Unknotting Number and Knot Diagram 1 Introduction
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Conservation of Complex Knotting and Slipknotting Patterns in Proteins
Conservation of complex knotting and PNAS PLUS slipknotting patterns in proteins Joanna I. Sułkowskaa,1,2, Eric J. Rawdonb,1, Kenneth C. Millettc, Jose N. Onuchicd,2, and Andrzej Stasiake aCenter for Theoretical Biological Physics, University of California at San Diego, La Jolla, CA 92037; bDepartment of Mathematics, University of St. Thomas, Saint Paul, MN 55105; cDepartment of Mathematics, University of California, Santa Barbara, CA 93106; dCenter for Theoretical Biological Physics , Rice University, Houston, TX 77005; and eCenter for Integrative Genomics, Faculty of Biology and Medicine, University of Lausanne, 1015 Lausanne, Switzerland Edited by* Michael S. Waterman, University of Southern California, Los Angeles, CA, and approved May 4, 2012 (received for review April 17, 2012) While analyzing all available protein structures for the presence of ers fixed configurations, in this case proteins in their native folded knots and slipknots, we detected a strict conservation of complex structures. Then they may be treated as frozen and thus unable to knotting patterns within and between several protein families undergo any deformation. despite their large sequence divergence. Because protein folding Several papers have described various interesting closure pathways leading to knotted native protein structures are slower procedures to capture the knot type of the native structure of and less efficient than those leading to unknotted proteins with a protein or a subchain of a closed chain (1, 3, 29–33). In general, similar size and sequence, the strict conservation of the knotting the strategy is to ensure that the closure procedure does not affect patterns indicates an important physiological role of knots and slip- the inherent entanglement in the analyzed protein chain or sub- knots in these proteins. -
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. -
CROSSING CHANGES and MINIMAL DIAGRAMS the Unknotting
CROSSING CHANGES AND MINIMAL DIAGRAMS ISABEL K. DARCY The unknotting number of a knot is the minimal number of crossing changes needed to convert the knot into the unknot where the minimum is taken over all possible diagrams of the knot. For example, the minimal diagram of the knot 108 requires 3 crossing changes in order to change it to the unknot. Thus, by looking only at the minimal diagram of 108, it is clear that u(108) ≤ 3 (figure 1). Nakanishi [5] and Bleiler [2] proved that u(108) = 2. They found a non-minimal diagram of the knot 108 in which two crossing changes suffice to obtain the unknot (figure 2). Lower bounds on unknotting number can be found by looking at how knot invariants are affected by crossing changes. Nakanishi [5] and Bleiler [2] used signature to prove that u(108)=2. Figure 1. Minimal dia- Figure 2. A non-minimal dia- gram of knot 108 gram of knot 108 Bernhard [1] noticed that one can determine that u(108) = 2 by using a sequence of crossing changes within minimal diagrams and ambient isotopies as shown in figure. A crossing change within the minimal diagram of 108 results in the knot 62. We then use an ambient isotopy to change this non-minimal diagram of 62 to a minimal diagram. The unknot can then be obtained by changing one crossing within the minimal diagram of 62. Bernhard hypothesized that the unknotting number of a knot could be determined by only looking at minimal diagrams by using ambient isotopies between crossing changes. -
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]. -
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. -
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. -
Jhep07(2018)145
