The Conway Polynomial and Amphicheiral Knots
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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. -
A Knot-Vice's Guide to Untangling Knot Theory, Undergraduate
A Knot-vice’s Guide to Untangling Knot Theory Rebecca Hardenbrook Department of Mathematics University of Utah Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 1 / 26 What is Not a Knot? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 2 / 26 What is a Knot? 2 A knot is an embedding of the circle in the Euclidean plane (R ). 3 Also defined as a closed, non-self-intersecting curve in R . 2 Represented by knot projections in R . Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 3 / 26 Why Knots? Late nineteenth century chemists and physicists believed that a substance known as aether existed throughout all of space. Could knots represent the elements? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 4 / 26 Why Knots? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 5 / 26 Why Knots? Unfortunately, no. Nevertheless, mathematicians continued to study knots! Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 6 / 26 Current Applications Natural knotting in DNA molecules (1980s). Credit: K. Kimura et al. (1999) Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 7 / 26 Current Applications Chemical synthesis of knotted molecules – Dietrich-Buchecker and Sauvage (1988). Credit: J. Guo et al. (2010) Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 8 / 26 Current Applications Use of lattice models, e.g. the Ising model (1925), and planar projection of knots to find a knot invariant via statistical mechanics. Credit: D. Chicherin, V.P. -
Tait's Flyping Conjecture for 4-Regular Graphs
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Combinatorial Theory, Series B 95 (2005) 318–332 www.elsevier.com/locate/jctb Tait’s flyping conjecture for 4-regular graphs Jörg Sawollek Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany Received 8 July 1998 Available online 19 July 2005 Abstract Tait’s flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S3. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes with respect to ambient isotopy and rigid vertex isotopy of graph embeddings are identical on the class of diagrams considered. © 2005 Elsevier Inc. All rights reserved. Keywords: Knotted graph; Alternating diagram; Flyping conjecture 0. Introduction Very early in the history of knot theory attention has been paid to alternating diagrams of knots and links. At the end of the 19th century Tait [21] stated several famous conjectures on alternating link diagrams that could not be verified for about a century. The conjectures concerning minimal crossing numbers of reduced, alternating link diagrams [15, Theorems A, B] have been proved independently by Thistlethwaite [22], Murasugi [15], and Kauffman [6]. Tait’s flyping conjecture, claiming that two reduced, alternating, prime diagrams of a given link can be connected by a finite sequence of so-called flypes (see [4, p. 311] for Tait’s original terminology), has been shown by Menasco and Thistlethwaite [14], and for a special case, namely, for well-connected diagrams, also by Schrijver [20]. -
Alternating Knots
ALTERNATING KNOTS WILLIAM W. MENASCO Abstract. This is a short expository article on alternating knots and is to appear in the Concise Encyclopedia of Knot Theory. Introduction Figure 1. P.G. Tait's first knot table where he lists all knot types up to 7 crossings. (From reference [6], courtesy of J. Hoste, M. Thistlethwaite and J. Weeks.) 3 ∼ A knot K ⊂ S is alternating if it has a regular planar diagram DK ⊂ P(= S2) ⊂ S3 such that, when traveling around K , the crossings alternate, over-under- over-under, all the way along K in DK . Figure1 show the first 15 knot types in P. G. Tait's earliest table and each diagram exhibits this alternating pattern. This simple arXiv:1901.00582v1 [math.GT] 3 Jan 2019 definition is very unsatisfying. A knot is alternating if we can draw it as an alternating diagram? There is no mention of any geometric structure. Dissatisfied with this characterization of an alternating knot, Ralph Fox (1913-1973) asked: "What is an alternating knot?" black white white black Figure 2. Going from a black to white region near a crossing. 1 2 WILLIAM W. MENASCO Let's make an initial attempt to address this dissatisfaction by giving a different characterization of an alternating diagram that is immediate from the over-under- over-under characterization. As with all regular planar diagrams of knots in S3, the regions of an alternating diagram can be colored in a checkerboard fashion. Thus, at each crossing (see figure2) we will have \two" white regions and \two" black regions coming together with similarly colored regions being kitty-corner to each other. -
