Narrow Results

In [Ilyutko and Safina, Graph-links: nonrealizability, orientation, and Jones polynomial, J. Math. Sc.214(5) (2016) 632–664], the first named author defined the notion of an oriented graph-link and constructed a writhe number of a vertex of an oriented graph-link, which equals the “real” writhe number of a crossing in the realizable case. As a result the Jones polynomial was defined for oriented graph-links and the first example of a non-realizable graph-link with more than one component was found. Despite the fact that all necessary definitions were given the authors did not define the notion of linking number for graph-links. In this paper, we are eliminating this deficiency. Moreover, we classify all graph-links having a representative with less than $4$ vertices.

We prove that for every distributive 〈∨,0〉-semilattice S, there are a meet-semilattice P with zero and a map μ: P × P → S such that μ(x,z) ≤ μ(x,y) ∨ μ(y,z) and x ≤ y implies that μ(x,y) = 0, for all x, y, z ∈ P, together with the following conditions:.

(P1) μ(v,u) = 0 implies that u = v, for all u ≤ v in P.

(P2) For all u ≤ v in P and all a, b ∈ S, if μ(v,u) ≤ a ∨ b, then there are a positive integer n and a decomposition u = x₀ ≤ x₁ ≤ ⋯ ≤ x_n = v such that either μ(x_{i + 1}, x_i) ≤ a or μ(x_i+1, x_i) ≤ b, for each i < n.

(P3) The subset {μ(x,0) | x ∈ P} generates the semilattice S.

Furthermore, every finite, bounded subset of P has a join, and P is bounded in case S is bounded. Furthermore, the construction is functorial on lattice-indexed diagrams of finite distributive 〈∨,0,1〉-semilattices.

For a class of algebras, denote by Con_c the class of all (∨, 0)-semilattices isomorphic to the semilattice Con_cA of all compact congruences of A, for some A in . For classes and of algebras, we denote by the smallest cardinality of a (∨, 0)-semilattices in Con_c which is not in Con_c if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties and , is either finite, or ℵ_n for some natural number n, or ∞. We also find two finitely generated modular lattice varieties and such that , thus answering a question by J. Tůma and F. Wehrung.

The singularity set of a generic standard projection to the three space of a closed surface linked in four space, consists of at most three types: double points, triple points or branch points. We say that this generic projection image is p-diagram if it does not contain any triple point. Two p-diagrams of equivalent surface links are called p-equivalent if there exist a finite sequence of local moves, such that each of them is one of the four moves taken from the seven on the well known Roseman list, that connects only p-diagrams. It is natural to ask that whether any of two p-diagrams of equivalent surface links always p-equivalent? We introduce an invariant of p-equivalent diagrams and an example of linked surfaces that answers our question negatively.

We show that the Wirtinger presentation of a prime alternating link group satisfies a generalized small cancellation condition. This new version of Weinbaum's solution to the word and conjugacy problems for these groups easily extends to finite sums of alternating links.

For a 1- or 2-dimensional knot, we give a lower bound log₂ n + 1 of the minimum number of distinct colors for all effective n-colorings. In particular, we prove that any effectively 9-colorable 1- or ribbon 2-knot is presented by a diagram where exactly five colors of nine are assigned to the arcs or sheets.

We prove that any 11-colorable knot is presented by an 11-colored diagram where exactly five colors of eleven are assigned to the arcs. The number five is the minimum for all non-trivially 11-colored diagrams of the knot. We also prove a similar result for any 11-colorable ribbon 2-knot.

This paper has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram $D$ is a sequence of moves that transform $D$ into a diagram $D^{\prime }$ with the minimal possible number of crossings for the isotopy class of the knot represented by $D$. The simplification is monotonic if the number of crossings never increases along the sequence. Our moves are certain $Z_{1,2,3}$ generalizing the classical Reidemeister moves $R_{1,2,3}$, and another one $C$ (together with a variant $\tilde{C}$) aimed at detecting whether a knot diagram can be viewed as a connected sum of two easier ones. We present an accurate description of the moves and several results of our implementation of the simplification procedure based on them, publicly available on the web.

