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Arxiv:Math/0307077V4
Table of Contents for the Handbook of Knot Theory William W. Menasco and Morwen B. Thistlethwaite, Editors (1) Colin Adams, Hyperbolic Knots (2) Joan S. Birman and Tara Brendle, Braids: A Survey (3) John Etnyre Legendrian and Transversal Knots (4) Greg Friedman, Knot Spinning (5) Jim Hoste, The Enumeration and Classification of Knots and Links (6) Louis Kauffman, Knot Diagramitics (7) Charles Livingston, A Survey of Classical Knot Concordance (8) Lee Rudolph, Knot Theory of Complex Plane Curves (9) Marty Scharlemann, Thin Position in the Theory of Classical Knots (10) Jeff Weeks, Computation of Hyperbolic Structures in Knot Theory arXiv:math/0307077v4 [math.GT] 26 Nov 2004 A SURVEY OF CLASSICAL KNOT CONCORDANCE CHARLES LIVINGSTON In 1926 Artin [3] described the construction of certain knotted 2–spheres in R4. The intersection of each of these knots with the standard R3 ⊂ R4 is a nontrivial knot in R3. Thus a natural problem is to identify which knots can occur as such slices of knotted 2–spheres. Initially it seemed possible that every knot is such a slice knot and it wasn’t until the early 1960s that Murasugi [86] and Fox and Milnor [24, 25] succeeded at proving that some knots are not slice. Slice knots can be used to define an equivalence relation on the set of knots in S3: knots K and J are equivalent if K# − J is slice. With this equivalence the set of knots becomes a group, the concordance group of knots. Much progress has been made in studying slice knots and the concordance group, yet some of the most easily asked questions remain untouched. -
Introduction to Vassiliev Knot Invariants First Draft. Comments
Introduction to Vassiliev Knot Invariants First draft. Comments welcome. July 20, 2010 S. Chmutov S. Duzhin J. Mostovoy The Ohio State University, Mansfield Campus, 1680 Univer- sity Drive, Mansfield, OH 44906, USA E-mail address: [email protected] Steklov Institute of Mathematics, St. Petersburg Division, Fontanka 27, St. Petersburg, 191011, Russia E-mail address: [email protected] Departamento de Matematicas,´ CINVESTAV, Apartado Postal 14-740, C.P. 07000 Mexico,´ D.F. Mexico E-mail address: [email protected] Contents Preface 8 Part 1. Fundamentals Chapter 1. Knots and their relatives 15 1.1. Definitions and examples 15 § 1.2. Isotopy 16 § 1.3. Plane knot diagrams 19 § 1.4. Inverses and mirror images 21 § 1.5. Knot tables 23 § 1.6. Algebra of knots 25 § 1.7. Tangles, string links and braids 25 § 1.8. Variations 30 § Exercises 34 Chapter 2. Knot invariants 39 2.1. Definition and first examples 39 § 2.2. Linking number 40 § 2.3. Conway polynomial 43 § 2.4. Jones polynomial 45 § 2.5. Algebra of knot invariants 47 § 2.6. Quantum invariants 47 § 2.7. Two-variable link polynomials 55 § Exercises 62 3 4 Contents Chapter 3. Finite type invariants 69 3.1. Definition of Vassiliev invariants 69 § 3.2. Algebra of Vassiliev invariants 72 § 3.3. Vassiliev invariants of degrees 0, 1 and 2 76 § 3.4. Chord diagrams 78 § 3.5. Invariants of framed knots 80 § 3.6. Classical knot polynomials as Vassiliev invariants 82 § 3.7. Actuality tables 88 § 3.8. Vassiliev invariants of tangles 91 § Exercises 93 Chapter 4. -
Order 2 Algebraically Slice Knots 1 Introduction