Published for SISSA by Springer Received: April 13, 2018 Accepted: July 16, 2018 Published: July 23, 2018 Symmetry enhancement and closing of knots in 3d/3d JHEP07(2018)145 correspondence Dongmin Ganga and Kazuya Yonekurab aCenter for Theorectical Physics, Seoul National University, Seoul 08826, Korea bKavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8583, Japan E-mail: [email protected], [email protected] Abstract: We revisit Dimofte-Gaiotto-Gukov's construction of 3d gauge theories asso- ciated to 3-manifolds with a torus boundary. After clarifying their construction from a viewpoint of compactification of a 6d N = (2; 0) theory of A1-type on a 3-manifold, we propose a topological criterion for SU(2)=SO(3) flavor symmetry enhancement for the u(1) symmetry in the theory associated to a torus boundary, which is expected from the 6d viewpoint. Base on the understanding of symmetry enhancement, we generalize the con- struction to closed 3-manifolds by identifying the gauge theory counterpart of Dehn filling operation. The generalized construction predicts infinitely many 3d dualities from surgery calculus in knot theory. Moreover, by using the symmetry enhancement criterion, we show that theories associated to all hyperboilc twist knots have surprising SU(3) symmetry en- hancement which is unexpected from the 6d viewpoint. Keywords: Chern-Simons Theories, Conformal Field Theory, Duality in Gauge Field Theories, M-Theory ArXiv ePrint: 1803.04009 Open Access, c The Authors. -
Results on Nonorientable Surfaces for Knots and 2-Knots
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Papers in Mathematics Mathematics, Department of 8-2021 Results on Nonorientable Surfaces for Knots and 2-knots Vincent Longo University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathstudent Part of the Geometry and Topology Commons Longo, Vincent, "Results on Nonorientable Surfaces for Knots and 2-knots" (2021). Dissertations, Theses, and Student Research Papers in Mathematics. 111. https://digitalcommons.unl.edu/mathstudent/111 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Dissertations, Theses, and Student Research Papers in Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. RESULTS ON NONORIENTABLE SURFACES FOR KNOTS AND 2-KNOTS by Vincent Longo A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professors Alex Zupan and Mark Brittenham Lincoln, Nebraska May, 2021 RESULTS ON NONORIENTABLE SURFACES FOR KNOTS AND 2-KNOTS Vincent Longo, Ph.D. University of Nebraska, 2021 Advisor: Alex Zupan and Mark Brittenham A classical knot is a smooth embedding of the circle S1 into the 3-sphere S3. We can also consider embeddings of arbitrary surfaces (possibly nonorientable) into a 4-manifold, called knotted surfaces. In this thesis, we give an introduction to some of the basics of the studies of classical knots and knotted surfaces, then present some results about nonorientable surfaces bounded by classical knots and embeddings of nonorientable knotted surfaces. -
Maximum Knotted Switching Sequences
Maximum Knotted Switching Sequences Tom Milnor August 21, 2008 Abstract In this paper, we define the maximum knotted switching sequence length of a knot projection and investigate some of its properties. In par- ticular, we examine the effects of Reidemeister and flype moves on maxi- mum knotted switching sequences, show that every knot has projections having a full sequence, and show that all minimal alternating projections of a knot have the same length maximum knotted switching sequence. Contents 1 Introduction 2 1.1 Basic definitions . 2 1.2 Changing knot projections . 3 2 Maximum knotted switching sequences 4 2.1 Immediate bounds . 4 2.2 Not all maximal knotted switching sequences are maximum . 5 2.3 Composite knots . 5 2.4 All knots have a projection with a full sequence . 6 2.5 An infinite family of knots with projections not having full se- quences . 6 3 The effects of Reidemeister moves on the maximum knotted switching sequence length of a knot projection 7 3.1 Type I Reidemeister moves . 7 3.2 Type II Reidemeister moves . 8 3.3 Alternating knots and Type II Reidemeister moves . 9 3.4 The Conway polynomial and Type II Reidemeister moves . 13 3.5 Type III Reidemeister moves . 15 4 The maximum sequence length of minimal alternating knot pro- jections 16 4.1 Flype moves . 16 1 1 Introduction 1.1 Basic definitions A knot is a smooth embedding of S1 into R3 or S3. A knot is said to be an unknot if it is isotopic to S1 ⊂ R3. Otherwise it is knotted. -
Knot and Link Tricolorability Danielle Brushaber Mckenzie Hennen Molly Petersen Faculty Mentor: Carolyn Otto University of Wisconsin-Eau Claire
