arXiv subjects

Masahico Saito

Publications and source records attributed to Masahico Saito.

At least 19 recordsLinked to original sources

arXiv2026

Surface Braid Invariants from Yang-Baxter Operators and Their Cohomology

Surface braids are a generalization of braids to surfaces in the 4-disk. They are described by planar graphs called charts, and sequences of braid words called braid systems, that represent monodromies of branch points in projections. Yang-Baxter operators (YBOs) have been used extensively in knot theory for constructions of Jones and other polynomials, defined by braid group representations. We propose to use YBOs for constructing surface braid invariants through braid systems. The intersections of eigenspaces are used for this construction. We further use 3-cocycles of the Yang-Baxter cohomology theories via charts, in an analogue of quandle cocycle invariants but more generally utilizing tensor products of modules. An approach towards defining knotted surface invariants through closures of surface braids is discussed.

math.GT↗

arXiv2026

Quantum Cocycle Invariants of Knots from Yang-Baxter Cohomology

Yang-Baxter operators (YBOs) have been employed to construct quantum knot invariants. More recently, cohomology theories for YBOs have been independently developed, drawing inspiration from analogous theories for quandles and other discrete algebraic structures. Quandle cohomology, in particular, gives rise to cocycle invariants for knots via $2$-cocycles, which are closely related to quandle extensions, and knotted surfaces via $3$-cocycles. These quandle cocycle knot invariants have also been shown to admit interpretations as quantum invariants. Similarly, $2$-cocycles in Yang-Baxter cohomology can be interpreted in terms of deformations of YBOs. Building on these parallels, we introduce quantum cocycle invariants of knots using $2$-cocycles of Yang-Baxter cohomology, from the perspective of deformation theory. In particular, we demonstrate that the quandle cocycle invariant can be interpreted in this framework, while the quantum version yields stronger invariants in certain examples. We develop our theory along two primary approaches to quantum invariants: via trace constructions and through (co)pairings. Explicit examples are provided, including those based on the Kauffman bracket. Furthermore, we show that both the Jones and Alexander polynomials can be derived within this framework as invariants arising from higher-order formal Laurent polynomial deformations via Yang-Baxter cohomology.

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arXiv2025

The Insertion Chain Complex: A Topological Approach to the Structure of Word Sets

We introduce the Insertion Chain Complex, a higher-dimensional extension of insertion graphs, as a new framework for analyzing finite sets of words. We study its topological and combinatorial properties, in particular its homology groups, as measures of the complexity for word sets. After establishing its theoretical foundations, we investigate the computational and combinatorial aspects of these complexes. Among other results, we classify minimal 1-dimensional cycles and prove that every finitely generated abelian group can be realized as the homology of the insertion complex for some set of words. We also identify conditions that guarantee vanishing homology. These results provide new invariants for characterizing finite sets of words through word-based topological structures and their properties.

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arXiv2025

Yang-Baxter Hochschild Cohomology

Braided algebras are associative algebras endowed with a Yang-Baxter operator that satisfies certain compatibility conditions involving the multiplication. Along with Hochschild cohomology of algebras, there is also a notion of Yang-Baxter cohomology, which is associated to any Yang-Baxter operator. In this article, we introduce and study a cohomology theory for braided algebras in dimensions 2 and 3, that unifies Hochschild and Yang-Baxter cohomology theories, and generalizes to all dimensions in characteristic $2$. We show that its second cohomology group classifies infinitesimal deformations of braided algebras. We provide infinite families of examples of braided algebras, including Hopf algebras, tensorized multiple conjugation quandles, and braided Frobenius algebras. Moreover, we derive the obstructions to higher deformations, which lie in the third cohomology group. Relations to Hopf algebra cohomology are also discussed.

math.QA↗

arXiv2025

Deformation Cohomology for Braided Commutativity

Braided algebras are algebraic structures consisting of an algebra endowed with a Yang-Baxter operator, satisfying some compatibility conditions.Yang-Baxter Hochschild cohomology was introduced by the authors to classify infinitesimal deformations of braided algebras, and determine obstructions to higher order deformations. Several examples of braided algebras satisfy a weaker version of commutativity, which is called braided commutativity and involves the Yang-Baxter operator of the algebra. We extend the theory of Yang-Baxter Hochschild cohomology to study braided commutative deformations of braided algebras. The resulting cohomology theory classifies infinitesimal deformations of braided algebras that are braided commutative, and provides obstructions for braided commutative higher order deformations. We consider braided commutativity for Hopf algebras in detail, and obtain some classes of nontrivial examples.

