DOCSLIB.ORG

Monopole Floer Homology, Link Surgery, and Odd Khovanov Homology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Monopole Floer Homology, Link Surgery, and Odd Khovanov Homology Jonathan Michael Bloom Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2011 c 2011 Jonathan Michael Bloom All Rights Reserved ABSTRACT Monopole Floer Homology, Link Surgery, and Odd Khovanov Homology Jonathan Michael Bloom We construct a link surgery spectral sequence for all versions of monopole Floer homology with mod 2 coefficients, generalizing the exact triangle. The spectral sequence begins with the monopole Floer homology of a hypercube of surgeries on a 3-manifold Y , and converges to the monopole Floer homology of Y itself. This allows one to realize the latter group as the homology of a complex over a combinatorial set of generators. Our construction relates the topology of link surgeries to the combinatorics of graph associahedra, leading to new inductive realizations of the latter. As an application, given a link L in the 3-sphere, we prove that the monopole Floer homology of the branched double-cover arises via a filtered perturbation of the differential on the reduced Khovanov complex of a diagram of L. The associated spectral sequence carries a filtration grading, as well as a mod 2 grading which interpolates between the delta grading on Khovanov homology and the mod 2 grading on Floer homology. Furthermore, the bigraded isomorphism class of the higher pages depends only on the Conway-mutation equivalence class of L. We constrain the existence of an integer bigrading by considering versions of the spectral sequence with non-trivial Uy action, and determine all monopole Floer groups of branched double-covers of links with thin Khovanov homology. Motivated by this perspective, we show that odd Khovanov homology with integer coef- ficients is mutation invariant. The proof uses only elementary algebraic topology and leads to a new formula for link signature that is well-adapted to Khovanov homology. Table of Contents 1 Introduction 1 1.1 Background on monopole Floer homology . 4 2 The link surgery spectral sequence 13 2.1 Hypercubes and permutohedra . 15 2.2 The link surgery spectral sequence: construction . 22 2.3 Product lattices and graph associahedra . 32 2.4 The surgery exact triangle . 42 2.5 The link surgery spectral sequence: convergence . 47 2.5.1 Grading . 48 2.5.2 Invariance . 49 2.6 The Uy map and HMg •(Y )............................ 56 3 Realizations of graph associahedra 69 4 Odd Khovanov homology and Conway mutation 76 4.1 A thriftier construction of reduced odd Khovanov homology . 78 4.2 The original construction of reduced odd Khovanov homology . 82 4.3 Branched double-covers, mutation, and link surgery . 86 5 Link signature and homological width 89 6 From Khovanov homology to monopole Floer homology 96 6.1 Khovanov homology and branched double-covers . 98 i 6.1.1 Grading . 102 6.1.2 Invariance . 107 6.2 The spectral sequence for a family of torus knots? . 108 7 Donaldson's TQFT 114 7.1 The algebraic perspective . 114 7.2 The geometric perspective . 117 7.3 Monopole Floer homology and positive scalar curvature . 120 8 Khovanov homology and Uy 127 8.1 An integer bigrading? . 131 8.2 The Brieskorn sphere −Σ(2; 3; 7) . 134 8.3 Beyond branched double-covers . 141 9 Appendix: Morse homology with boundary via path algebras 144 Bibliography 149 ii List of Figures 2.1 The cobordism for the hypercube f0; 1g3. ................... 17 2.2 The hexagon of metrics for f0; 1g3........................ 21 2.3 The permutohedron of metrics for f0; 1g4.................... 21 2.4 The associahedron of metrics for f0; 1; 1g2................... 35 2.5 The polyhedron of metrics for f0; 1; 1g × f0; 1g2. 36 2.6 The graph associahedron of the 3-clique with one leaf. 38 2.7 From link surgeries to lattices, graphs, and polytopes. 40 2.8 The cobordism for f0; 1; 1; 00g and the pentagon of metrics. 44 2.9 The cobordism used to the construct the homotopy equivalence for f0; 1g2. 50 2.10 The hexagon of metrics used in the homotopy equivalence for f0; 1g2. 52 2.11 The cobordism used to construct the Uy map andm ~ (W ). 58 2 2.12 The cobordism and hexagon of metrics for the HMg • version of f0; 1g . 60 2.13 The associahedron of metrics for f0; 1; 1; 00g × f0; 1g. 64 3.1 Sliding the sphere to construct realizations of graph associahedra. 70 3.2 Realizations of the permutohedron and a graph associahedron. 70 3.3 The graphs G and G0 for Λ and Λ × f0; 1g. .................. 71 3.4 Realizations of the associahedra. 72 4.1 Oriented resolution conventions. 79 4.2 Linking number conventions. 80 4.3 The Kinoshita-Terasaka and Conway knots. 83 4.4 Constructing a surgery diagram for the branched double-cover. 