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The Borromean Rings: a Video About the New IMU Logo
The Borromean Rings: A Video about the New IMU Logo Charles Gunn and John M. Sullivan∗ Technische Universitat¨ Berlin Institut fur¨ Mathematik, MA 3–2 Str. des 17. Juni 136 10623 Berlin, Germany Email: {gunn,sullivan}@math.tu-berlin.de Abstract This paper describes our video The Borromean Rings: A new logo for the IMU, which was premiered at the opening ceremony of the last International Congress. The video explains some of the mathematics behind the logo of the In- ternational Mathematical Union, which is based on the tight configuration of the Borromean rings. This configuration has pyritohedral symmetry, so the video includes an exploration of this interesting symmetry group. Figure 1: The IMU logo depicts the tight config- Figure 2: A typical diagram for the Borromean uration of the Borromean rings. Its symmetry is rings uses three round circles, with alternating pyritohedral, as defined in Section 3. crossings. In the upper corners are diagrams for two other three-component Brunnian links. 1 The IMU Logo and the Borromean Rings In 2004, the International Mathematical Union (IMU), which had never had a logo, announced a competition to design one. The winning entry, shown in Figure 1, was designed by one of us (Sullivan) with help from Nancy Wrinkle. It depicts the Borromean rings, not in the usual diagram (Figure 2) but instead in their tight configuration, the shape they have when tied tight in thick rope. This IMU logo was unveiled at the opening ceremony of the International Congress of Mathematicians (ICM 2006) in Madrid. We were invited to produce a short video [10] about some of the mathematics behind the logo; it was shown at the opening and closing ceremonies, and can be viewed at www.isama.org/jms/Videos/imu/. -
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. -
On Spectral Sequences from Khovanov Homology 11
ON SPECTRAL SEQUENCES FROM KHOVANOV HOMOLOGY ANDREW LOBB RAPHAEL ZENTNER Abstract. There are a number of homological knot invariants, each satis- fying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain com- plex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies. In this paper we focus on the spectral sequences due to Kronheimer-Mrowka from Khovanov homology to instanton knot Floer homology, and on that due to Ozsv´ath-Szab´oto the Heegaard-Floer homology of the branched double cover. For example, we use the 1-handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the (4; 5)-torus knot, determining these up to an ambiguity in a pair of degrees; to deter- mine the Ozsv´ath-Szab´ospectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer-Mrowka and the Ozsv´ath-Szab´ospectral sequences necessarily lower the delta grading for all pretzel knots. 1. Introduction Recent work in the area of the 3-manifold invariants called knot homologies has il- luminated the relationship between Floer-theoretic knot homologies and `quantum' knot homologies. The relationships observed take the form of spectral sequences starting with a quantum invariant and abutting to a Floer invariant. A primary ex- ample is due to Ozsv´athand Szab´o[15] in which a spectral sequence is constructed from Khovanov homology of a knot (with Z=2 coefficients) to the Heegaard-Floer homology of the 3-manifold obtained as double branched cover over the knot. -
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. -
Arxiv:1502.03029V2 [Math.GT] 16 Apr 2015 De.Then Edges
COUNTING GENERIC QUADRISECANTS OF POLYGONAL KNOTS ALDO-HILARIO CRUZ-COTA, TERESITA RAMIREZ-ROSAS Abstract. Let K be a polygonal knot in general position with vertex set V . A generic quadrisecant of K is a line that is disjoint from the set V and intersects K in exactly four distinct points. We give an upper bound for the number of generic quadrisecants of a polygonal knot K in general position. This upper bound is in terms of the number of edges of K. 1. Introduction In this article, we study polygonal knots in three dimensional space that are in general position. Given such a knot K, we define a quadrisecant of K as an unoriented line that intersects K in exactly four distinct points. We require that these points are not vertices of the knot, in which case we say that the quadrisecant is generic. Using geometric and combinatorial arguments, we give an upper bound for the number of generic quadrisecants of a polygonal knot K in general position. This bound is in terms of the number n ≥ 3 of edges of K. More precisely, we prove the following. Main Theorem 1. Let K be a polygonal knot in general position, with exactly n n edges. Then K has at most U = (n − 3)(n − 4)(n − 5) generic quadrisecants. n 12 Applying Main Theorem 1 to polygonal knots with few edges, we obtain the following. (1) If n ≤ 5, then K has no generic quadrisecant. (2) If n = 6, then K has at most three generic quadrisecants. (3) If n = 7, then K has at most 14 generic quadrisecants. -
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. -
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. -
Introduction to Geometric Knot Theory 2: Ropelength and Tight Knots
