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The Khovanov Homology of Rational Tangles
The Khovanov Homology of Rational Tangles Benjamin Thompson A thesis submitted for the degree of Bachelor of Philosophy with Honours in Pure Mathematics of The Australian National University October, 2016 Dedicated to my family. Even though they’ll never read it. “To feel fulfilled, you must first have a goal that needs fulfilling.” Hidetaka Miyazaki, Edge (280) “Sleep is good. And books are better.” (Tyrion) George R. R. Martin, A Clash of Kings “Let’s love ourselves then we can’t fail to make a better situation.” Lauryn Hill, Everything is Everything iv Declaration Except where otherwise stated, this thesis is my own work prepared under the supervision of Scott Morrison. Benjamin Thompson October, 2016 v vi Acknowledgements What a ride. Above all, I would like to thank my supervisor, Scott Morrison. This thesis would not have been written without your unflagging support, sublime feedback and sage advice. My thesis would have likely consisted only of uninspired exposition had you not provided a plethora of interesting potential topics at the start, and its overall polish would have likely diminished had you not kept me on track right to the end. You went above and beyond what I expected from a supervisor, and as a result I’ve had the busiest, but also best, year of my life so far. I must also extend a huge thanks to Tony Licata for working with me throughout the year too; hopefully we can figure out what’s really going on with the bigradings! So many people to thank, so little time. I thank Joan Licata for agreeing to run a Knot Theory course all those years ago. -
View Given in Figure 7
New York Journal of Mathematics New York J. Math. 26 (2020) 149{183. Arithmeticity and hidden symmetries of fully augmented pretzel link complements Jeffrey S. Meyer, Christian Millichap and Rolland Trapp Abstract. This paper examines number theoretic and topological prop- erties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and ex- actly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant trace fields, which we explicitly compute. Further, we construct two infinite fami- lies of non-arithmetic fully augmented link complements: one that has no hidden symmetries and the other where the number of hidden sym- metries grows linearly with volume. This second family realizes the maximal growth rate for the number of hidden symmetries relative to volume for non-arithmetic hyperbolic 3-manifolds. Our work requires a careful analysis of the geometry of these link complements, including their cusp shapes and totally geodesic surfaces inside of these manifolds. Contents 1. Introduction 149 2. Geometric decomposition of fully augmented pretzel links 154 3. Hyperbolic reflection orbifolds 163 4. Cusp and trace fields 164 5. Arithmeticity 166 6. Symmetries and hidden symmetries 169 7. Half-twist partners with many hidden symmetries 174 References 181 1. Introduction 3 Every link L ⊂ S determines a link complement, that is, a non-compact 3 3-manifold M = S n L. If M admits a metric of constant curvature −1, we say that both M and L are hyperbolic, and in fact, if such a hyperbolic Received June 5, 2019. -
A Note on Quasi-Alternating Montesinos Links
A NOTE ON QUASI-ALTERNATING MONTESINOS LINKS ABHIJIT CHAMPANERKAR AND PHILIP ORDING Abstract. Quasi-alternating links are a generalization of alternating links. They are ho- mologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi- alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we also establish con- ditions on the rational parameters of a Montesinos link to be non-quasi-alternating. We discuss examples which are not covered by the above results. 1. Introduction The set Q of quasi-alternating links was defined by Ozsv´athand Szab´o[17] as the smallest set of links satisfying the following: • the unknot is in Q • if link L has a diagram L with a crossing c such that (1) both smoothings of c, L0 and L1 are in Q (2) det(L0) 6= 0 6= det(L1) L L0 L• (3) det(L) = det(L0) + det(L1) then L is in Q. The set Q includes the class of non-split alternating links. Like alternating links, quasi- alternating links are homologically thin for both Khovanov homology and knot Floer ho- mology [14]. The branched double covers of quasi-alternating links are L-spaces [17]. These properties make Q an interesting class to study from the knot homological point of view. -
Khovanov Homology, Its Definitions and Ramifications
