On Monoids Related to Braid Groups and Transformation Semigroups. James East

Transcription

1 On Monoids Related to Braid Groups and Transformation Semigroups James East School of Mathematics and Statistics University of Sydney September 2005 A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy

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3 Contents List of Figures Acknowledgements iv vii Chapter 1. Introduction 1 Overview of Thesis 7 General notation 8 Chapter 2. Preliminaries Semigroups and Groups Transformation Semigroups Presentations The Braid Group 19 Chapter 3. Factorisable Inverse Monoids Factorisable Inverse Monoids Coset Monoids and Embeddings The Structure of Factorisable Inverse Monoids Presentations of Factorisable Inverse Monoids 37 Chapter 4. Factorisable Inverse Braid Monoids The Inverse Braid Monoid The Factorisable Braid Monoid The Permeable Braid Monoid Visualising the Elements of FB n and PB n 58 Chapter 5. Presentations of Factorisable Inverse Braid Monoids The Inverse Braid Monoid The Factorisable Braid Monoid The Permeable Braid Monoid 93 Chapter 6. Pure Factorisable Inverse Braid Monoids The Pure Inverse Braid Monoid The Pure Factorisable Braid Monoid The Pure Permeable Braid Monoid 111 ii

4 CONTENTS iii Chapter 7. Presentations of Pure Factorisable Inverse Braid Monoids The Pure Inverse Braid Monoid The Pure Factorisable Braid Monoid The Pure Permeable Braid Monoid 130 Chapter 8. Applications of Order-Preserving Partial Permutations The Monoid of Order-Preserving Partial Permutations The Inverse Braid Monoid The Monoid of Order-Preserving Partial Braids The Singular Part of the Inverse Braid Monoid The Dual Symmetric Inverse Semigroup The Factorisable and Permeable Braid Monoids 162 Appendix A. Catalogue of Presentations 163 A.1. Braid Monoids 163 A.2. Transformation Semigroups 165 A.3. Pure Braid Monoids 168 Bibliography 170 Index of Notation 173 Index 176

5 List of Figures 1.1 Representing the trefoil knot (left) as a closed braid (right) Un-knotting the trefoil Representing the singular trefoil knot (left) as a closed singular braid (right) Two resolutions β 1 B 2 (left) and β 2 B 2 (right) of the singular braid β SB 2 (centre) The braid β 2 B 2 closes to give the unknot A partial braid in IB The braid β B The trivial braid 1 B The modified braids β (left) and 1 (right) after removing the second and third strings Projections of β (left) and 1 (right) onto a horizontal plane The braid γ B Representing γ as a commutator A merge-and-part homotopy from γ to The element α PT 8 defined by 1α = 3, 3α = 4, 6α = 8, 7α = The product (composite) of two elements α, β PT The elements of the semigroup S = {a, b, c, d, e} I The generator E ij Eq n Relation (E3): E ij E jk = E jk E ik = E ik E ij if i < j < k A braid on {1, 2, 3, 4} The product of two braids β, γ B A braid equivalent to αβ from Figure The identity braid 1 B The inverse of a braid: ββ 1 β 1 β The braids ς i (left) and ς 1 i (right) in B n. 23 iv

6 LIST OF FIGURES v 2.12 The braids ς ij (left) and ς 1 ij (right) in B n The braids α ij (left) and α 1 ij (right) in B n An element of I X \ F X where X = {1, 2, 3,... } A picture of a block bijection θ I The product of two block bijections θ 1, θ 2 I Two pictures of a uniform block bijection in F The product of two uniform block bijections θ 1, θ 2 F The equivalences E 1 (top), E 2 (middle), and E 1 E 2 (bottom) in Eq X The braids δ ij (left) and γ ij (right) in B n Some partial braids on {1, 2, 3, 4} The product of two partial braids The inverse of a partial braid: ββ 1 β β and β 1 ββ 1 β The partial braids 1, 1 A IB 6 where A = {2, 3, 5} {1, 2, 3, 4, 5, 6} The braids β 1 ς i β (left) and β 1 ςi 2 β (right) for the case in which k = i β < l = (i + 1) β The strings s (left) and t (right) The strings merging Possible configurations before and after merge-and-part The homotopies H 3, H 4, and H Possible configurations before and after permeating A picture of an element of Eq 6 B The action of a braid β B 6 on an equivalence E Eq The product of two elements (E 1, β), (E 2, γ) Eq 4 B The partial braid 1 {i} c IB n Relation (IB3): τς 1 τς 1 = ς 1 τς 1 τ = τς 1 τ, where τ = 1 {1} c The partial braids (ς 1 ) {1} c (left) and (ς 1 ) {2} c (right) in IB n The partial braid θ i = (ς i ) {i} c in IB n Relation (IB1) : θ i ς i θ i = θ i for all i Relation (IB2) : ς i θ i = θ i+1 ς i+1 for all i n The relation σ i f i = f i σ i is not in (R IB ) for any i The relation f i f j = f j f i is in (R IB ) whenever i j > 1. 71

7 LIST OF FIGURES vi 5.9 The partial braid (ς i ) {i+1} c in IB n Relation (FB3): xyxy = yxyx where x = (E 12, 1) and y = (1, ς 2 ) Relation (FB4): xyxy = yxyx where x = (E 12, 1) and y = (1, ς 2 ς 3 ς 1 ς 2 ) The singular braid τ i SB n Possible configurations before and after move (ii) Possible configurations before and after move (iii) An intermediate triple singular point created during move (iii) The relation ς 2 i τ i+1 τ i+1 ς 2 i does not appear to hold in FSB n An (E, 2)-homotopy from ς 2 2 to ς 1 ς 2 ς 2 ς Relation (PB6) : xyz = zyx where x = [1] Ei,i+1, y = ς j, and z = [1] Ej,j+1 in the case j = i The braid β i1 i 2 i 3 B An example calculation: ς1 1 α 13α23 1 ς 1 = α12 1 α 1 13 α 23α 12 = α12 1 α13 1 (α 23 α13 1 )α 13 α The maps ˇλ A, ˇρ A POI 8 where A = {2, 3, 5, 8} The maps ˇλ 4 L 8 and ˇρ 4 R Relation (L1): ˇλ iˇλj = ˇλ j+1ˇλi if 1 i j n Relation (L2): ˇλ iˇλn = ˇλ i if 1 i n Relation (BL2): ς iˇλj = ˇλ j 1 if 1 i = j 1 n Relation (BL4): ς iˇλj = ˇλ j ς i 1 if 1 j < i n Relation (PL3): α ijˇλk = ˇλ k α i 1,j 1 if 1 k < i < j n A planar block bijection in I A picture of α POI 8 (grey) and θ α POI 9 (black) The images ˇλ i f (left) and ˇρ i f (right) of the generators of POI n

8 Acknowledgements First and foremost I wish to thank my supervisor David Easdown and express my gratitude for his guidance and support. David was always willing to give freely of his time and experience, and was enormously helpful in the shaping and developing of this thesis. I also benefited greatly from discussions with Des FitzGerald, Bob Howlett, Andrew Mathas, James Parkinson, and Jono Kusilek. My time at Sydney University was made all the more enjoyable by my friends and colleagues, 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 my wife Roslyn for her constant support and understanding. Finally, I would like to thank God for the ability and opportunity to have pursued my interest 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 published by any other person except where due acknowledgment has been made. vii