East, James Phillip Hinton

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Sydney eScholarship

On Monoids Related to Braid Groups andTransformation SemigroupsJames EastSchool of Mathematics and StatisticsUniversity of SydneySeptember 2005A thesis submitted in fulfilment of the requirementsfor the degree of Doctor of PhilosophyContentsList of Figures ivAcknowledgements viiChapter 1. Introduction 1Overview of Thesis 7General notation 8Chapter 2. Preliminaries 92.1. Semigroups and Groups 92.2. Transformation Semigroups 112.3. Presentations 132.4. The Braid Group 19Chapter 3. Factorisable Inverse Monoids 273.1. Factorisable Inverse Monoids 273.2. Coset Monoids and Embeddings 313.3. The Structure of Factorisable Inverse Monoids 353.4. Presentations of Factorisable Inverse Monoids 37Chapter 4. Factorisable Inverse Braid Monoids 414.1. The Inverse Braid Monoid 434.2. The Factorisable Braid Monoid 494.3. The Permeable Braid Monoid 544.4. Visualising the Elements of FBn and PBn 58Chapter 5. Presentations of Factorisable Inverse Braid Monoids 605.1. The Inverse Braid Monoid 615.2. The Factorisable Braid Monoid 755.3. The Permeable Braid Monoid 93Chapter 6. Pure Factorisable Inverse Braid Monoids 996.1. The Pure Inverse Braid Monoid 1026.2. The Pure Factorisable Braid Monoid 1066.3. The Pure Permeable Braid Monoid 111iiCONTENTS iiiChapter 7. Presentations of Pure Factorisable Inverse Braid Monoids 1207.1. The Pure Inverse Braid Monoid 1217.2. The Pure Factorisable Braid Monoid 1237.3. The Pure Permeable Braid Monoid 130Chapter 8. Applications of Order-Preserving Partial Permutations 1318.1. The Monoid of Order-Preserving Partial Permutations 1328.2. The Inverse Braid Monoid 1418.3. The Monoid of Order-Preserving Partial Braids 1518.4. The Singular Part of the Inverse Braid Monoid 1578.5. The Dual Symmetric Inverse Semigroup 1608.6. The Factorisable and Permeable Braid Monoids 162Appendix A. Catalogue of Presentations 163A.1. Braid Monoids 163A.2. Transformation Semigroups 165A.3. Pure Braid Monoids 168Bibliography 170Index of Notation 173Index 176List of Figures1.1 Representing the trefoil knot (left) as a closed braid (right). 11.2 Un-knotting the trefoil. 11.3 Representing the singular trefoil knot (left) as a closed singular braid(right). 21.4 Two resolutions β1 ∈ B2 (left) and β2 ∈ B2 (right) of the singular braidβ ∈ SB2 (centre). 21.5 The braid β2 ∈ B2 closes to give the unknot. 21.6 A partial braid in IB3. 31.7 The braid β ∈ B4. 41.8 The trivial braid 1 ∈ B4. 41.9 The modified braids β ′ (left) and 1′ (right) after removing the secondand third strings. 51.10 Projections of β ′ (left) and 1′ (right) onto a horizontal plane. 51.11 The braid γ ∈ B4. 51.12 Representing γ as a commutator. 61.13 A merge-and-part homotopy from γ to 1. 62.1 The element α ∈ PT8 defined by 1α = 3, 3α = 4, 6α = 8, 7α = 3. 122.2 The product (composite) of two elements α, β ∈ PT4. 132.3 The elements of the semigroup S = {a, b, c, d, e} ⊆ I2. 132.4 The generator Eij ∈ Eqn. 192.5 Relation (E3): Eij ∨ Ejk = Ejk ∨ Eik = Eik ∨ Eij if i < j < k. 192.6 A braid on {1, 2, 3, 4}. 202.7 The product of two braids β, γ ∈ B3. 212.8 A braid equivalent to αβ from Figure 2.7. 212.9 The identity braid 1 ∈ B5. 222.10 The inverse of a braid: ββ−1 ∼ β−1β ∼ 1. 222.11 The braids ςi (left) and ς−1i (right) in Bn. 23ivLIST OF FIGURES v2.12 The braids ςij (left) and ς−1ij (right) in Bn. 232.13 The braids αij (left) and α−1ij (right) in Bn. 243.1 An element of IX \ FX where X = {1, 2, 3, . . . }. 283.2 A picture of a block bijection θ ∈ I∗8 . 303.3 The product of two block bijections θ1, θ2 ∈ I∗4 . 303.4 Two pictures of a uniform block bijection in F∗8. 313.5 The product of two uniform block bijections θ1, θ2 ∈ F∗4. 313.6 The equivalences E1 (top), E2 (middle), and E1 ∨ E2 (bottom) in EqX . 