KHOVANOV HOMOLOGY, ITS DEFINITIONS AND RAMIFICATIONS OLEG VIRO Uppsala University, Uppsala, Sweden POMI, St. Petersburg, Russia Abstract. Mikhail Khovanov defined, for a diagram of an oriented clas- sical link, a collection of groups labelled by pairs of integers. These groups were constructed as homology groups of certain chain complexes. The Euler characteristics of these complexes are coefficients of the Jones polynomial of the link. The original construction is overloaded with algebraic details. Most of specialists use adaptations of it stripped off the details. The goal of this paper is to overview these adaptations and show how to switch between them. We also discuss a version of Khovanov homology for framed links and suggest a new grading for it. 1. Introduction For a diagram D of an oriented link L, Mikhail Khovanov [7] constructed a collection of groups Hi;j(D) such that X j i i;j K(L)(q) = q (−1) dimQ(H (D) ⊗ Q); i;j where K(L) is a version of the Jones polynomial of L. These groups are constructed as homology groups of certain chain complexes. The Khovanov homology has proved to be a powerful and useful tool in low dimensional topology. Recently Jacob Rasmussen [16] has applied it to problems of estimating the slice genus of knots and links. He defied a knot invariant which gives a low bound for the slice genus and which, for knots with only positive crossings, allowed him to find the exact value of the slice genus. Using this technique, he has found a new proof of Milnor conjecture on the slice genus of toric knots. -
Computational Complexity of Problems in 3-Dimensional Topology
[NoVIKOV 1962] For n > 5, THERE DOES NOT EXIST AN ALGORITHM WHICH solves: n Computational COMPLEXITY ISSPHERE: Given A TRIANGULATED M IS IT n OF PROBLEMS IN HOMEOMORPHIC TO S ? 3-dimensional TOPOLOGY Thm (Geometrization + MANY Results) TherE IS AN ALGORITHM TO DECIDE IF TWO COMPACT 3-mflDS ARE homeomorphic. Nathan DunfiELD University OF ILLINOIS SLIDES at: http://dunfield.info/preprints/ Today: HoW HARD ARE THESE 3-manifold questions? HoW QUICKLY CAN WE SOLVE them? NP: YES ANSWERS HAVE PROOFS THAT CAN BE Decision Problems: YES OR NO answer. CHECKED IN POLYNOMIAL time. k x F pi(x) = 0 SORTED: Given A LIST OF integers, IS IT sorted? SAT: Given ∈ 2 , CAN CHECK ALL IN LINEAR time. SAT: Given p1,... pn F2[x1,..., xk] IS k ∈ x F pi(x) = 0 i UNKNOTTED: A DIAGRAM OF THE UNKNOT WITH THERE ∈ 2 WITH FOR ALL ? 11 c CROSSINGS CAN BE UNKNOTTED IN O(c ) UNKNOTTED: Given A PLANAR DIAGRAM FOR K 3 Reidemeister MOves. [LackENBY 2013] IN S IS K THE unknot? A Mn(Z) INVERTIBLE: Given ∈ DOES IT HAVE AN INVERSE IN Mn(Z)? coNP: No ANSWERS CAN BE CHECKED IN POLYNOMIAL time. UNKNOTTED: Yes, ASSUMING THE GRH P: Decision PROBLEMS WHICH CAN BE SOLVED [KuperberG 2011]. IN POLYNOMIAL TIME IN THE INPUT size. SORTED: O(LENGTH OF LIST) 3.5 1.1 INVERTIBLE: O n log(LARGEST ENTRY) NKNOTTED IS IN . Conj: U P T KNOTGENUS: Given A TRIANGULATION , A KNOT b1 = 0 (1) IS KNOTGENUS IN P WHEN ? K T g Z 0 K ⊂ , AND A ∈ > , DOES BOUND AN ORIENTABLE SURFACE OF GENUS 6 g? IS THE HOMEOMORPHISM PROBLEM FOR [Agol-Hass-W.Thurston 2006] 3-manifolds IN NP? KNOTGENUS IS NP-complete. -
The Multivariable Alexander Polynomial on Tangles by Jana
The Multivariable Alexander Polynomial on Tangles by Jana Archibald A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright c 2010 by Jana Archibald Abstract The Multivariable Alexander Polynomial on Tangles Jana Archibald Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2010 The multivariable Alexander polynomial (MVA) is a classical invariant of knots and links. We give an extension to regular virtual knots which has simple versions of many of the relations known to hold for the classical invariant. By following the previous proofs that the MVA is of finite type we give a new definition for its weight system which can be computed as the determinant of a matrix created from local information. This is an improvement on previous definitions as it is directly computable (not defined recursively) and is computable in polynomial time. We also show that our extension to virtual knots is a finite type invariant of virtual knots. We further explore how the multivariable Alexander polynomial takes local infor- mation and packages it together to form a global knot invariant, which leads us to an extension to tangles. To define this invariant we use so-called circuit algebras, an exten- sion of planar algebras which are the ‘right’ setting to discuss virtual knots. Our tangle invariant is a circuit algebra morphism, and so behaves well under tangle operations and gives yet another definition for the Alexander polynomial. The MVA and the single variable Alexander polynomial are known to satisfy a number of relations, each of which has a proof relying on different approaches and techniques. -
Categorified Invariants and the Braid Group