ISSN 1464-8997 (on line) 1464-8989 (printed) 335 Geometry & Topology Monographs Volume 2: Proceedings of the Kirbyfest Pages 335–342 Order 2 Algebraically Slice Knots Charles Livingston Abstract The concordance group of algebraically slice knots is the sub- group of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebraically slice knots also contain elements of finite order; in fact it contains an infinite subgroup generated by elements of order 2. AMS Classification 57M25; 57N70, 57Q20 Keywords Concordance, concordance group, slice, algebraically slice 1 Introduction The classical knot concordance group, C , was defined by Fox [5] in 1962. The work of Fox and Milnor [6], along with that of Murasugi [18] and Levine [14, 15], revealed fundamental aspects of the structure of C . Since then there has been tremendous progress in 3– and 4–dimensional geometric topology, yet nothing more is now known about the underlying group structure of C than was known in 1969. In this paper we will describe new and unexpected classes of order 2 in C . What is known about C is quickly summarized. It is a countable abelian group. ∞ ∞ ∞ According to [15] there is a surjective homomorphism of φ: C → Z ⊕Z2 ⊕Z4 . The results of [6] quickly yield an infinite set of elements of order 2 in C , all of which are mapped to elements of order 2 by φ. The results just stated, and their algebraic consequences, present all that is known concerning the purely algebraic structure of C in either the smooth or topological locally flat category. -
On Signatures of Knots
1 ON SIGNATURES OF KNOTS Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/eaar For Cherry Kearton Durham, 20 June 2010 2 MR: Publications results for "Author/Related=(kearton, C*)" http://www.ams.org/mathscinet/search/publications.html?arg3=&co4... Matches: 48 Publications results for "Author/Related=(kearton, C*)" MR2443242 (2009f:57033) Kearton, Cherry; Kurlin, Vitaliy All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron. Algebr. Geom. Topol. 8 (2008), no. 3, 1223--1247. (Reviewer: J. P. E. Hodgson) 57Q37 (57Q35 57Q45) MR2402510 (2009k:57039) Kearton, C.; Wilson, S. M. J. New invariants of simple knots. J. Knot Theory Ramifications 17 (2008), no. 3, 337--350. 57Q45 (57M25 57M27) MR2088740 (2005e:57022) Kearton, C. $S$-equivalence of knots. J. Knot Theory Ramifications 13 (2004), no. 6, 709--717. (Reviewer: Swatee Naik) 57M25 MR2008881 (2004j:57017)( Kearton, C.; Wilson, S. M. J. Sharp bounds on some classical knot invariants. J. Knot Theory Ramifications 12 (2003), no. 6, 805--817. (Reviewer: Simon A. King) 57M27 (11E39 57M25) MR1967242 (2004e:57029) Kearton, C.; Wilson, S. M. J. Simple non-finite knots are not prime in higher dimensions. J. Knot Theory Ramifications 12 (2003), no. 2, 225--241. 57Q45 MR1933359 (2003g:57008) Kearton, C.; Wilson, S. M. J. Knot modules and the Nakanishi index. Proc. Amer. Math. Soc. 131 (2003), no. 2, 655--663 (electronic). (Reviewer: Jonathan A. Hillman) 57M25 MR1803365 (2002a:57033) Kearton, C. Quadratic forms in knot theory.t Quadratic forms and their applications (Dublin, 1999), 135--154, Contemp. Math., 272, Amer. Math. Soc., Providence, RI, 2000. -
Textile Research Journal
Textile Research Journal http://trj.sagepub.com A Topological Study of Textile Structures. Part II: Topological Invariants in Application to Textile Structures S. Grishanov, V. Meshkov and A. Omelchenko Textile Research Journal 2009; 79; 822 DOI: 10.1177/0040517508096221 The online version of this article can be found at: http://trj.sagepub.com/cgi/content/abstract/79/9/822 Published by: http://www.sagepublications.com Additional services and information for Textile Research Journal can be found at: Email Alerts: http://trj.sagepub.com/cgi/alerts Subscriptions: http://trj.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.co.uk/journalsPermissions.nav Citations http://trj.sagepub.com/cgi/content/refs/79/9/822 Downloaded from http://trj.sagepub.com by Andrew Ranicki on October 11, 2009 Textile Research Journal Article A Topological Study of Textile Structures. Part II: Topological Invariants in Application to Textile Structures S. Grishanov1 Abstract This paper is the second in the series TEAM Research Group, De Montfort University, Leicester, on topological classification of textile structures. UK The classification problem can be resolved with the aid of invariants used in knot theory for classi- V. Meshkov and A. Omelchenko fication of knots and links. Various numerical and St Petersburg State Polytechnical University, St polynomial invariants are considered in application Petersburg, Russia to textile structures. A new Kauffman-type polyno- mial invariant is constructed for doubly-periodic textile structures. The values of the numerical and polynomial invariants are calculated for some sim- plest doubly-periodic interlaced structures and for some woven and knitted textiles. -
