Related papers: The Word Problem for Braided Monoidal Categories i…
In this paper we investigate the decidability and complexity of problems related to braid composition. While all known problems for a class of braids with three strands, $B_3$, have polynomial time solutions we prove that a very natural…
For two given $\omega$-terms $\alpha$ and $\beta$, the word problem for $\omega$-terms over a variety $\boldsymbol{\mathrm{V}}$ asks whether $\alpha=\beta$ in all monoids in $\boldsymbol{\mathrm{V}}$. We show that the word problem for…
We prove that the word problem in the mapping class group of the once-punctured surface of genus g has complexity O(|w|^2 g for |w| > log(g) where |w| is the length of the word in a (standard) set of generators. The corresponding bound in…
The word problem is an old and central problem in (computational) group theory. It is well-known that the word problem is undecidable in general, but decidable for specific types of presentations. Consistent polycyclic presentations are an…
Hard instances of natural computational problems are often elusive. In this note we present an example of a natural decision problem, the word problem for a certain finitely presented group, whose hard instances are easy to find. More…
We show a simple and easily implementable solution to the word problem for virtual braid groups.
Given a category with a bifunctor and natural isomorphisms for associativity, commutativity and left and right identity we do not assume that extra constraining diagrams hold. We introduce groupoids of coupling trees to describe a version…
A braided monoidal category may be considered a $3$-category with one object and one $1$-morphism. In this paper, we show that, more generally, $3$-categories with one object and $1$-morphisms given by elements of a group $G$ correspond to…
A PROB is a "product and braid" category. Such categories can be used to encode the structure borne by an object in a braided monoidal category. In this paper we provide PROBs whose categories of algebras in a braided monoidal category are…
It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure. We make this precise by establishing undesirable properties that any such monoidal biclosed structure…
The avoidability, or unavoidability of patterns in words over finite alphabets has been studied extensively. A word (pattern) over a finite set is said to be unavoidable if, for all but finitely many words, there exists a morphism mapping…
We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and…
Presentations for unbraided, braided and symmetric pseudomonoids are defined. Biequivalences characterising the semistrict bicategories generated by these presentations are proven. It is shown that these biequivalences categorify results in…
We give an alternative presentation of braided monoidal categories. Instead of the usual associativity and braiding we have just one constraint (the b-structure). In the unital case, the coherence conditions for a b-structure are shown to…
The Diophantine problem for a monoid $M$ is the decision problem to decide whether any given system of equations has a solution in $M$. In this note, we give a simple example of a context-free, word-hyperbolic, finitely presented, special…
Braided monoidal categories arise naturally as centres of monoidal categories and have been the focus of much recent attention in both mathematics and physics. By suitably restricting the use of the exchange rule, we obtain a sequent…
The satisfiability problem of hybrid logics with the downarrow binder is known to be undecidable. This initiated a research program on decidable and tractable fragments. In this paper, we investigate the effect of restricting the…
We consider classes of fundamental groups of complements of various kinds of codimension 2 embeddings and show that, in general, the problem of deciding whether or not a group in one class belongs to a smaller class is algorithmically…
Under appropriate conditions, if one picks a commutative algebra A with action of group G in braided monoidal category C, the category of A modules in C obtains a natural crossed G-braided structure. In the case of general commutative…
In this paper we investigate the descriptional complexity of knot theoretic problems and show upper bounds for planarity problem of signed and unsigned knot diagrams represented by Gauss words. Since a topological equivalence of knots can…