Knot Cobordisms, Bridge Index, and Torsion in Floer Homology
KNOT COBORDISMS, BRIDGE INDEX, AND TORSION IN FLOER HOMOLOGY ANDRAS´ JUHASZ,´ MAGGIE MILLER AND IAN ZEMKE Abstract. Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a knot. It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for Tp;q and Tp;q#T p;q, respectively. We also show that the bridge index of Tp;q is minimal within its concordance class. The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. We show knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot. 1. Introduction The slice-ribbon conjecture is one of the key open problems in knot theory. It states that every slice knot is ribbon; i.e., admits a slice disk on which the radial function of the 4-ball induces no local maxima. -
Crossing Number of Alternating Knots in S × I
Pacific Journal of Mathematics CROSSING NUMBER OF ALTERNATING KNOTS IN S × I Colin Adams, Thomas Fleming, Michael Levin, and Ari M. Turner Volume 203 No. 1 March 2002 PACIFIC JOURNAL OF MATHEMATICS Vol. 203, No. 1, 2002 CROSSING NUMBER OF ALTERNATING KNOTS IN S × I Colin Adams, Thomas Fleming, Michael Levin, and Ari M. Turner One of the Tait conjectures, which was stated 100 years ago and proved in the 1980’s, said that reduced alternating projections of alternating knots have the minimal number of crossings. We prove a generalization of this for knots in S ×I, where S is a surface. We use a combination of geometric and polynomial techniques. 1. Introduction. A hundred years ago, Tait conjectured that the number of crossings in a reduced alternating projection of an alternating knot is minimal. This state- ment was proven in 1986 by Kauffman, Murasugi and Thistlethwaite, [6], [10], [11], working independently. Their proofs relied on the new polynomi- als generated in the wake of the discovery of the Jones polynomial. We usually think of this result as applying to knots in the 3-sphere S3. However, it applies equally well to knots in S2×I (where I is the unit interval [0, 1]). Indeed, if one removes two disjoint balls from S3, the resulting space is homeomorphic to S2 × I. It is not hard to see that these two balls do not affect knot equivalence. We conclude that the theory of knot equivalence in S2 × I is the same as in S3. With this equivalence in mind, it is natural to ask if the Tait conjecture generalizes to knots in spaces of the form S × I where S is any compact surface. -
Preprint of an Article Published in Journal of Knot Theory and Its Ramifications, 26 (10), 1750055 (2017)
Preprint of an article published in Journal of Knot Theory and Its Ramifications, 26 (10), 1750055 (2017) https://doi.org/10.1142/S0218216517500559 © World Scientific Publishing Company https://www.worldscientific.com/worldscinet/jktr INFINITE FAMILIES OF HYPERBOLIC PRIME KNOTS WITH ALTERNATION NUMBER 1 AND DEALTERNATING NUMBER n MARIA´ DE LOS ANGELES GUEVARA HERNANDEZ´ Abstract For each positive integer n we will construct a family of infinitely many hyperbolic prime knots with alternation number 1, dealternating number equal to n, braid index equal to n + 3 and Turaev genus equal to n. 1 Introduction After the proof of the Tait flype conjecture on alternating links given by Menasco and Thistlethwaite in [25], it became an important question to ask how a non-alternating link is “close to” alternating links [15]. Moreover, recently Greene [11] and Howie [13], inde- pendently, gave a characterization of alternating links. Such a characterization shows that being alternating is a topological property of the knot exterior and not just a property of the diagrams, answering an old question of Ralph Fox “What is an alternating knot? ”. Kawauchi in [15] introduced the concept of alternation number. The alternation num- ber of a link diagram D is the minimum number of crossing changes necessary to transform D into some (possibly non-alternating) diagram of an alternating link. The alternation num- ber of a link L, denoted alt(L), is the minimum alternation number of any diagram of L. He constructed infinitely many hyperbolic links with Gordian distance far from the set of (possibly, splittable) alternating links in the concordance class of every link. -