344.1 The braids δij (left) and γij (right) in Bn. 424.2 Some partial braids on {1, 2, 3, 4}. 434.3 The product of two partial braids. 444.4 The inverse of a partial braid: ββ−1β ∼ β and β−1ββ−1 ∼ β−1. 454.5 The partial braids 1, 1A ∈ IB6 where A = {2, 3, 5} ⊆ {1, 2, 3, 4, 5, 6}. 454.6 The braids β−1ςiβ (left) and β−1ς2i β (right) for the case in whichk = iβ¯ < l = (i+ 1)β¯. 464.7 The strings s (left) and t (right). 494.8 The strings merging. 494.9 Possible configurations before and after merge-and-part. 494.10 The homotopies H3, H4, and H5. 524.11 Possible configurations before and after permeating. 554.12 A picture of an element of Eq6 o B6. 584.13 The action of a braid β ∈ B6 on an equivalence E ∈ Eq6. 594.14 The product of two elements (E1, β), (E2, γ) ∈ Eq4 o B4. 595.1 The partial braid 1{i}c ∈ IBn. 625.2 Relation (IB3): τς1τς1 = ς1τς1τ = τς1τ , where τ = 1{1}c . 635.3 The partial braids (ς1){1}c (left) and (ς1){2}c (right) in IBn. 695.4 The partial braid θi = (ςi){i}c in IBn. 695.5 Relation (IB1)′′: θiςiθi = θi for all i. 705.6 Relation (IB2)′′: ςiθi = θi+1ςi+1 for all i ≤ n− 2. 705.7 The relation σifi = fiσi is not in (R′′IB)] for any i. 715.8 The relation fifj = fjfi is in (R′′IB)] whenever |i− j| > 1. 71LIST OF FIGURES vi5.9 The partial braid (ςi){i+1}c in IBn. 715.10 Relation (FB3): xyxy = yxyx where x = (E12, 1) and y = (1, ς2). 815.11 Relation (FB4): xyxy = yxyx where x = (E12, 1) and y = (1, ς2ς3ς1ς2). 815.12 The singular braid τi ∈ SBn. 865.13 Possible configurations before and after move (ii). 875.14 Possible configurations before and after move (iii). 875.15 An intermediate triple singular point created during move (iii). 875.16 The relation ς2i τi+1  τi+1ς2i does not appear to hold in FSBn. 895.17 An (E , 2)-homotopy from ς22 to ς1ς2ς2ς1. 955.18 Relation (PB6)′: xyz = zyx where x = [[1]]Ei,i+1 , y = ςj, and z = [[1]]Ej,j+1in the case j = i + 1. 977.1 The braid βi1i2i3 ∈ B. 1237.2 An example calculation: ς−11 α13α−123 ς1 = α−112 α−113 α23α12 =α−112 α−113 (α23α−113 )α13α12. 1248.1 The maps λˇA, ρˇA ∈ POI8 where A = {2, 3, 5, 8}. 1328.2 The maps λˇ4 ∈ L8 and ρˇ4 ∈ R8. 1338.3 Relation (L1): λˇiλˇj = λˇj+1λˇi if 1 ≤ i ≤ j ≤ n− 1. 1348.4 Relation (L2): λˇiλˇn = λˇi if 1 ≤ i ≤ n. 1348.5 Relation (BL2): ςiλˇj = λˇj−1 if 1 ≤ i = j − 1 ≤ n− 1. 1438.6 Relation (BL4): ςiλˇj = λˇjςi−1 if 1 ≤ j < i ≤ n− 1. 1438.7 Relation (PL3): αijλˇk = λˇkαi−1,j−1 if 1 ≤ k < i < j ≤ n. 1538.8 A planar block bijection in I∗8 . 1608.9 A picture of α ∈ POI8 (grey) and θα ∈ POI∗9 (black). 1618.10 The images λˇif (left) and ρˇif (right) of the generators of POIn−1. 161AcknowledgementsFirst and foremost I wish to thank my supervisor David Easdown and express my gratitudefor his guidance and support. David was always willing to give freely of his time andexperience, and was enormously helpful in the shaping and developing of this thesis.I also benefited greatly from discussions with Des FitzGerald, Bob Howlett, AndrewMathas, James Parkinson, and Jono Kusilek.My time at Sydney University was made all the more enjoyable by my friends and col-leagues, especially Tim and Brad who I had the pleasure of sharing an office with.Thanks go to my family for their encouragement over the years, and in particular to mywife Roslyn for her constant support and understanding.Finally, I would like to thank God for the ability and opportunity to have pursued myinterest in mathematics.This thesis contains no material which has been accepted for the award of any other degree.To the best of my knowledge and belief this thesis contains no material previously publishedby any other person except where due acknowledgment has been made.vii

On Monoids Related to Braid Groups and Transformation Semigroups

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On Monoids Related to Braid Groups and Transformation Semigroups

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