Knot and Link Tricolorability Danielle Brushaber McKenzie Hennen Molly Petersen Faculty Mentor: Carolyn Otto University of Wisconsin-Eau Claire Problem & Importance Colorability Tables of Characteristics Theorem: For WH 51 with n twists, WH 5 is tricolorable when Knot Theory, a field of Topology, can be used to model Original Knot The unknot is not tricolorable, therefore anything that is tri- 1 colorable cannot be the unknot. The prime factors of the and understand how enzymes (called topoisomerases) work n = 3k + 1 where k ∈ N ∪ {0}. in DNA processes to untangle or repair strands of DNA. In a determinant of the knot or link provides the colorability. For human cell nucleus, the DNA is linear, so the knots can slip off example, if a knot’s determinant is 21, it is 3-colorable (tricol- orable), and 7-colorable. This is known by the theorem that Proof the end, and it is difficult to recognize what the enzymes do. Consider WH 5 where n is the number of the determinant of a knot is 0 mod n if and only if the knot 1 However, the DNA in mitochon- full positive twists. n is n-colorable. dria is circular, along with prokary- Link Colorability Det(L) Unknot/Link (n = 1...n = k) otic cells (bacteria), so the enzyme L Number Theorem: If det(L) = 0, then L is WOLOG, let WH 51 be colored in this way, processes are more noticeable in 3 3 3 1 n 1 n-colorable for all n. excluding coloring the twist component. knots in this type of DNA. -
The Kauffman Bracket and Genus of Alternating Links
California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 6-2016 The Kauffman Bracket and Genus of Alternating Links Bryan M. Nguyen Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Other Mathematics Commons Recommended Citation Nguyen, Bryan M., "The Kauffman Bracket and Genus of Alternating Links" (2016). Electronic Theses, Projects, and Dissertations. 360. https://scholarworks.lib.csusb.edu/etd/360 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. The Kauffman Bracket and Genus of Alternating Links A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Bryan Minh Nhut Nguyen June 2016 The Kauffman Bracket and Genus of Alternating Links A Thesis Presented to the Faculty of California State University, San Bernardino by Bryan Minh Nhut Nguyen June 2016 Approved by: Dr. Rolland Trapp, Committee Chair Date Dr. Gary Griffing, Committee Member Dr. Jeremy Aikin, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract Giving a knot, there are three rules to help us finding the Kauffman bracket polynomial. Choosing knot's orientation, then applying the Seifert algorithm to find the Euler characteristic and genus of its surface. Finally finding the relationship of the Kauffman bracket polynomial and the genus of the alternating links is the main goal of this paper. -
Arxiv:2108.09698V1 [Math.GT] 22 Aug 2021
Submitted to Topology Proceedings THE TABULATION OF PRIME KNOT PROJECTIONS WITH THEIR MIRROR IMAGES UP TO EIGHT DOUBLE POINTS NOBORU ITO AND YUSUKE TAKIMURA Abstract. This paper provides the complete table of prime knot projections with their mirror images, without redundancy, up to eight double points systematically thorough a finite procedure by flypes. In this paper, we show how to tabulate the knot projec- tions up to eight double points by listing tangles with at most four double points by an approach with respect to rational tangles of J. H. Conway. In other words, for a given prime knot projection of an alternating knot, we show how to enumerate possible projec- tions of the alternating knot. Also to tabulate knot projections up to ambient isotopy, we introduce arrow diagrams (oriented Gauss diagrams) of knot projections having no over/under information of each crossing, which were originally introduced as arrow diagrams of knot diagrams by M. Polyak and O. Viro. Each arrow diagram of a knot projection completely detects the difference between the knot projection and its mirror image. 1. Introduction arXiv:2108.09698v1 [math.GT] 22 Aug 2021 Arnold ([2, Figure 53], [3, Figure 15]) obtained a table of reduced knot projections (equivalently, reduced generic immersed spherical curves) up to seven double points. In Arnold’s table, the number of prime knot pro- jections with seven double points is six. However, this table is incomplete (see Figure 1, which equals [2, Figure 53]). Key words and phrases. knot projection; tabulation; flype MSC 2010: 57M25, 57Q35. 1 2 NOBORU ITO AND YUSUKE TAKIMURA Figure 1.