Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Framed Blob Monoids

Jesús Juyumaya, Diego Lobos

TL;DR

This work develops a comprehensive framework for blob and framed blob monoids, culminating in diagrammatic and partition-monoid framizations $Bl_n$, $Bl_{d,n}$, and their abacus/connected variants. It introduces indexing-matrix parametrizations and Martin-Saleur diagrams to obtain normal forms, enabling explicit cardinality calculations through the counts $\Omega_k^{(n)}$ and $\chi_k^{(n)}$, with recursive relations and connections to Catalan numbers. The paper further constructs framed versions by introducing beads and framing generators, proving isomorphisms between algebraic and diagrammatic framings, and deriving alternative combinatorial formulas for cardinalities. A separate connected framization $Bl_{d,n}^c$ is defined and realized via abacus hook diagrams, yielding distinct cardinalities from $Bl_{d,n}$ for $d\ge 2$, and thus non-isomorphism of these framizations. Overall, the results provide a detailed combinatorial and diagrammatic toolkit for framizations of knot-related monoids, with explicit enumeration formulas and valuable links to TL diagrams and planar partitions.

Abstract

We introduce and study blob and framed blob monoids. In particular, several realizations of these monoids are given. We compute the cardinality of the framed blob monoid and derive some combinatorial formulas involving this cardinality.

Framed Blob Monoids

TL;DR

This work develops a comprehensive framework for blob and framed blob monoids, culminating in diagrammatic and partition-monoid framizations , , and their abacus/connected variants. It introduces indexing-matrix parametrizations and Martin-Saleur diagrams to obtain normal forms, enabling explicit cardinality calculations through the counts and , with recursive relations and connections to Catalan numbers. The paper further constructs framed versions by introducing beads and framing generators, proving isomorphisms between algebraic and diagrammatic framings, and deriving alternative combinatorial formulas for cardinalities. A separate connected framization is defined and realized via abacus hook diagrams, yielding distinct cardinalities from for , and thus non-isomorphism of these framizations. Overall, the results provide a detailed combinatorial and diagrammatic toolkit for framizations of knot-related monoids, with explicit enumeration formulas and valuable links to TL diagrams and planar partitions.

Abstract

We introduce and study blob and framed blob monoids. In particular, several realizations of these monoids are given. We compute the cardinality of the framed blob monoid and derive some combinatorial formulas involving this cardinality.
Paper Structure (16 sections, 42 theorems, 105 equations, 11 figures)

This paper contains 16 sections, 42 theorems, 105 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Blob monoid
  5. Indexing matrices
  6. Martin-Saleur blob monoid
  7. Blob monoid as partition monoid
  8. Partition and abacus monoids
  9. Motzkin blob monoid
  10. Hook blob monoid
  11. Abacus blob monoid
  12. Framed blob monoid
  13. Some combinatorics
  14. Some formulas for $\Omega_k^{(n)}$
  15. Alternative formula
  16. ...and 1 more sections

Key Result

Proposition 2.3

For $j\leq i$, define $U_{i,j} :=u_i u_{i-1}\cdots u_j$. Every element of $Bl_n$ can be written in normal form, that is, in the form where

Figures (11)

  • Figure 1: On the left is $\mathfrak{U}\left[i0\right]$ and on the right is $\mathfrak{U}\left[ij\right]$ with $0\not= j<i.$
  • Figure 2: The product $\alpha\beta$.
  • Figure 3: The elementary tangle $U_i$.
  • Figure 4: The diagram for $U_0$.
  • Figure 5: Martin-Saleur blob diagram $D$ of (\ref{['eq-tikz-blobbed-diagram']}) as an element of $\mathfrak{M}_9^{(1)}$.
  • ...and 6 more figures

Theorems & Definitions (114)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Example 2.9
  • ...and 104 more