88 iii 5.1 A dealternator connected, 2-almost alternating diagram of T (3; 7). 92 5.2 A local move to ensure dealternator connectedness. 93 6.1 The short arc between the strands of a resolved crossing. 100 6.2 A surgery diagram for the branched double cover of the trefoil. 101 6.3 The branched double-cover of the cube of resolutions of the trefoil. 103 6.4 Four types of crossings in an oriented diagram with checkerboard coloring. 106 6.5 Conjectural HMg • spectral sequences for T (3; 6n ± 1). 110 7.1 Commutative diagram relating the functors HMg • and CKh. 114 7.2 Commutative diagram for the functor Λ∗H1. 116 8.1 Conjectural HMd • and other spectral sequences for T (3; 7). 137 8.2 Conjectural HMd • spectral sequences for T (3; 6n ± 1) . 139 8.3 Conjectural HMd • spectral sequences for T (3; 19) . 140 8.4 Comparison of Khf (T (3; 7)) and Khf (P (−2; 3; 7)). 141 9.1 Path algebras and Morse homology for manifolds with boundary. 145 9.2 The differential and identities as elements of a path algebra. 147 9.3 The path algebra of a cobordism with fixed metric. 148 9.4 Chain map for a cobordism with three ends. 150 iv Acknowledgments I am deeply grateful to my advisor, Peter Ozsv´ath,for his unwavering and enthusiastic support over the last five years, and for providing wonderful problems to explore. Even before I arrived at Columbia, he arranged for me to attend the 2006 Summer School at the Park City Mathematical Institute, a formative experience that kindled my interest in low-dimensional topology and geometry. I started graduate school with a focus on Heegaard Floer homology and Khovanov homology. At Professor Ozsv´ath'ssuggestion, and with his invaluable guidance, I begin to pursue connections with monopole Floer theory. This has taken me down an exciting and rewarding path. I am indebted to Tim Perutz and Tom Mrowka for generous mentoring in monopole Floer theory. I have also benefitted from the expertise of Adam Knapp, Peter Kronheimer, and Max Lipyanskiy. I want to thank Peter Kronheimer and Tom Mrowka for investing the time and energy to write their foundational book Monopoles and Three-Manifolds. My copy has surely seen better days. It has been a privilege to be a member of the mathematical community at Columbia. I am grateful for the friendship and support of John Baldwin, Ben Elias, Allison Gilmore, Josh Greene, Eli Grigsby, Adam Levine, Tom Peters, Ina Petkova, Thibaut Pugin, Vera Vertesi, and Rumen Zarev. My research has benefitted from discussions with Professors Dave Bayer, Mikhail Khovanov, Robert Lipshitz, and Dylan Thurston at Columbia, and Professors Satyan Devadoss, Danny Ruberman, James Stasheff, and Zolt´anSz´abo abroad, among many others. I had the good fortune of spending the spring of 2010 at the Mathematical Sciences Research Institute for the program Homology Theories of Knots and Links. I want to thank my uncle Steven Greenfield for providing a warm home away from home during my stay in Berkeley. I have spent my final year of graduate school as an exchange scholar at MIT, and v would like to thank the department for welcoming me. I am also grateful for support from the National Science Foundation (grant DMS-0739392). I want to express my appreciation to several people who nurtured my interest in math- ematics in my youth: my mentor Greg Bachelis, my calculus teacher Joe Brandell, and my cousin David Goss, who inspired me to attend the Ross Program. Through it all, my parents, sister, and grandparents have been a constant source of love and support. Finally, I want to thank Masha for her endless encouragement and so much more. vi CHAPTER 1. INTRODUCTION 1 Chapter 1 Introduction Monopole Floer homology is a gauge-theoretic invariant defined via Morse theory on the Chern-Simons-Dirac functional. As such, the underlying chain complex is generated by Seiberg-Witten monopoles over a 3-manifold, and the differential counts monopoles over the product of the 3-manifold with R. We review this construction in Section 1.1. In [27], a surgery exact triangle is associated to a triple of surgeries on a knot in a 3-manifold (for a precursor in instanton Floer homology, see [11], [18]). In Chapter 2, we construct a link surgery spectral sequence in monopole Floer homology, generalizing the exact triangle. This is a spectral sequence which starts at the monopole Floer homology of a hypercube of surgeries on Y along L, and converges to the monopole Floer homology of Y itself. The differentials count monopoles on 2-handle cobordisms equipped with families of metrics parameterized by polytopes called permutohedra. Those metrics parameterized by the boundary of the permutohedra are stretched to infinity along collections of hypersur- faces representing surgered 3-manifolds.
Recommended publications
  • Geometry of Generalized Permutohedra