TANGLES for KNOTS and LINKS 1. Introduction It Is
TANGLES FOR KNOTS AND LINKS ZACHARY ABEL 1. Introduction It is often useful to discuss only small \pieces" of a link or a link diagram while disregarding everything else. For example, the Reidemeister moves describe manipulations surrounding at most 3 crossings, and the skein relations for the Jones and Conway polynomials discuss modifications on one crossing at a time. Tangles may be thought of as small pieces or a local pictures of knots or links, and they provide a useful language to describe the above manipulations. But the power of tangles in knot and link theory extends far beyond simple diagrammatic convenience, and this article provides a short survey of some of these applications. 2. Tangles As Used in Knot Theory Formally, we define a tangle as follows (in this article, everything is in the PL category): Definition 1. A tangle is a pair (A; t) where A =∼ B3 is a closed 3-ball and t is a proper 1-submanifold with @t 6= ;. Note that t may be disconnected and may contain closed loops in the interior of A. We consider two tangles (A; t) and (B; u) equivalent if they are isotopic as pairs of embedded manifolds. An n-string tangle consists of exactly n arcs and 2n boundary points (and no closed loops), and the trivial n-string 2 tangle is the tangle (B ; fp1; : : : ; png) × [0; 1] consisting of n parallel, unintertwined segments. See Figure 1 for examples of tangles. Note that with an appropriate isotopy any tangle (A; t) may be transformed into an equivalent tangle so that A is the unit ball in R3, @t = A \ t lies on the great circle in the xy-plane, and the projection (x; y; z) 7! (x; y) is a regular projection for t; thus, tangle diagrams as shown in Figure 1 are justified for all tangles. -
An Introduction to the Theory of Knots
An Introduction to the Theory of Knots Giovanni De Santi December 11, 2002 Figure 1: Escher’s Knots, 1965 1 1 Knot Theory Knot theory is an appealing subject because the objects studied are familiar in everyday physical space. Although the subject matter of knot theory is familiar to everyone and its problems are easily stated, arising not only in many branches of mathematics but also in such diverse fields as biology, chemistry, and physics, it is often unclear how to apply mathematical techniques even to the most basic problems. We proceed to present these mathematical techniques. 1.1 Knots The intuitive notion of a knot is that of a knotted loop of rope. This notion leads naturally to the definition of a knot as a continuous simple closed curve in R3. Such a curve consists of a continuous function f : [0, 1] → R3 with f(0) = f(1) and with f(x) = f(y) implying one of three possibilities: 1. x = y 2. x = 0 and y = 1 3. x = 1 and y = 0 Unfortunately this definition admits pathological or so called wild knots into our studies. The remedies are either to introduce the concept of differentiability or to use polygonal curves instead of differentiable ones in the definition. The simplest definitions in knot theory are based on the latter approach. Definition 1.1 (knot) A knot is a simple closed polygonal curve in R3. The ordered set (p1, p2, . , pn) defines a knot; the knot being the union of the line segments [p1, p2], [p2, p3],..., [pn−1, pn], and [pn, p1]. -
The Geometric Content of Tait's Conjectures
