles in links.

A knot $K_1$ is a minor of a knot $K_2$ if any regular projection of $K_2$ is also a regular projection of $K_1$. This defines a pre-ordering on the set of all knots. For each knot of five or less crossings, the set of all regular projections of it is determined by Taniyama [A partial order of knots, Tokyo J. Math.12(1) (1989) 205–229]. Thus, the pre-ordering is determined up to five crossing knots. In this paper, we determine the set of all regular projections of the knot $6_2$.

Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to 24 crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.

In this paper, we show that every finite spatial graph is a connected sum of a planar graph, which is a forest, and a tangle. As a consequence, we get that any finite spatial graph is a connected sum of a planar graph and a braid. Using these decompositions it is not difficult to find a set of generators and defining relations for the fundamental group of compliment of a spatial graph in 3-space ${ℝ}^3$.

The space of Gauss diagram formulas that are knot invariants is introduced by Goussarov–Polyak–Viro in 2000; it is extended to nanophrases by Gibson–Ito in 2011. However, known invariants in concrete presentations of Gauss diagram formulas are very limited, even in the one-component case. This paper gives a recipe to obtain explicit forms of Gauss diagram formulas that are invariants of virtual links with base points or tangles. As an application, we introduce a new construction of Gauss diagram formulas of $3$-bouquets and how to give link invariants that do not change with base point moves, including a reconstruction of the Milnor’s triple linking number.

We define a new hierarchy of isotopy invariants of colored oriented links through oriented tangle diagrams. We prove the colored braid relation and the Markov trace property explicitly.

A method of coding diagrams of knots, links and tangles is introduced. Also, how to draw a diagram for a given code is explained.

We discuss new methods of changing knot and link diagrams in such a way that their Jones polynomial is unaltered. It is similar to Conway's mutation, but applies to more complicated "tangles," and is performed simultaneously at multiple sites in the diagram — thus "global mutation." The method of achieving this can be regarded as communication via "messenger tangles."

Using the vertex model interpretation of the coloured (generalised) Jones polynomial of a link L, we show that if the colour of the i^th component is N_i+m_ir, then modulo t^r−1 this coloured Jones polynomial is congruent, up to a product of calculable factors, to the coloured Jones polynomial with the colour of the i^th component N_i, where N_i and r are positive integers, and m_i is a non-negative integer.

The proof depends on the fact that, up to a known factor, the coloured Jones polynomial of a link may be calculated from a (1, 1)-tangle, the closure of which represents the link.

Embeddings of 4-regular graphs into 3-space are examined by studying graph diagrams, i.e. projections of embedded graphs to an appropriate plane. New diagrams can be constructed from the old ones by replacing graph vertices with rational tangles, and these diagrams lead to topological invariants of embedded graphs. The new invariants are calculated for some examples, in particular for classes of alternating diagrams of the figure-eight graph. As an application, it is shown that these diagrams have minimal crossing number, which gives generalizations to some of the so-called Tait conjectures.

We consider the parallelism of two strings in alternating tangles. We show that if there is a pair of parallel strings in an alternating tangle then its alternating diagrams satify certain conditions. As a corollary, for a knot admitting a decomposition into two alternating tangles with two or three strings, we prove that its non-trivial Dehn surgery yields a 3-manifold with an essential lamination. Hence such a knot has property P and satisfy the cabling conjecture.

It is shown that the proportion of alternating n-crossing, prime link types amongst all n-crossing prime link types tends to zero exponentially with increasing n. Also, a characterization is established for essential annuli in alternating tangles, and a simple criterion is given for equivalence of alternating tangles.

We will discuss tangle decompositions of tunnel number one links, and show that any tangle composite tunnel number one link can be obtained as the union of a quasi Hopf tangle and a trivial tangle. We also consider the uniqueness of tangle decompositions of such links, and show that there are infinitely many tunnel number one links that admit non-isotopic essential tangle decompositions.

We prove that the skein polynomial and the Jones polynomial of oriented links are not changed be the rotation operation of Anstee, Przytycki and Rolfsen [1], provided the rotor part of the diagram is of a certain special type.

We show how to build tangles T in a 3-ball with the property that any knot obtained by tangle sum with T has a persistent lamination in its exterior, and therefore has property P. The construction is based on an example of a persistent lamination in the exterior of the twist knot 6₁, due to Ulrich Oertel. We also show how the construction can be generalized to n-string tangles.

We characterize composite double torus knots and links, 2-string composite double torus knots.

Let t be a virtual tangle embedded in a virtual knot or link k. If a prime p divides the determinants of both the numerator and the denominator closure of t, then p divides the determinant of k.

The IOTA Tangle, a Directed Acyclic Graph (DAG)-based distributed ledger, is popular for its scalability and suitability for IoT applications, offering fee-less transactions. A critical component of IOTA’s architecture is the Cumulative Weight Calculation (CWC), essential for its tip selection mechanism. This paper introduces an optimization of the IOTA Reference Implementation (IRI) CWC process originally implemented using Breadth-First Search (BFS) by employing Depth-First Search (DFS) and Iterative Deepening Search (IDS) algorithms. We present a comparative analysis of these methods, demonstrating that DFS and IDS provide significant improvements in computational efficiency, particularly beneficial for IoT devices with limited processing capabilities. Our findings are substantiated through a series of experiments on a Tangle snapshot, highlighting the enhanced performance and reduced resource utilization of the proposed methods. This study contributes to the ongoing development of DAG-based distributed ledgers, offering insights into more efficient algorithmic solutions for large-scale, decentralized networks.