    Geometry of Generalized Permutohedra

    Geometry of Generalized Permutohedra by Jeffrey Samuel Doker A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Federico Ardila, Co-chair Lior Pachter, Co-chair Matthias Beck Bernd Sturmfels Lauren Williams Satish Rao Fall 2011 Geometry of Generalized Permutohedra Copyright 2011 by Jeffrey Samuel Doker 1 Abstract Geometry of Generalized Permutohedra by Jeffrey Samuel Doker Doctor of Philosophy in Mathematics University of California, Berkeley Federico Ardila and Lior Pachter, Co-chairs We study generalized permutohedra and some of the geometric properties they exhibit. We decompose matroid polytopes (and several related polytopes) into signed Minkowski sums of simplices and compute their volumes. We define the associahedron and multiplihe- dron in terms of trees and show them to be generalized permutohedra. We also generalize the multiplihedron to a broader class of generalized permutohedra, and describe their face lattices, vertices, and volumes. A family of interesting polynomials that we call composition polynomials arises from the study of multiplihedra, and we analyze several of their surprising properties. Finally, we look at generalized permutohedra of different root systems and study the Minkowski sums of faces of the crosspolytope. i To Joe and Sue ii Contents List of Figures iii 1 Introduction 1 2 Matroid polytopes and their volumes 3 2.1 Introduction . .3 2.2 Matroid polytopes are generalized permutohedra . .4 2.3 The volume of a matroid polytope . .8 2.4 Independent set polytopes . 11 2.5 Truncation flag matroids . 14 3 Geometry and generalizations of multiplihedra 18 3.1 Introduction .
  • The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing

    The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing

    The Topology of Bendless Orthogonal Three-Dimensional Graph Drawing David Eppstein Computer Science Dept. Univ. of California, Irvine What’s in this talk? Unexpected equivalence between a style of graph drawing and a type of topological embedding 3d grid drawings in which each vertex has three perpendicular edges 2d surface embeddings in which the faces meet nicely and may be 3-colored ...and its algorithmic consequences xyz graphs Let S be a set of points in three dimensions such that each axis-aligned line contains zero or two points of S Draw an edge between any two points on an axis-aligned line Three xyz graphs within a 3 x 3 x 3 grid Note that edges are allowed to cross Crossings differ visually from vertices as vertices never have two parallel edges The permutohedron Convex hull of all permutations of (1,2,3,4) in 3-space x+y+z+w=10 Forms a truncated octahedron (4,1,2,3) (3,1,2,4) (4,2,1,3) (3,2,1,4) (4,1,3,2) (2,1,3,4) (4,3,1,2) (2,3,1,4) (3,1,4,2) (4,2,3,1) (2,1,4,3) (4,3,2,1) (1,2,3,4) (3,4,1,2) (1,3,2,4) (2,4,1,3) (3,2,4,1) (1,2,4,3) (3,4,2,1) (1,4,2,3) (2,3,4,1) (1,3,4,2) (2,4,3,1) (1,4,3,2) Inverting the permutohedron Move each permutation vertex to its inverse permutation affine transform so that the edges are axis-aligned A polyhedron for the inverse permutohedron Rearrange face planes to form nonconvex topological sphere A different xyz graph on 4-element permutations Project (x,y,z,w) to (x,y,z) xyz graphs with many vertices in a small bounding box In n x n x n box, place points such that x+y+z = 0 or 1 mod n n = 4, the
  • Günther's Negation Cycles and Morphic Palidromes Applications of Morphic Palindromes to the Study of Günther’S Negative Language

