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Framization and Deframization

Francesca Aicardi, Jesús Juyumaya, Paolo Papi

TL;DR

The paper develops a unified framework for framization and deframization of knot-theoretic monoids and their algebras. It introduces framed versions of classical knot monoids (symmetric, braid, set partition, Jones, Brauer, and rook) via abacus constructions with modular bead counts, and defines a systematic deframization procedure using bridges to produce ramified (tied) monoids and algebras. Key contributions include explicit framizations (e.g., $S_{d,n}$, $ ext{F}_{d,n}$, $P_{d,n}$, $J_{d,n}$, $Br_{d,n}$, $R_{d,n}$, $R'_{d,n}$) along with their abacus realizations and cardinalities, plus a concrete recipe to obtain tied monoids (e.g., $tS_n$, $tJ_n$, $tBr_n$, $tR_n$) from framed ones. The authors extend these notions to algebras, deriving framed and tied versions of Hecke, bt, TL, and BMW algebras, and illustrate how deframization preserves dimension under suitable constructions, yielding diagrammatically interpretable invariants for tied/framed links. The framework thus links diagrammatic knot theory with algebraic deformational families, enabling new invariants and a deeper understanding of the relationship between framed, tied, and ramified structures. Overall, the work provides a cohesive methodology to transition between framed and ramified objects, with potential applications to knot invariants and topological quantum algebra.

Abstract

Starting from the geometric construction of the framed braid group, we define and study the framization of several Brauer-type monoids and also the set partition monoid, all of which appear in knot theory. We introduce the concept of deframization, which is a procedure to obtain a tied monoid from a given framed monoid. Furthermore, we show in detail how this procedure works on the monoids mentioned above. We also discuss the framization and deframization of some algebras, which are deformations, respectively, of the framized and deframized monoids discussed here.

Framization and Deframization

TL;DR

The paper develops a unified framework for framization and deframization of knot-theoretic monoids and their algebras. It introduces framed versions of classical knot monoids (symmetric, braid, set partition, Jones, Brauer, and rook) via abacus constructions with modular bead counts, and defines a systematic deframization procedure using bridges to produce ramified (tied) monoids and algebras. Key contributions include explicit framizations (e.g., , , , , , , ) along with their abacus realizations and cardinalities, plus a concrete recipe to obtain tied monoids (e.g., , , , ) from framed ones. The authors extend these notions to algebras, deriving framed and tied versions of Hecke, bt, TL, and BMW algebras, and illustrate how deframization preserves dimension under suitable constructions, yielding diagrammatically interpretable invariants for tied/framed links. The framework thus links diagrammatic knot theory with algebraic deformational families, enabling new invariants and a deeper understanding of the relationship between framed, tied, and ramified structures. Overall, the work provides a cohesive methodology to transition between framed and ramified objects, with potential applications to knot invariants and topological quantum algebra.

Abstract

Starting from the geometric construction of the framed braid group, we define and study the framization of several Brauer-type monoids and also the set partition monoid, all of which appear in knot theory. We introduce the concept of deframization, which is a procedure to obtain a tied monoid from a given framed monoid. Furthermore, we show in detail how this procedure works on the monoids mentioned above. We also discuss the framization and deframization of some algebras, which are deformations, respectively, of the framized and deframized monoids discussed here.
Paper Structure (40 sections, 20 theorems, 78 equations, 21 figures)

This paper contains 40 sections, 20 theorems, 78 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Notational convention.
  4. Braid and symmetric groups
  5. The rook monoid $R_n$
  6. The monoid $P_n$ of set partitions
  7. Jones and Brauer monoids
  8. The Hecke algebra
  9. The rook algebra
  10. The Temperley-Lieb algebra
  11. The BMW algebra
  12. Framization
  13. Framization of symmetric and braid group
  14. The framed set partition monoid
  15. Framed Jones monoid
  16. ...and 25 more sections

Key Result

Proposition 3.1

If $g$ is a gap of $w\in J_n$, then the diagram of $w$ has a line with endpoint $g$ in both top and bottom segment, and the diagram of the normal form has a vertical line at $g$. Moreover, the integers $\underline j_1,\dots,\underline j_k$ are either the left points of down-brackets or the lower poi

Figures (21)

  • Figure 1: Two elements of $C_{7,5}$ and their product.
  • Figure 2: Diagrams of the generators $\mathsf{o}_i$, $\mathsf{o}_i^3$ and $\mathsf{o}_i^{-1}= \mathsf{o}_i^{5}$ in $C_{6,n}$.
  • Figure 3: Two elements of $\mathcal{F}_{3,5}$ and their product.
  • Figure 4: Two elements of $P_5$ and their product.
  • Figure 5: Two elements of $P_{7,5}$ and their product.
  • ...and 16 more figures

Theorems & Definitions (81)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Definition 3.1
  • Definition 3.2
  • Example 3.1
  • Remark 3.2
  • Proposition 3.1
  • proof
  • Example 3.2
  • ...and 71 more