Grid Homology for Knots and Links
Grid Homology for Knots and Links Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo This is a preliminary version of the book Grid Homology for Knots and Links published by the American Mathematical Society (AMS). This preliminary version is made available with the permission of the AMS and may not be changed, edited, or reposted at any other website without explicit written permission from the author and the AMS. Contents Chapter 1. Introduction 1 1.1. Grid homology and the Alexander polynomial 1 1.2. Applications of grid homology 3 1.3. Knot Floer homology 5 1.4. Comparison with Khovanov homology 7 1.5. On notational conventions 7 1.6. Necessary background 9 1.7. The organization of this book 9 1.8. Acknowledgements 11 Chapter 2. Knots and links in S3 13 2.1. Knots and links 13 2.2. Seifert surfaces 20 2.3. Signature and the unknotting number 21 2.4. The Alexander polynomial 25 2.5. Further constructions of knots and links 30 2.6. The slice genus 32 2.7. The Goeritz matrix and the signature 37 Chapter 3. Grid diagrams 43 3.1. Planar grid diagrams 43 3.2. Toroidal grid diagrams 49 3.3. Grids and the Alexander polynomial 51 3.4. Grid diagrams and Seifert surfaces 56 3.5. Grid diagrams and the fundamental group 63 Chapter 4. Grid homology 65 4.1. Grid states 65 4.2. Rectangles connecting grid states 66 4.3. The bigrading on grid states 68 4.4. The simplest version of grid homology 72 4.5. -
The Concordance Genus of Knots 1 Introduction and Basic Results
ISSN 1472-2739 (on-line) 1472-2747 (printed) 1 Algebraic & Geometric Topology Volume 4 (2004) 1–22 ATG Published: 9 January 2004 The concordance genus of knots Charles Livingston Abstract In knot concordance three genera arise naturally, g(K),g4(K), and gc(K): these are the classical genus, the 4–ball genus, and the concor- dance genus, defined to be the minimum genus among all knots concordant to K . Clearly 0 ≤ g4(K) ≤ gc(K) ≤ g(K). Casson and Nakanishi gave examples to show that g4(K) need not equal gc(K). We begin by reviewing and extending their results. For knots representing elements in A, the concordance group of algebraically slice knots, the relationships between these genera are less clear. Casson and Gordon’s result that A is nontrivial implies that g4(K) can be nonzero for knots in A. Gilmer proved that g4(K) can be arbitrarily large for knots in A. We will prove that there are knots K in A with g4(K) = 1 and gc(K) arbitrarily large. Finally, we tabulate gc for all prime knots with 10 crossings and, with two exceptions, all prime knots with fewer than 10 crossings. This requires the description of previously unnoticed concordances. AMS Classification 57M25, 57N70 Keywords Concordance, knot concordance, genus, slice genus 1 Introduction and basic results For a knot K ⊂ S3 , three genera arise naturally: g(K), the genus of K , is the 3 minimum genus among surfaces bounded by K in S ; g4(K) is the minimum 4 genus among surfaces bounded by K in B ; gc(K) is the minimum value of g(K′) among all knots K′ concordant to K . -
Applied Topology Methods in Knot Theory