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arXiv2025

Structures of Monoids Motivated by DNA Origami

We construct a class of monoids, called origami monoids, motivated by Jones monoids and by strand organization in DNA origami structures. Two types of basic building blocks of DNA origami closely associated with the graphical representation of Jones monoids are identified and are taken as generators for the origami monoid. Motivated by plausible modifications of the DNA origami structures and the relations of the well studied Jones monoids, we then identify a set of relations that characterize the origami monoid. These relations expand the relations of the Jones monoids and include a new set of relations called contextual commutation. With contextual commutation, certain generators commute only when found within a given context. We prove that the origami monoids are finite and propose a normal form representation of their elements. We establish a correspondence between the Green's classes of the origami monoid and the Green's classes of a direct product of Jones monoids.

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arXiv2024

Planar Equivalence of Knotoids and Quandle Invariants

While knotoids on the sphere are well-understood by a variety of invariants, knotoids on the plane have proven more subtle to classify due to their multitude over knotoids on the sphere and a lack of invariants that detect a diagram's planar nature. In this paper, we investigate equivalence of planar knotoids using quandle colorings and cocycle invariants. These quandle invariants are able to detect planarity by considering quandle colorings that are restricted at distinguished points in the diagram, namely the endpoints and the point-at-infinity. After defining these invariants we consider their applications to symmetry properties of planar knotoids such as invertibility and chirality. Furthermore we introduce an invariant called the triangular quandle cocycle invariant and show that it is a stronger invariant than the end specified quandle colorings.

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arXiv2023

Yang-Baxter Solutions from Categorical Augmented Racks

An augmented rack is a set with a self-distributive binary operation induced by a group action, and has been extensively used in knot theory. Solutions to the Yang-Baxter equation (YBE) have been also used for knots, since the discovery of the Jones polynomial. In this paper, an interpretation of augmented racks in tensor categories is given for coalgebras that are Hopf algebra modules, and associated solutions to the YBE are constructed. Explicit constructions are given using quantum heaps and the adjoint of Hopf algebras. Furthermore, an inductive construction of Yang-Baxter solutions is given by means of the categorical augmented racks, yielding infinite families of solutions. Constructions of braided monoidal categories are also provided using categorical augmented racks.

math.QA↗

arXiv2023

Betti Numbers of Prodsimplicial Complexes for Directed Graphs with Applications to Word Reductions

We propose custom made cell complexes, in particular prodsimplicial complexes, in order to analyze data consisting of directed graphs. These are constructed by attaching cells that are products of simplices and are suited to study data of acyclic directed graphs, called here consistently directed graphs. We investigate possible values of the first and second Betti numbers and the types of cycles that generate nontrivial homology. We apply these tools to directed graphs associated with reductions of double occurrence words, words that are associated with DNA recombination processes in certain species of ciliates. We study the effects of word operations on the homology for these graphs.

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arXiv2022

Extensions of Augmented Racks and Surface Ribbon Cocycle Invariants

A rack is a set with a binary operation that is right-invertible and self-distributive, properties diagrammatically corresponding to Reidemeister moves II and III, respectively. A rack is said to be an {\it augmented rack} if the operation is written by a group action. Racks and their cohomology theories have been extensively used for knot and knotted surface invariants. Similarly to group cohomology, rack 2-cocycles relate to extensions, and a natural question that arises is to characterize the extensions of augmented racks that are themselves augmented racks. In this paper, we characterize such extensions in terms of what we call {\it fibrant and additive} cohomology of racks. Simultaneous extensions of racks and groups are considered, where the respective $2$-cocycles are related through a certain formula. Furthermore, we construct coloring and cocycle invariants for compact orientable surfaces with boundary in ribbon forms embedded in $3$-space.

math.GT↗

arXiv2022

Fundamental Heap for Framed Links and Ribbon Cocycle Invariants

A heap is a set with a certain ternary operation that is self-distributive (TSD) and exemplified by a group with the operation $(x,y,z)\mapsto xy^{-1}z$. We introduce and investigate framed link invariants using heaps. In analogy with the knot group, we define the fundamental heap of framed links using group presentations. The fundamental heap is determined for some classes of links such as certain families of torus and pretzel links. We show that for these families of links there exist epimorphisms from fundamental heaps to Vinberg and Coxeter groups, implying that corresponding groups are infinite. A relation to the Wirtinger presentation is also described. The cocycle invariant is defined using ternary self-distributive (TSD) cohomology, by means of a state sum that uses ternary heap $2$-cocycles as weights. This invariant corresponds to a rack cocycle invariant for the rack constructed by doubling of a heap, while colorings can be regarded as heap morphisms from the fundamental heap. For the construction of the invariant, first computational methods for the heap cohomology are developed. It is shown that the cohomology splits into two types, called degenerate and nondegenerate, and that the degenerate part is one dimensional. Subcomplexes are constructed based on group cosets, that allow computations of the nondegenerate part. Computations of the cocycle invariants are presented using the cocycles constructed, and conversely, it is proved that the invariant values can be used to derive algebraic properties of the cohomology.