    Günther's Negation Cycles and Morphic Palidromes Applications of Morphic Palindromes to the Study of Günther’S Negative Language

    Summer-Edition 2017 — vordenker-archive — Rudolf Kaehr (1942-2016) Title Günther's Negation Cycles and Morphic Palidromes Applications of morphic palindromes to the study of Günther’s negative language Archive-Number / Categories 3_35 / K01, K08, K09, K10 Publication Date 2014 Keywords / Topics o GÜNTHER’S NEGATIVE LANGUAGE : Cycles as palindromes, Palindromes as Cycles, Morphic even palin- dromes, o COMBINATORIAL ASPECTS OF GÜNTHER’S NEGATIVE LANGUAGES : Günther’s approaches to permuta- tions, Hamilton paths vs. Heideggerian journeys, Combinatorial knots and braids Disciplines Computer Science, Logic and Foundations of Mathematics, Cybernetics, Theory of Science, Memristive Systems Abstract Günther’s concept of technology is based on a ‘melting’ of number and notion, Begriff und Zahl, in a polycontextural setting. A key to its study is established by the negation-cycles of polycontextural (meontic) logics that are establishing a negative language. It is proposed that a morphogrammatic understanding of technology is uncovering a level deeper than polycontexturality and is connecting numbers and concepts not just with the will and its praxeological actions but with ‘labour’ (Arbeit) in the sense of Marx’s Grundrisse ("Arbeit als absolute Armut”.) Palin- dromic cycles are offering a deeper access for the definition of negative languages than the meontic cycles. Morphogrammatics is opening up languages, i.e. writting systems of creativity and labour. For a interactive reading, download the CDF reader for free: http://www.wolfram.com/cdf-player/
  • Faces of Generalized Permutohedra

    Faces of Generalized Permutohedra

    Documenta Math. 207 Faces of Generalized Permutohedra Alex Postnikov 1, Victor Reiner 2, Lauren Williams 3 Received: May 5, 2007 Revised: June 16, 2008 Communicated by G¨unter Ziegler Abstract. The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and γ- vectors. These polytopes include permutohedra, associahedra, graph- associahedra, simple graphic zonotopes, nestohedra, and other inter- esting polytopes. We give several explicit formulas for h-vectors and γ-vectors involv- ing descent statistics. This includes a combinatorial interpretation for γ-vectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal’s conjecture on the nonnegativity of γ-vectors. We calculate explicit generating functions and formulae for h- polynomials of various families of graph-associahedra, including those corresponding to all Dynkin diagrams of finite and affine types. We also discuss relations with Narayana numbers and with Simon New- comb’s problem. We give (and conjecture) upper and lower bounds for f-, h-, and γ-vectors within several classes of generalized permutohedra. An appendix discusses the equivalence of various notions of deforma- tions of simple polytopes. 2000 Mathematics Subject Classification: Primary 05Axx. Keywords and Phrases: polytopes, face numbers, permutohedra. 1supported in part by NSF CAREER Award DMS-0504629 2supported in part by NSF Grant DMS-0601010 3supported in part by an NSF Postdoctoral Fellowship Documenta Mathematica 13 (2008)
  • Simplex and Polygon Equations