An OU sequence is a cyclically ordered sequence in symbols $O$ and $U$ such that the number of $O$’s is equal to that of $U$’s. Every knot diagram defines an O’U sequence by reading $O$ and $U$ at over- and under-crossings, respectively, appeared along the diagram. In this note, we determine the OU sequences for a $2$-bridge knot arising from its diagrams with two over-bridges.

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 $2+\lfloor {log}_{2}n\rfloor$.

For an effectively $n$-colorable knot $K$, the palette number ${\mathrm{\text{C}}}_n^{\ast }(K)$ is the minimum number of distinct colors for all effective $n$-colorings of $K$. It is known that ${\mathrm{\text{C}}}_n^{\ast }(K)\ge 2+\lfloor {log}_{2}n\rfloor$ for any effectively $n$-colorable knot $K$. In this paper, we show that for any odd $n\ge 3$ and effectively $n$-colorable torus knot $K$ it holds that ${\mathrm{\text{C}}}_n^{\ast }(K)=2+\lfloor {log}_{2}n\rfloor$ namely, any effectively $n$-colorable torus knot has an effectively $n$-colored diagram with $2+\lfloor {log}_{2}n\rfloor$ colors.

It is known that there is no 2-knot with triple point number two. The present paper shows that there is no surface-knot of genus one with triple point number two.

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of $n$-crossing diagrams for every $n$ greater than one allows the definition of the $n$-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.

In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant $[\cdot ]$ of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams $K$, the following formula holds: $[K]=\tilde{K}$, where $\tilde{K}$ is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson–Orrison–Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.

In 2013, Adams introduced the $n$-crossing number of a knot $K$, denoted by $c_n(K)$. Inequalities between the $2$-, $3$-, $4$- and $5$-crossing numbers have been previously established. We prove $c_9(K)\le c_3(K)-2$ for all knots $K$ that are not the trivial, trefoil, or figure-eight knot. We show this inequality is optimal and obtain previously unknown values for $c_9(K)$.

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

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.

The arc index of a link is closely related to its crossing number. An algorithm is known which produces an arc-presentation by drawing the binding circle on a link diagram. When the algorithm can be applied and the link is alternating, the presentation is known to be minimal. Here we exhibit an example which shows that no such algorithm can be valid for all link diagrams.

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form ${ℒ}_{\infty \lambda }$. We prove that many naturally defined classes are anti-elementary, including the following:

• the class of all lattices of finitely generated convex $\ell$-subgroups of members of any class of $\ell$-groups containing all Archimedean $\ell$-groups;
• the class of all semilattices of finitely generated $\ell$-ideals of members of any nontrivial quasivariety of $\ell$-groups;
• the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem;
• the class of all semilattices of finitely generated two-sided ideals of rings;
• the class of all semilattices of finitely generated submodules of modules;
• the class of all monoids encoding the nonstable K₀-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero;
• (assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large $4$-frame.

The main underlying principle is that under quite general conditions, for a functor $\mathrm{\Phi }:{𝒜}\to {ℬ}$, if there exists a noncommutative diagram $\overrightarrow{D}$ of ${𝒜}$, indexed by a common sort of poset called an almost join-semilattice, such that

• $\mathrm{\Phi }{\overrightarrow{D}}^I$ is a commutative diagram for every set $I$,
• $\mathrm{\Phi }\overrightarrow{D}\ncong \mathrm{\Phi }\overrightarrow{X}$ for any commutative diagram $\overrightarrow{X}$ in ${𝒜}$,

then the range of $\mathrm{\Phi }$ is anti-elementary.

A ring R is said to be right serial, if it is a direct sum of right ideals which are uniserial. A ring that is right serial need not be left serial. Right artinian, right serial ring naturally arise in the study of artinian rings satisfying certain conditions. For example, if an artinian ring R is such that all finitely generated indecomposable right R-modules are uniform or all finitely generated indecomposable left R-modules are local, then R is right serial. Such rings have been studied by many authors including Ivanov, Singh and Bleehed, and Tachikawa. In this paper, a universal construction of a class of indecomposable, non-local, basic, right artinian, right serial rings is given. The construction depends on a right artinian, right serial ring generating system X, which gives rise to a tensor ring T(L). It is proved that any basic right artinian, right serial ring is a homomorphic image of one such T(L).