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 7, July 2015, Pages 2801–2814 S 0002-9939(2015)12482-3 Article electronically published on February 26, 2015 CATEGORIFIED INVARIANTS AND THE BRAID GROUP JOHN A. BALDWIN AND J. ELISENDA GRIGSBY (Communicated by Daniel Ruberman) Abstract. We investigate two “categorified” braid conjugacy class invariants, one coming from Khovanov homology and the other from Heegaard Floer ho- mology. We prove that each yields a solution to the word problem but not the conjugacy problem in the braid group. In particular, our proof in the Khovanov case is completely combinatorial. 1. Introduction Recall that the n-strand braid group Bn admits the presentation σiσj = σj σi if |i − j|≥2, Bn = σ1,...,σn−1 , σiσj σi = σjσiσj if |i − j| =1 where σi corresponds to a positive half twist between the ith and (i + 1)st strands. Given a word w in the generators σ1,...,σn−1 and their inverses, we will denote by σ(w) the corresponding braid in Bn. Also, we will write σ ∼ σ if σ and σ are conjugate elements of Bn. As with any group described in terms of generators and relations, it is natural to look for combinatorial solutions to the word and conjugacy problems for the braid group: (1) Word problem: Given words w, w as above, is σ(w)=σ(w)? (2) Conjugacy problem: Given words w, w as above, is σ(w) ∼ σ(w)? The fastest known algorithms for solving Problems (1) and (2) exploit the Gar- side structure(s) of the braid group (cf. -
Dehn Filling of the “Magic” 3-Manifold
communications in analysis and geometry Volume 14, Number 5, 969–1026, 2006 Dehn filling of the “magic” 3-manifold Bruno Martelli and Carlo Petronio We classify all the non-hyperbolic Dehn fillings of the complement of the chain link with three components, conjectured to be the smallest hyperbolic 3-manifold with three cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many one-cusped and two-cusped hyperbolic manifolds, including most of those with smallest known volume. Among other consequences of this classification, we mention the following: • for every integer n, we can prove that there are infinitely many hyperbolic knots in S3 having exceptional surgeries {n, n +1, n +2,n+3}, with n +1,n+ 2 giving small Seifert manifolds and n, n + 3 giving toroidal manifolds. • we exhibit a two-cusped hyperbolic manifold that contains a pair of inequivalent knots having homeomorphic complements. • we exhibit a chiral 3-manifold containing a pair of inequivalent hyperbolic knots with orientation-preservingly homeomorphic complements. • we give explicit lower bounds for the maximal distance between small Seifert fillings and any other kind of exceptional filling. 0. Introduction We study in this paper the Dehn fillings of the complement N of the chain link with three components in S3, shown in figure 1. The hyperbolic structure of N was first constructed by Thurston in his notes [28], and it was also noted there that the volume of N is particularly small. The relevance of N to three-dimensional topology comes from the fact that by filling N, one gets most of the hyperbolic manifolds known and most of the interesting non-hyperbolic fillings of cusped hyperbolic manifolds. -
Research Statement 1 TQFT and Khovanov Homology
Miller Fellow 1071 Evans Hall, University of California, Berkeley, CA 94720 +1 805 453-7347 [email protected] http://tqft.net/ Scott Morrison - Research statement 1 TQFT and Khovanov homology During the 1980s and 1990s, surprising and deep connections were found between topology and algebra, starting from the Jones polynomial for knots, leading on through the representation theory of a quantum group (a braided tensor category) and culminating in topological quantum field theory (TQFT) invariants of 3-manifolds. In 1999, Khovanov [Kho00] came up with something entirely new. He associated to each knot diagram a chain complex whose homology was a knot invariant. Some aspects of this construction were familiar; the Euler characteristic of his complex is the Jones polynomial, and just as we had learnt to understand the Jones polynomial in terms of a braided tensor category, Khovanov homology can now be understood in terms of a certain braided tensor 2-category. But other aspects are more mysterious. The categories associated to quantum knot invariants are essentially semisimple, while those arising in Khovanov homology are far from it. Instead they are triangulated, and exact triangles play a central role in both definitions and calculations. My research on Khovanov homology attempts to understand the 4-dimensional geometry of Khovanov homology ( 1.1). Modulo a conjecture on the functoriality of Khovanov homology in 3 § S , we have a construction of an invariant of a link in the boundary of an arbitrary 4-manifold, generalizing the usual invariant in the boundary of the standard 4-ball. While Khovanov homology and its variations provide a ‘categorification’ of quantum knot invariants, to date there has been no corresponding categorification of TQFT 3-manifold in- variants. -