    Simplex and Polygon Equations

    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 042, 49 pages Simplex and Polygon Equations Aristophanes DIMAKIS y and Folkert MULLER-HOISSEN¨ z y Department of Financial and Management Engineering, University of the Aegean, 82100 Chios, Greece E-mail: [email protected] z Max Planck Institute for Dynamics and Self-Organization, 37077 G¨ottingen,Germany E-mail: [email protected] Received October 23, 2014, in final form May 26, 2015; Published online June 05, 2015 http://dx.doi.org/10.3842/SIGMA.2015.042 Abstract. It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a \mixed order". We describe simplex equations (including the Yang{Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of \polygon equations" realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-ske- letons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N-simplex equation to the (N + 1)-gon equation, its dual, and a compatibility equation. Key words: higher Bruhat order; higher Tamari order; pentagon equation; simplex equation 2010 Mathematics Subject Classification: 06A06; 06A07; 52Bxx; 82B23 1 Introduction The famous (quantum) Yang{Baxter equation is R^12R^13R^23 = R^23R^13R^12; where R^ 2 End(V ⊗ V ), for a vector space V , and boldface indices specify the two factors of a threefold tensor product on which R^ acts.
  • Hyperbolic Structures from Link Diagrams

    Hyperbolic Structures from Link Diagrams

    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2012 Hyperbolic Structures from Link Diagrams Anastasiia Tsvietkova [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Geometry and Topology Commons Recommended Citation Tsvietkova, Anastasiia, "Hyperbolic Structures from Link Diagrams. " PhD diss., University of Tennessee, 2012. https://trace.tennessee.edu/utk_graddiss/1361 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Anastasiia Tsvietkova entitled "Hyperbolic Structures from Link Diagrams." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. Morwen B. Thistlethwaite, Major Professor We have read this dissertation and recommend its acceptance: Conrad P. Plaut, James Conant, Michael Berry Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Hyperbolic Structures from Link Diagrams A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Anastasiia Tsvietkova May 2012 Copyright ©2012 by Anastasiia Tsvietkova. All rights reserved. ii Acknowledgements I am deeply thankful to Morwen Thistlethwaite, whose thoughtful guidance and generous advice made this research possible.
  • Knots with Unknotting Number 1 and Essential Conway Spheres 1

    Knots with Unknotting Number 1 and Essential Conway Spheres 1

    Algebraic & Geometric Topology 6 (2006) 2051–2116 2051 arXiv version: fonts, pagination and layout may vary from AGT published version Knots with unknotting number 1 and essential Conway spheres CMCAGORDON JOHN LUECKE For a knot K in S3 , let T(K) be the characteristic toric sub-orbifold of the orbifold (S3; K) as defined by Bonahon–Siebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T(K), unless either K is an EM–knot (of Eudave-Munoz)˜ or (S3; K) contains an EM– tangle after cutting along T(K). As a consequence, we describe exactly which large algebraic knots (ie, algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway’s table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsvath–Szab´ o,´ this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram. As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation. 57N10; 57M25 1 Introduction Montesinos showed [24] that if a knot K has unknotting number 1 then its double branched cover M can be obtained by a half-integral Dehn surgery on some knot K∗ in S3 .
  • Associahedron

