Narrow Results

In this paper, we generalize virtual knot theory to multi-virtual knot theory where there are a multiplicity of virtual crossings. Each virtual crossing type can detour over the other virtual crossing types, and over classical or immersed crossings. New invariants of multi-virtual knots and links are introduced and new problems that arise are described. We show how the extensions of the Penrose coloring evaluation for trivalent plane graphs and our generalizations of this to non-planar graphs and arbitrary numbers of colors acts as a motivation for the construction of the multi-virtual theory.

We introduce the pole diagram, which helps to retrieve information from a knot diagram when we smooth crossings. By using the notion, we define a bracket polynomial for the Miyazawa polynomial. The bracket polynomial gives a simple definition and evaluation for the Miyazawa polynomial. Then we show that the virtual crossing number of a virtualized alternating link is determined by its diagram. Furthermore, we construct infinitely many virtual link diagrams which attain the minimal real and virtual crossing numbers together.

We study the structure of the periodic orbits on a simple branched manifold which is a subtemplate of the branched manifold of the chaotic attractor that is obtained from a cancer model. We indicate the conditions for the knotted and linked periodic orbits by using symbolic dynamics, then we extend the results to the periodic solutions of the chaotic attractor. Furthermore, we compare the knots obtained by using same symbol sequences for the simple branched manifold and the attractor's branched manifold by using a knot invariant, specifically, we calculate the Kauffman bracket polynomial. In order to count the number of closed curves which is required to calculate the bracket polynomial we propose a new method which uses cyclic permutations.

For knots in S³ criteria for free periodicity are obtained in terms of the bracket and the Jones polynomials. The criteria introduced in this paper generalize the result obtained by Murasugi for the Jones polynomials of periodic knots.

We adapt Thistlethwaite's alternating tangle decomposition of a knot diagram to identify the potential extreme terms in its bracket polynomial, and give a simple combinatorial calculation for their coefficients, based on the intersection graph of certain chord diagrams.

An important problem of knot theory is to find or estimate the extreme coefficients of the Jones–Kauffman polynomial for (virtual) links with a given number of classical crossings. This problem has been studied by Morton and Bae [1] and Manchón [11] for the case of classical links. It turns out that the general case can be reduced to the case when the extreme coefficient function is expressible in terms of chord diagrams (previous authors consider only d-diagrams which correspond to the classical case [9]). We find the maximal absolute values for generic chord diagrams, thus, for generic virtual knots. Also we consider the "next" coefficient of the Jones–Kauffman polynomial in terms of framed chord diagrams and find its maximal value for a given number of chords. These two functions on chord diagrams are of their own interest because there are related to the Vassiliev invariants of classical knots and J-invariants of planar curves, as mentioned in [10].

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of three families of 2-braid virtual knots. Each of them has a certain feature; for example, we can show: for any positive integer N, there exists a virtual knot group with an element of order N. It is known that the collection of the virtual knot groups is the same as that of the ribbon T²-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon S²-knot groups, S²-knot groups, T²-knot groups, and S³-knot groups.

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c₃(K), which is the minimum number of triple crossings in a projection of K. We obtain upper and lower bounds on c₃(K) in terms of the traditional crossing number and show that both are realized. We also relate triple crossing number to the span of the bracket polynomial and use this to determine c₃(K) for a variety of knots and links. We then use c₃(K) to obtain bounds on the volume of a hyperbolic knot or link. We also consider extensions to c_n(K).

Pseudo links have two crossing types: classical crossings and indeterminate crossings. They were first introduced by Hanaki as a possible tool for analyzing images produced by electron microscopy of DNA. A normalized bracket polynomial is defined for pseudo links and then used to construct an obstruction to cosmetic crossings in classical links.

We consider $2$-tangle replacements of link diagrams, i.e. replacing a crossing of a link diagram $D$ with a $2$-tangle diagram $T$. We show that if $D$ and $T$ are adequate then the $2$-tangle replacement, denoted by $D_T$, is also adequate. This gives a way to obtain infinitely many new minimal crossing diagrams.

For a signed cyclic graph $G$, we can construct a unique virtual link $L$ by taking the medial construction and converting 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In this paper, we shall prove that a virtual link $L$ is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial $F[G]$ for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between $F[G]$ of a signed cyclic graph $G$ and the bracket polynomial of one of the virtual link diagrams associated with $G$. Finally, we give a spanning subgraph expansion for $F[G]$.

A multi-variable polynomial invariant for knotoids and linkoids, which is an enhancement of the bracket polynomial for knotoids introduced by Turaev, is given by using the concept of a pole diagram which originates in constructing a virtual link invariant. Several features of the polynomial are revealed.

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics.

We present simple congruences modulo $p$ of the Jones polynomial for the torus link $T(p,\lambda )$, where $p$ is a prime number and $\lambda$ is a nonzero integer. These congruences are expressed in terms of Chebyshev polynomials and quantum numbers. The approach makes elementary use of the Kauffman bracket and does not require any representation theory.

The Alexander–Conway polynomial is reconstructed in a manner similar to the way the Jones polynomial is constructed by using the Kauffman bracket polynomial. This is a summary of the reconstruction.