Applied topology methods in knot theory Radmila Sazdanovic NC State University Optimal Transport, Topological Data Analysis and Applications to Shape and Machine Learning//Mathematical Bisciences Institute OSU 29 July 2020 big data tools in pure mathematics Joint work with P. Dlotko, J. Levitt. M. Hajij • How to vectorize data, how to get a point cloud out of the shapes? Idea: Use a vector consisting of numerical descriptors • Methods: • Machine learning: How to use ML with infinite data that is hard to sample in a reasonable way. a • Topological data analysis • Hybrid between TDA and statistics: PCA combined with appropriate filtration • Goals: improving identification and circumventing obstacles from computational complexity of shape descriptors. big data tools in knot theory • Input: point clouds obtained from knot invariants • Machine learning • PCA Principal Component Analysis with different filtrations • Topological data analysis • Goals: • characterizing discriminative power of knot invariants for knot detection • comparing knot invariants • experimental evidence for conjectures knots and links 1 3 Knot is an equivalence class of smooth embeddings f : S ! R up to ambient isotopy. knot theory • Problems are easy to state but a remarkable breath of techniques are employed in answering these questions • combinatorial • algebraic • geometric • Knots are interesting on their own but they also provide information about 3- and 4-dimensional manifolds Theorem (Lickorish-Wallace) Every closed, connected, oriented 3-manifold can be obtained by doing surgery on a link in S3: knots are `big data' #C 0 3 4 5 6 7 8 9 10 #PK 1 1 1 2 3 7 21 49 165 #C 11 12 13 14 15 #PK 552 2,176 9,988 46,972 253,293 #C 16 17 18 19 #PK 1,388,705 8,053,393 48,266,466 294,130,458 Table: Number of prime knots for a given crossing number. -
An Exploration Into Knot Theory Application Regarding Enzymatic
AN EXPLORATION INTO KNOT THEORY APPLICATIONS REGARDING ENZYMATIC ACTION ON DNA MOLECULES A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfllment Of the Requirements for the Degree Master of Science In Mathematics By Albert Joseph Jefferson IV 2019 SIGNATURE PAGE THESIS: AN EXPLORATION INTO KNOT THEORY APPLICATIONS REGARDING ENZYMATIC ACTION ON DNA MOLECULES AUTHOR: Albert Joseph Jefferson IV DATE SUBMITTED: Fall 2019 Department of Mathematics and Statistics Dr. Robin Wilson Thesis Committee Chair Mathematics & Statistics Dr. John Rock Mathematics & Statistics Dr. Amber Rosin Mathematics & Statistics ii ACKNOWLEDGMENTS I would frst like to thank my mother, who tirelessly continues to support every goal of mine, regardless of the feld, and whose guidance has been invaluable in my life. I also would like to thank my 3 sisters, each of whom are excellent role models and have given me so much inspiration. A giant thank you to my advisor, Robin Wilson, who has dealt with my procrastination gracefully, as well to my thesis committee for being so fexible with the defense (each of you are forever my mentors and I owe so much to you all). Finally, I’d like to thank my boyfriend who has been my rock through this entire process, I love you so so much. iii ABSTRACT This thesis will explore the applications of knot theory in biology at the graduate level, specifcally knot theory’s application on DNA super coiling, site specifc recombination, and the overall topology of DNA. We will start with a short introduction to knot theory, introduce rational tangles (the foundation for our tangle model for enzymatic reactions on DNA), review a very specifc overview of DNA and site specifc recombination as it pertains to our motivations, and fnally introduce the tangle model for a Tyrosine Recom- binase and its infuence on the topology of DNA following its action on the molecule. -
On the Additivity of Crossing Numbers