    Associahedron

    Associahedron Jean-Louis Loday CNRS, Strasbourg April 23rd, 2005 Clay Mathematics Institute 2005 Colloquium Series Polytopes 7T 4 × 7TTTT 4 ×× 77 TT 44 × 7 simplex 4 ×× 77 44 × 7 ··· 4 ×× 77 ×× 7 o o ooo ooo cube o o ··· oo ooo oo?? ooo ?? ? ?? ? ?? OOO OO ooooOOO oooooo OO? ?? oOO ooo OOO o OOO oo?? permutohedron OOooo ? ··· OO o ? O O OOO ooo ?? OOO o OOO ?? oo OO ?ooo n = 2 3 ··· Permutohedron := convex hull of (n+1)! points n+1 (σ(1), . , σ(n + 1)) ∈ R Parenthesizing X= topological space with product (a, b) 7→ ab Not associative but associative up to homo- topy (ab)c • ) • a(bc) With four elements: ((ab)c)d H jjj HH jjjj HH jjjj HH tjj HH (a(bc))d HH HH HH HH H$ vv (ab)(cd) vv vv vv vv (( ) ) TTT vv a bc d TTTT vv TTT vv TTTz*vv a(b(cd)) We suppose that there is a homotopy between the two composite paths, and so on. Jim Stasheff Staheff’s result (1963): There exists a cellular complex such that – vertices in bijection with the parenthesizings – edges in bijection with the homotopies – 2-cells in bijection with homotopies of com- posite homotopies – etc, and which is homeomorphic to a ball. Problem: construct explicitely the Stasheff com- plex in any dimension. oo?? ooo ?? ?? ?? • OOO OO n = 0 1 2 3 Planar binary trees (see R. Stanley’s notes p. 189) Planar binary trees with n + 1 leaves, that is n internal vertices: n o ? ?? ? ?? ?? ?? Y0 = { | } ,Y1 = ,Y2 = ? , ? , ( ) ? ? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ?? ? Y3 = ? , ? , ? , ? , ? . Bijection between planar binary trees and paren- thesizings: x0 x1 x2 x3 x4 RRR ll ll RRR ll RRlRll lll RRlll RRR lll lll lRRR lll RRR lll RRlll (((x0x1)x2)(x3x4)) The notion of grafting s t 3 33 t ∨ s = 3 Associahedron n To t ∈ Yn we associate M(t) ∈ R : n M(t) := (a1b1, ··· , aibi, ··· , anbn) ∈ R ai = # leaves on the left side of the ith vertex bi = # leaves on the right side of the ith vertex Examples: ? ? ?? ?? ?? ?? M( ) = (1),M ? = (1, 2),M ? = (2, 1), ? ? ? ?? ?? ?? ?? M ? = (1, 2, 3),M ? = (1, 4, 1).
  • Monoids in Representation Theory, Markov Chains, Combinatorics and Operator Algebras

    Monoids in Representation Theory, Markov Chains, Combinatorics and Operator Algebras

    Monoids in Representation Theory, Markov Chains, Combinatorics and Operator Algebras Benjamin Steinberg, City College of New York Australian Mathematical Society Meeting December 2019 Group representation theory and Markov chains • Group representation theory is sometimes a valuable tool for analyzing Markov chains. Group representation theory and Markov chains • Group representation theory is sometimes a valuable tool for analyzing Markov chains. • The method (at least for nonabelian groups) was perhaps first popularized in a paper of Diaconis and Shahshahani from 1981. Group representation theory and Markov chains • Group representation theory is sometimes a valuable tool for analyzing Markov chains. • The method (at least for nonabelian groups) was perhaps first popularized in a paper of Diaconis and Shahshahani from 1981. • They analyzed shuffling a deck of cards by randomly transposing cards. Group representation theory and Markov chains • Group representation theory is sometimes a valuable tool for analyzing Markov chains. • The method (at least for nonabelian groups) was perhaps first popularized in a paper of Diaconis and Shahshahani from 1981. • They analyzed shuffling a deck of cards by randomly transposing cards. • Because the transpositions form a conjugacy class, the eigenvalues and convergence time of the walk can be understood via characters of the symmetric group. Group representation theory and Markov chains • Group representation theory is sometimes a valuable tool for analyzing Markov chains. • The method (at least for nonabelian groups) was perhaps first popularized in a paper of Diaconis and Shahshahani from 1981. • They analyzed shuffling a deck of cards by randomly transposing cards. • Because the transpositions form a conjugacy class, the eigenvalues and convergence time of the walk can be understood via characters of the symmetric group.
  • [Math.CO] 6 Oct 2008 Egyfmnadnta Reading Nathan and Fomin Sergey Eeaie Associahedra Generalized Otssesand Systems Root

    [Math.CO] 6 Oct 2008 Egyfmnadnta Reading Nathan and Fomin Sergey Eeaie Associahedra Generalized Otssesand Systems Root