RASMUSSEN INVARIANTS of SOME 4-STRAND PRETZEL KNOTS Se
Honam Mathematical J. 37 (2015), No. 2, pp. 235{244 http://dx.doi.org/10.5831/HMJ.2015.37.2.235 RASMUSSEN INVARIANTS OF SOME 4-STRAND PRETZEL KNOTS Se-Goo Kim and Mi Jeong Yeon Abstract. It is known that there is an infinite family of general pretzel knots, each of which has Rasmussen s-invariant equal to the negative value of its signature invariant. For an instance, homo- logically σ-thin knots have this property. In contrast, we find an infinite family of 4-strand pretzel knots whose Rasmussen invariants are not equal to the negative values of signature invariants. 1. Introduction Khovanov [7] introduced a graded homology theory for oriented knots and links, categorifying Jones polynomials. Lee [10] defined a variant of Khovanov homology and showed the existence of a spectral sequence of rational Khovanov homology converging to her rational homology. Lee also proved that her rational homology of a knot is of dimension two. Rasmussen [13] used Lee homology to define a knot invariant s that is invariant under knot concordance and additive with respect to connected sum. He showed that s(K) = σ(K) if K is an alternating knot, where σ(K) denotes the signature of−K. Suzuki [14] computed Rasmussen invariants of most of 3-strand pret- zel knots. Manion [11] computed rational Khovanov homologies of all non quasi-alternating 3-strand pretzel knots and links and found the Rasmussen invariants of all 3-strand pretzel knots and links. For general pretzel knots and links, Jabuka [5] found formulas for their signatures. Since Khovanov homologically σ-thin knots have s equal to σ, Jabuka's result gives formulas for s invariant of any quasi- alternating− pretzel knot. -
CALIFORNIA STATE UNIVERSITY, NORTHRIDGE P-Coloring Of
CALIFORNIA STATE UNIVERSITY, NORTHRIDGE P-Coloring of Pretzel Knots A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics By Robert Ostrander December 2013 The thesis of Robert Ostrander is approved: |||||||||||||||||| |||||||| Dr. Alberto Candel Date |||||||||||||||||| |||||||| Dr. Terry Fuller Date |||||||||||||||||| |||||||| Dr. Magnhild Lien, Chair Date California State University, Northridge ii Dedications I dedicate this thesis to my family and friends for all the help and support they have given me. iii Acknowledgments iv Table of Contents Signature Page ii Dedications iii Acknowledgements iv Abstract vi Introduction 1 1 Definitions and Background 2 1.1 Knots . .2 1.1.1 Composition of knots . .4 1.1.2 Links . .5 1.1.3 Torus Knots . .6 1.1.4 Reidemeister Moves . .7 2 Properties of Knots 9 2.0.5 Knot Invariants . .9 3 p-Coloring of Pretzel Knots 19 3.0.6 Pretzel Knots . 19 3.0.7 (p1, p2, p3) Pretzel Knots . 23 3.0.8 Applications of Theorem 6 . 30 3.0.9 (p1, p2, p3, p4) Pretzel Knots . 31 Appendix 49 v Abstract P coloring of Pretzel Knots by Robert Ostrander Master of Science in Mathematics In this thesis we give a brief introduction to knot theory. We define knot invariants and give examples of different types of knot invariants which can be used to distinguish knots. We look at colorability of knots and generalize this to p-colorability. We focus on 3-strand pretzel knots and apply techniques of linear algebra to prove theorems about p-colorability of these knots. -
Khovanov Homology Is an Unknot-Detector
Khovanov homology is an unknot-detector P. B. Kronheimer and T. S. Mrowka Harvard University, Cambridge MA 02138 Massachusetts Institute of Technology, Cambridge MA 02139 Abstract. We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot. 1 Introduction 1.1 Statement of results This paper explores a relationship between the Khovanov cohomology of a knot, as defined in [16], and various homology theories defined using Yang-Mills instantons, of which the archetype is Floer’s instanton homology of 3-manifolds [8]. A consequence of this relationship is a proof that Khovanov cohomology detects the unknot. (For related results, see [10, 11, 12]). Theorem 1.1. A knot in S 3 is the unknot if and only if its reduced Khovanov cohomology is Z. In [23], the authors construct a Floer homology for knots and links in 3- manifolds using moduli spaces of connections with singularities in codimension 2. (The locus of the singularity is essentially the link K, or R K in a cylindrical 4-manifold.) Several variations of this construction are already considered in [23], but we will introduce here one more variation, which we call I \.K/. Our invariant I \.K/ is an invariant for unoriented links K S 3 with a marked point x K and a preferred normal vector v to K at x.