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arXiv2021

Fundamental Heaps for Surface Ribbons and Cocycle Invariants

We introduce the notion of fundamental heap for compact orientable surfaces with boundary embedded in $3$-space, which is an isotopy invariant of the embedding. It is a group, endowed with a ternary heap operation, defined using diagrams of surfaces in a form of thickened trivalent graphs called surface ribbons. We prove that the fundamental heap has a free part whose rank is given by the number of connected components of the surface. We study the behavior of the invariant under boundary connected sum, as well as addition/deletion of twisted bands, and provide formulas relating the number of generators of the fundamental heap to the Euler characteristics. We describe in detail the effect of stabilization on the fundamental heap, and determine that for each given finitely presented group there exists a surface ribbon whose fundamental heap is isomorphic to it, up to extra free factors. A relation between the fundamental heap and the Wirtinger presentation is also described. Moreover, we introduce cocycle invariants for surface ribbons using the notion of mutually distributive cohomology and heap colorings. Explicit computations of fundamental heap and cocycle invariants are presented.

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arXiv2021

Braided Frobenius Algebras from certain Hopf Algebras

A braided Frobenius algebra is a Frobenius algebra with braiding that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation $(x,y,z) \mapsto xy^{-1}z$, that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a braiding, by means of a Yang-Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between the braiding and Frobenius operations.

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arXiv2020

Skein theoretic approach to Yang-Baxter homology

We introduce skein theoretic techniques to compute the Yang-Baxter (YB) homology and cohomology groups of the R-matrix corresponding to the Jones polynomial. More specifically, we show that the YB operator $R$ for Jones, normalized for homology, admits a skein decomposition $R = I + βα$, where $α: V^{\otimes 2} \rightarrow k$ is a "cup" pairing map and $β: k \rightarrow V^{\otimes 2}$ is a "cap" copairing map, and differentials in the chain complex associated to $R$ can be decomposed into horizontal tensor concatenations of cups and caps. We apply our skein theoretic approach to determine the second and third YB homology groups, confirming a conjecture of Przytycki and Wang. Further, we compute the cohomology groups of $R$, and provide computations in higher dimensions that yield some annihilations of submodules.

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arXiv2019

Heap and Ternary Self-Distributive Cohomology

Heaps are para-associative ternary operations bijectively exemplified by groups via the operation $(x,y,z) \mapsto x y^{-1} z$. They are also ternary self-distributive, and have a diagrammatic interpretation in terms of framed links. Motivated by these properties, we define para-associative and heap cohomology theories and also a ternary self-distributive cohomology theory with abelian heap coefficients. We show that one of the heap cohomologies is related to group cohomology via a long exact sequence. Moreover we construct maps between second cohomology groups of normalized group cohomology and heap cohomology, and show that the latter injects into the ternary self-distributive second cohomology group. We proceed to study heap objects in symmetric monoidal categories providing a characterization of pointed heaps as involutory Hopf monoids in the given category. Finally we prove that heap objects are also "categorically" self-distributive in an appropriate sense.

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arXiv2019

Insertions Yielding Equivalent Double Occurrence Words

A double occurrence word (DOW) is a word in which every symbol appears exactly twice; two DOWs are equivalent if one is a symbol-to-symbol image of the other. We consider the so called repeat pattern ($αα$) and the return pattern ($αα^R$), with gaps allowed between the $α$'s. These patterns generalize square and palindromic factors of DOWs, respectively. We introduce a notion of inserting repeat/return words into DOWs and study how two distinct insertions into the same word can produce equivalent DOWs. Given a DOW $w$, we characterize the structure of $w$ which allows two distinct insertions to yield equivalent DOWs. This characterization depends on the locations of the insertions and on the length of the inserted repeat/return words and implies that when one inserted word is a repeat word and the other is a return word, then both words must be trivial (i.e., have only one symbol). The characterization also introduces a method to generate families of words recursively.

math.CO↗

arXiv2019

Higher Arity Self-Distributive Operations in Cascades and their Cohomology

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive $n$-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing $2$-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given.

math.GT↗

arXiv2019

Algebraic Systems for DNA Origami Motivated from Temperley-Lieb Algebras

We initiate an algebraic approach to study DNA origami structures by associating an element from a monoid to each structure. We identify two types of basic building blocks and describe an DNA origami structure with their composition. These building blocks are taken as generators of a monoid, called origami monoid, and, motivated by the well studied Temperley-Lieb algebras, we identify a set of relations that characterize the origami monoid. We also present several observations about the Green's relations for the origami monoid and study the relations to a cross product of Jones monoids that is a morphic image of an origami monoid.

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