Introduction to Geometric Knot Theory 2: Ropelength and Tight Knots Jason Cantarella University of Georgia ICTP Knot Theory Summer School, Trieste, 2009 Review from first lecture: Definition (Federer 1959) The reach of a space curve is the largest so that any point in an -neighborhood of the curve has a unique nearest neighbor on the curve. Idea reach(K ) (also called thickness) is controlled by curvature maxima (kinks) and self-distance minima (struts). Cantarella Geometric Knot Theory Ropelength Definition The ropelength of K is given by Rop(K ) = Len(K )/ reach(K ). Theorem (with Kusner, Sullivan 2002, Gonzalez, De la Llave 2003, Gonzalez, Maddocks, Schuricht, Von der Mosel 2002) Ropelength minimizers (called tight knots) exist in each knot and link type and are C1,1. We can bound Rop in terms of Cr. Cantarella Geometric Knot Theory Ropelength Definition The ropelength of K is given by Rop(K ) = Len(K )/ reach(K ). Theorem (with Kusner, Sullivan 2002, Gonzalez, De la Llave 2003, Gonzalez, Maddocks, Schuricht, Von der Mosel 2002) Ropelength minimizers (called tight knots) exist in each knot and link type and are C1,1. We can bound Rop in terms of Cr. Cantarella Geometric Knot Theory Ropelength Definition The ropelength of K is given by Rop(K ) = Len(K )/ reach(K ). Theorem (with Kusner, Sullivan 2002, Gonzalez, De la Llave 2003, Gonzalez, Maddocks, Schuricht, Von der Mosel 2002) Ropelength minimizers (called tight knots) exist in each knot and link type and are C1,1. We can bound Rop in terms of Cr. Cantarella Geometric Knot Theory Ropelength Definition The ropelength of K is given by Rop(K ) = Len(K )/ reach(K ). -
Total Curvature, Ropelength and Crossing Number of Thick Knots
Total Curvature, Ropelength and Crossing Number of Thick Knots Yuanan Diao and Claus Ernst Abstract. We first study the minimum total curvature of a knot when it is embedded on the cubic lattice. Let K be a knot or link with a lattice embedding of minimum total curvature ¿(K) among all possible lattice embeddingsp of K. We show that there exist positive constants c1 and c2 such that c1 Cr(K) · ¿(K) · c2Cr(K) for any knot type K. Furthermore we show that the powers of Cr(K) in the above inequalities are sharp hence cannot be improved in general. Our results and observations show that lattice embeddings with minimum total curvature are quite different from those with minimum or near minimum lattice embedding length. In addition, we discuss the relationship between minimal total curvature and minimal ropelength for a given knot type. At the end of the paper, we study the total curvatures of smooth thick knots and show that there are some essential differences between the total curvatures of smooth thick knots and lattice knots. 1. Introduction An essential field in the area of geometric knot theory concerns questions about the behavior of thick knots. One such question is about the ropelength of a knot. That is, if we are to tie a given knot with a rope of unit thickness, how long does the rope needs to be? In particular, for a given knot K whose crossing number is Cr(K), can we express the ropelength L(K) of K as a function of Cr(K)? Or alternatively, can we find a lower and upper bound of L(K) in terms of Cr(K)? For a general lower bound, it is shown in [3, 4] that there exists a constant a > 0 such that for any knot type K, L(K) ¸ a ¢ (Cr(K))3=4. -
The Linking Number and the Writhe of Uniform Random Walks And
The linking number and the writhe of uniform random walks and polygons in confined spaces E. Panagiotou ,∗ K. C. Millett ,† S. Lambropoulou ‡ November 11, 2018 Abstract Random walks and polygons are used to model polymers. In this pa- per we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configura- tions as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking num- ber of oriented uniform random walks or polygons of length n, in a convex 2 confined space, are of the form O(n ). Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form O(√n). Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of n edges each, is of the form O(n). Equilateral random walks and polygons are used to model polymers in θ-conditions. We use numerical simulations to investigate how the self-linking and linking number of equilateral random walks scale with their length. 1 Introduction A polymer melt may consist of ring polymers (closed chains), linear polymers (open chains), or a mixed collection of ring and linear polymers. Polymer chains are long flexible molecules that impose spatial constraints on each other because they cannot intersect (de Gennes 1979, Rubinstein and Colby 2003). -
Closed Curves with No Quadrisecants
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology Vol.‘ZI. No. 3. PP. 235-243. 1982 004&9383/82/030235-09SO3.00/0 Printed in Great Britain. @ 1982 Pergamon Press Ltd. CLOSED CURVES WITH NO QUADRISECANTS H. R. MORTON and D. M. Q. MONDt (Receioed 18 December 1980) $1. INTRODUCTION IN HIS PAPER[ 11, Wall discusses the idea of a generically embedded curve in BP’,related to the local appearance of the radially projected curve as viewed from different points in R3. Wall shows that a generic curve never meets a straight line in five points. There may, however, be finitely many straight lines, quadrisecants, which meet the curve in four points. The local view of the curve from distant points on a quadrisecant will have four branches crossing transversely. In this paper we ask how simple the embedding of a closed curve must be if the curve has no quadrisecants. Under a slight extension of Wall’s term generic to ensure a similar regularity of appearance of the curve when viewed from other points of the curve itself, we prove: THEOREM. A generically embedded closed curve with no quadrisecants is unknotted. We expect that in the non-generic case a knotted closed curve will have a genuine quadrisecant. (If we widened the definition of quadrisecant to allow the intersections of a line with the curve to be counted with multiplicity, including, e.g. a tangent line which meets the curve in two further points, then a limiting argument would produce at least a “fake” quadrisecant for a knotted closed curve.) Although the theorem deals with differentiable embeddings, it can be proved by a similar argument that a knotted PL curve in general position must have a quadrise- cant, other than one of its own straight edges.