ON THE ADDITIVITY OF CROSSING NUMBERS A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Alicia Arrua 2015 SIGNATURE PAGE THESIS: ON THE ADDITIVITY OF CROSSING NUMBERS AUTHOR: Alicia Arrua DATE SUBMITTED: Spring 2015 Mathematics and Statistics Department Dr. Robin Wilson Thesis Committee Chair Mathematics & Statistics Dr. Greisy Winicki-Landman Mathematics & Statistics Dr. Berit Givens Mathematics & Statistics ii ACKNOWLEDGMENTS This thesis would not have been possible without the invaluable knowledge and guidance from Dr. Robin Wilson. His support throughout this entire experience has been amazing and incredibly helpful. I’d also like to thank Dr. Greisy Winicki- Landman and Dr. Berit Givens for being a part of my thesis committee and offering their support. I’d also like to thank my family for putting up with my late nights of work and motivating me when I needed it. Lastly, thank you to the wonderful friends I’ve made during my time at Cal Poly Pomona, their humor and encouragement aided me more than they know. iii ABSTRACT The additivity of crossing numbers over a composition of links has been an open problem for over one hundred years. It has been proved that the crossing number over alternating links is additive independently in 1987 by Louis Kauffman, Kunio Murasugi, and Morwen Thistlethwaite. Further, Yuanan Diao and Hermann Gru ber independently proved that the crossing number is additive over a composition of torus links. In order to investigate the additivity of crossing numbers over a composition of a different class of links, we introduce a tool called the deficiency of a link. -
Linking Numbers in Three-Manifolds
LINKING NUMBERS IN THREE-MANIFOLDS PATRICIA CAHN AND ALEXANDRA KJUCHUKOVA Abstract. Let M be a connected, closed, oriented three-manifold and K, L two rationally null- homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral 3-fold cover of S3 branched along a knot α ⊂ S3. Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot α can be derived from dihedral covers of α. The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications. 1. Introduction The notion of linking number between knots in S3 dates back at least as far as Gauss [13]. More generally, given a closed, oriented three-manifold M and two rationally null-homologous, oriented, simple closed curves K; L ⊂ M, the linking number lk(K; L) is defined as well. It is given by 1 (K · C ), where C is a 2-chain in M with boundary nL, n 2 , and · denotes the signed n L L N intersection number. This linking number is well-defined and symmetric [3]. Let the three-manifold M be presented as a 3-fold irregular dihedral branched cover of S3, branched along a knot. Every closed oriented three-manifold admits such a presentation [16, 17, 21]. -
Cameron Mca. Gordon UT Austin, CNS September 19, 2018
1 KNOTS Cameron McA. Gordon UT Austin, CNS September 19, 2018 2 A knot is a closed loop in space. 3 A knot is a closed loop in space. unknot 4 c(K)= crossing number of K = minimum number of crossings in any diagram of K. 5 c(K)= crossing number of K = minimum number of crossings in any diagram of K. 6 Vortex Atoms (Lord Kelvin, 1867) c(K) = order of knottiness of K Peter Guthrie Tait (1831-1901) 7 8 9 10 c(K) # of knots c(K) # of knots 3 1 11 552 4 1 12 2, 176 5 2 13 9, 988 6 3 14 46, 972 7 7 15 253, 293 8 21 16 1, 388, 705 9 49 17 8, 053, 378 10 165 11 c(K) # of knots c(K) # of knots 3 1 11 552 4 1 12 2, 176 5 2 13 9, 988 6 3 14 46, 972 7 7 15 253, 293 8 21 16 1, 388, 705 9 49 17 8, 053, 378 10 165 9, 755, 313 prime knots with c(K) ≤ 17. 12 How can you prove that two knots are different? 13 How can you prove that two knots are different? The Perko Pair 14 How can you prove that two knots are different? The Perko Pair How can you determine whether a given knot is the unknot or not? 15 There are many knot invariants that help with these questions. 16 There are many knot invariants that help with these questions. Example. Alexander polynomial (1928) 17 There are many knot invariants that help with these questions.