    Root Systems and Generalized Associahedra Sergey Fomin and Nathan Reading arXiv:math/0505518v3 [math.CO] 6 Oct 2008 IAS/Park City Mathematics Series Volume 14, 2004 Root Systems and Generalized Associahedra Sergey Fomin and Nathan Reading These lecture notes provide an overview of root systems, generalized associahe- dra, and the combinatorics of clusters. Lectures 1-2 cover classical material: root systems, finite reflection groups, and the Cartan-Killing classification. Lectures 3–4 provide an introduction to cluster algebras from a combinatorial perspective. Lec- ture 5 is devoted to related topics in enumerative combinatorics. There are essentially no proofs but an abundance of examples. We label un- proven assertions as either “lemma” or “theorem” depending on whether they are easy or difficult to prove. We encourage the reader to try proving the lemmas, or at least get an idea of why they are true. For additional information on root systems, reflection groups and Coxeter groups, the reader is referred to [9, 25, 34]. For basic definitions related to convex polytopes and lattice theory, see [58] and [31], respectively. Primary sources on generalized associahedra and cluster combinatorics are [13, 19, 21]. Introductory surveys on cluster algebras were given in [22, 56, 57]. Note added in press (February 2007): Since these lecture notes were written, there has been much progress in the general area of cluster algebras and Catalan combinatorics of Coxeter groups and root systems. We have not attempted to update the text to reflect these most recent advances. Instead, we refer the reader to the online Cluster Algebras Portal, maintained by the first author.
  • On L-Space Knots Obtained from Unknotting Arcs in Alternating Diagrams

    On L-Space Knots Obtained from Unknotting Arcs in Alternating Diagrams

    New York Journal of Mathematics New York J. Math. 25 (2019) 518{540. On L-space knots obtained from unknotting arcs in alternating diagrams Andrew Donald, Duncan McCoy and Faramarz Vafaee Abstract. Let D be a diagram of an alternating knot with unknotting number one. The branched double cover of S3 branched over D is an L-space obtained by half integral surgery on a knot KD. We denote the set of all such knots KD by D. We characterize when KD 2 D is a torus knot, a satellite knot or a hyperbolic knot. In a different direction, we show that for a given n > 0, there are only finitely many L-space knots in D with genus less than n. Contents 1. Introduction 518 2. Almost alternating diagrams of the unknot 521 2.1. The corresponding operations in D 522 3. The geometry of knots in D 525 3.1. Torus knots in D 525 3.2. Constructing satellite knots in D 526 3.3. Substantial Conway spheres 534 References 538 1. Introduction A knot K ⊂ S3 is an L-space knot if it admits a positive Dehn surgery to an L-space.1 Examples include torus knots, and more broadly, Berge knots in S3 [Ber18]. In recent years, work by many researchers provided insight on the fiberedness [Ni07, Ghi08], positivity [Hed10], and various notions of simplicity of L-space knots [OS05b, Hed11, Krc18]. For more examples of L-space knots, see [Hom11, Hom16, HomLV14, Mot16, MotT14, Vaf15]. Received May 8, 2018. 2010 Mathematics Subject Classification. 57M25, 57M27.
  • The Conway Knot Is Not Slice

    The Conway Knot Is Not Slice

    THE CONWAY KNOT IS NOT SLICE LISA PICCIRILLO Abstract. A knot is said to be slice if it bounds a smooth properly embedded disk in B4. We demonstrate that the Conway knot, 11n34 in the Rolfsen tables, is not slice. This com- pletes the classification of slice knots under 13 crossings, and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot. 1. Introduction The classical study of knots in S3 is 3-dimensional; a knot is defined to be trivial if it bounds an embedded disk in S3. Concordance, first defined by Fox in [Fox62], is a 4-dimensional extension; a knot in S3 is trivial in concordance if it bounds an embedded disk in B4. In four dimensions one has to take care about what sort of disks are permitted. A knot is slice if it bounds a smoothly embedded disk in B4, and topologically slice if it bounds a locally flat disk in B4. There are many slice knots which are not the unknot, and many topologically slice knots which are not slice. It is natural to ask how characteristics of 3-dimensional knotting interact with concordance and questions of this sort are prevalent in the literature. Modifying a knot by positive mutation is particularly difficult to detect in concordance; we define positive mutation now. A Conway sphere for an oriented knot K is an embedded S2 in S3 that meets the knot 3 transversely in four points. The Conway sphere splits S into two 3-balls, B1 and B2, and ∗ K into two tangles KB1 and KB2 .