Journal of Knot Theory and Its Ramifications

We investigate algebraic and computability-theoretic properties of orderable magmas. A magma is an algebraic structure with a single binary operation. A right order on a magma is a linear ordering of its domain, which is right-invariant with respect to the magma operation. We use tools of computability theory to investigate Turing complexity of orders on computable orderable magmas. A magma is computable if it is finite, or if its domain can be identified with the set of natural numbers and the magma operation is computable. Interesting orderable magmas that are not even associative come from knot theory.

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.

An alternative framework underlying connection between tensor ${\mathrm{\text{sl}}}_2$-calculus and spin networks is suggested. New sign convention for the inner product in the dual spinor space leads to a simpler and direct set of initial rules for the diagrammatic recoupling methods. Yet, it preserves the standard chromatic graph evaluations. In contrast with the standard formulation, the background space is that of symmetric tensor spaces, which seems to be in accordance with the representation theory of $\mathrm{\text{SL}}(2)$. An example of Apollonian disk packing is shown to be a source of spin networks. The graph labeling is extended to non-integer values, resulting in the complex values of chromatic evaluations.

The original approach of Lie to the theory of transformation groups acting on smooth manifolds, on the basis of differential equations, being applied to smooth loops, has permitted the development of the infinitesimal theory of smooth loops generalizing the Lie group theory.

A loop with the law of associativity verified for its binary operation is a group. It has been shown that the system of differential equations characterizing a smooth loop with the right Bol identity and the integrability conditions lead to the binary-ternary algebra as a proper infinitesimal object, which turns out to be the Bol algebra (i.e. a Lie triple system with an additional bilinear skew-symmetric operation). There exist the analogous considerations for Moufang loops.

We will consider the differential equations of smooth loops, generalizing smooth left Bol loops, with the identities that are the characteristic identities for the algebraic description of some relativistic space-time models. Further examinations of the integrability conditions for the differential equations allow us to introduce the proper infinitesimal object for some subclass of loops under consideration.

The geometry of corresponding homogeneous spaces can be described in terms of tensors of curvature and torsion.

The most popular theories of everything are various versions of the superstring theory. The theories require existence of additional space dimensions, vibrations of which create the material particles in $3\mathrm{\text{D}}$ space. The additional space dimensions are understood as being currently smaller than the Planck Length and due to this not directly observable. We search for multidimensional models of the Universe (one time dimension; three isotropic, flat external dimensions, and $n$-internal dimensions), which satisfy the multidimensional Einstein equations and which started from the same radius of all of the internal and external dimensions, with an anisotropic energy–momentum tensor. Analytical solution of $(N+1)$-dimensional Einstein equation in a reparameterized time is reminded and discussed. The energy–momentum tensor is solely responsible for expansion of the external dimensions and shrinking of the internal ones; and to obtain this behavior of the space the tensor needs to fulfill some conditions i.e. the energy–momentum tensor cannot include only radiation, vacuum and baryonic matter. For the behavior of the physical space consistent with the one observed in our Universe, the dark energy and/or dark matter have to exist.

This paper addresses parameter identification of signals that are smooth and have compact support, but are otherwise arbitrary. We propose a definition of the time reference that results in the time uncertainty estimates that are much smaller than what one might expect with a casual analysis. These estimates are easily calculable given the noise level of the recording device and the signal shape.

The Chebyshev polynomials appear somewhat mysteriously in the theory of the skein modules. A generalization of the Chebyshev polynomials is proposed so that it includes both Chebyshev and Fibonacci and Lucas polynomials as special cases. Then, since it requires relaxation of a condition for traces of matrix powers and matrix representations, similar relaxation leads to a generalization of the Jones polynomial via reinterpretation of the Kauffman bracket construction. Moreover, the Witten’s approach via counting solutions of the Kapustin–Witten equation to get the Jones polynomial is simplified in the trivial knots case to studying solutions of a Laplace operator. Thus, harmonic ideas may be of importance in knot theory. Speculations on extension(s) of the latter approach via consideration of spin structures and related operators are given.

Frobenius algebra is formulated within the Abelian monoidal category of operad of graphs. A not necessarily associative algebra $\mathsf{Y}$ is said to be a Frobenius algebra if there exists a $\mathsf{Y}$-module isomorphism. A new concept of a solvable Frobenius algebra is introduced: an algebra $\mathsf{Y}$ is said to be a solvable Frobenius algebra if it possesses a nonzero one-sided $\mathsf{Y}$-module morphism with nontrivial radical. In the category of operad of graphs, we can express the necessary and sufficient conditions for an algebra to be a solvable Frobenius algebra. The notion of a solvable Frobenius algebra makes it possible to find all commutative nonassociative Frobenius algebras (Conjecture 10.1), and to find all Frobenius structures for commutative associative Frobenius algebras. Frobenius algebra allows $S_2$-permuted opposite algebra to be extended to $S_3$-permuted algebras.

We classify rooted trees which have strictly unimodal plucking polynomials ($q$-polynomials). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of $q$-binomial coefficients.

The algebraic part of an approach to groupoids started by Zakrzewski is presented together with some new results on formal properties of morphisms.

We study the asymptotic behavior of integer sequences related to knots that are generated by linear recurrences. We determine which of the 85 such sequences cataloged in Online Encyclopedia of Integer Sequences have ratios of their consecutive terms converging to a limit. We show that all but one of the ratio limits can be expressed by means of the $(m,n)$-anacci constants with $n$ equal to 1 or 2. Finally, we demonstrate how the $(m,2)$-anacci constants are linked to affine geometry.

We investigate (2,1):1 structures, which consist of a countable set $A$ together with a function $f:A\to A$ such that for every element $x$ in $A$, $f$ maps either exactly one element or exactly two elements of $A$ to $x$. These structures extend the notions of injection structures, 2:1 structures, and (2,0):1 structures studied by Cenzer, Harizanov, and Remmel, all of which can be thought of as infinite directed graphs. We look at various computability-theoretic properties of (2,1):1 structures, most notably that of computable categoricity. We say that a structure ${𝒜}$ is computably categorical if there exists a computable isomorphism between any two computable copies of ${𝒜}$. We give a sufficient condition under which a (2,1):1 structure is computably categorical, and present some examples of (2,1):1 structures with different computability-theoretic properties.

In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang–Baxter operators (we will call it the “algebraic” version in this paper). In 2012, Przytycki defined another homology theory for pre-Yang–Baxter operators which has a nice graphic visualization (we will call it the “graphic” version in this paper). We show that they are equivalent. The “graphic” homology is also defined for pre-Yang–Baxter operators, and we give some examples of its one-term and two-term homologies. In the two-term case, we have found torsion in homology of Yang–Baxter operator that yields the Jones polynomial.

A brief history of finite semifields upto 1965 is followed by a more detailed examination of recent discoveries with particular emphasis on constructions, fractional dimensions and cyclic semifields. Other results are also developed.