The geometric content of Tait's conjectures Ohio State CKVK* seminar Thomas Kindred, University of Nebraska-Lincoln Monday, November 9, 2020 Historical background: Tait's conjectures, Fox's question Tait's conjectures (1898) Let D and D0 be reduced alternating diagrams of a prime knot L. (Prime implies 6 9 T1 T2 ; reduced means 6 9 T .) Then: 0 (1) D and D minimize crossings: j jD = j jD0 = c(L): 0 0 (2) D and D have the same writhe: w(D) = w(D ) = j jD0 − j jD0 : (3) D and D0 are related by flype moves: T 2 T1 T2 T1 Question (Fox, ∼ 1960) What is an alternating knot? Tait's conjectures all remained open until the 1985 discovery of the Jones polynomial. Fox's question remained open until 2017. Historical background: Proofs of Tait's conjectures In 1987, Kauffman, Murasugi, and Thistlethwaite independently proved (1) using the Jones polynomial, whose degree span is j jD , e.g. V (t) = t + t3 − t4. Using the knot signature σ(L), (1) implies (2). In 1993, Menasco-Thistlethwaite proved (3), using geometric techniques and the Jones polynomial. Note: (3) implies (2) and part of (1). They asked if purely geometric proofs exist. The first came in 2017.... Tait's conjectures (1898) T 2 GivenT1 reducedT2 alternatingT1 diagrams D; D0 of a prime knot L: 0 (1) D and D minimize crossings: j jD = j jD0 = c(L): 0 0 (2) D and D have the same writhe: w(D) = w(D ) = j jD0 − j jD0 : (3) D and D0 are related by flype moves: Historical background: geometric proofs Question (Fox, ∼ 1960) What is an alternating knot? Theorem (Greene; Howie, 2017) 3 A knot L ⊂ S is alternating iff it has spanning surfaces F+ and F− s.t.: • Howie: 2(β1(F+) + β1(F−)) = s(F+) − s(F−). -
Thin Position and Bridge Number for Knots in the 3-Sphere
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Pergamon Topology Vol. 36, No. 2, pp. 505-507, 1997 Copyright 0 1996 Elsevier Science Ltd Printed in Gnat Britain. AU rights reserved 0040-9383/96/$15.00 + 0.W THIN POSITION AND BRIDGE NUMBER FOR KNOTS IN THE 3-SPHERE ABIGAILTHOMPSON’ (Received 23 August 1995) The bridge number of a knot in S3, introduced by Schubert [l] is a classical and well-understood knot invariant. The concept of thin position for a knot was developed fairly recently by Gabai [a]. It has proved to be an extraordinarily useful notion, playing a key role in Gabai’s proof of property R as well as Gordon-Luecke’s solution of the knot complement problem [3]. The purpose of this paper is to examine the relation between bridge number and thin position. We show that either a knot in thin position is also in the position which realizes its bridge number or the knot has an incompressible meridianal planar surface properly imbedded in its complement. The second possibility implies that the knot has a generalized tangle decomposition along an incompressible punctured 2-sphere. Using results from [4,5], we note that if thin position for a knot K is not bridge position, then there exists a closed incompressible surface in the complement of the knot (Corollary 3) and that the tunnel number of the knot is strictly greater than one (Corollary 4). We need a few definitions prior to starting the main theorem. -
Transverse Unknotting Number and Family Trees
Transverse Unknotting Number and Family Trees Blossom Jeong Senior Thesis in Mathematics Bryn Mawr College Lisa Traynor, Advisor May 8th, 2020 1 Abstract The unknotting number is a classical invariant for smooth knots, [1]. More recently, the concept of knot ancestry has been defined and explored, [3]. In my research, I explore how these concepts can be adapted to study trans- verse knots, which are smooth knots that satisfy an additional geometric condition imposed by a contact structure. 1. Introduction Smooth knots are well-studied objects in topology. A smooth knot is a closed curve in 3-dimensional space that does not intersect itself anywhere. Figure 1 shows a diagram of the unknot and a diagram of the positive trefoil knot. Figure 1. A diagram of the unknot and a diagram of the positive trefoil knot. Two knots are equivalent if one can be deformed to the other. It is well-known that two diagrams represent the same smooth knot if and only if their diagrams are equivalent through Reidemeister moves. On the other hand, to show that two knots are different, we need to construct an invariant that can distinguish them. For example, tricolorability shows that the trefoil is different from the unknot. Unknotting number is another invariant: it is known that every smooth knot diagram can be converted to the diagram of the unknot by changing crossings. This is used to define the smooth unknotting number, which measures the minimal number of times a knot must cross through itself in order to become the unknot. This thesis will focus on transverse knots, which are smooth knots that satisfy an ad- ditional geometric condition imposed by a contact structure.