Tait's Flyping Conjecture for 4-Regular Graphs
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Arxiv:1006.4176V4 [Math.GT] 4 Nov 2011 Nepnigo Uho Ntter.Freape .W Alexander Mo W
UNKNOTTING UNKNOTS A. HENRICH AND L. KAUFFMAN Abstract. A knot is an an embedding of a circle into three–dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of unknotting a knot diagram when we know that it is unknotted. This problem is surprisingly difficult, since it has been shown that knot diagrams may need to be made more complicated before they may be simplified. We do not yet know, however, how much more complicated they must get. We give an introduction to the work of Dynnikov who discovered the key use of arc–presentations to solve the problem of finding a way to de- tect the unknot directly from a diagram of the knot. Using Dynnikov’s work, we show how to obtain a quadratic upper bound for the number of crossings that must be introduced into a sequence of unknotting moves. We also apply Dynnikov’s results to find an upper bound for the number of moves required in an unknotting sequence. 1. Introduction When one first delves into the theory of knots, one learns that knots are typically studied using their diagrams. The first question that arises when considering these knot diagrams is: how can we tell if two knot diagrams represent the same knot? Fortunately, we have a partial answer to this question. Two knot diagrams represent the same knot in R3 if and only if they can be related by the Reidemeister moves, pictured below. -
Knots: a Handout for Mathcircles
Knots: a handout for mathcircles Mladen Bestvina February 2003 1 Knots Informally, a knot is a knotted loop of string. You can create one easily enough in one of the following ways: • Take an extension cord, tie a knot in it, and then plug one end into the other. • Let your cat play with a ball of yarn for a while. Then find the two ends (good luck!) and tie them together. This is usually a very complicated knot. • Draw a diagram such as those pictured below. Such a diagram is a called a knot diagram or a knot projection. Trefoil and the figure 8 knot 1 The above two knots are the world's simplest knots. At the end of the handout you can see many more pictures of knots (from Robert Scharein's web site). The same picture contains many links as well. A link consists of several loops of string. Some links are so famous that they have names. For 2 2 3 example, 21 is the Hopf link, 51 is the Whitehead link, and 62 are the Bor- romean rings. They have the feature that individual strings (or components in mathematical parlance) are untangled (or unknotted) but you can't pull the strings apart without cutting. A bit of terminology: A crossing is a place where the knot crosses itself. The first number in knot's \name" is the number of crossings. Can you figure out the meaning of the other number(s)? 2 Reidemeister moves There are many knot diagrams representing the same knot. For example, both diagrams below represent the unknot. -
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. -
On Marked Braid Groups
July 30, 2018 6:43 WSPC/INSTRUCTION FILE braids_and_groups Journal of Knot Theory and Its Ramifications c World Scientific Publishing Company ON MARKED BRAID GROUPS DENIS A. FEDOSEEV Moscow State University, Chair of Differential Geometry and Applications VASSILY O. MANTUROV Bauman Moscow State Technical University, Chair FN{12 ZHIYUN CHENG School of Mathematical Sciences, Beijing Normal University Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China ABSTRACT In the present paper, we introduce Z2-braids and, more generally, G-braids for an arbitrary group G. They form a natural group-theoretic counterpart of G-knots, see [2]. The underlying idea, used in the construction of these objects | decoration of crossings with some additional information | generalizes an important notion of parity introduced by the second author (see [1]) to different combinatorically{geometric theories, such as knot theory, braid theory and others. These objects act as natural enhancements of classical (Artin) braid groups. The notion of dotted braid group is introduced: classical (Artin) braid groups live inside dotted braid groups as those elements having presentation with no dots on the strands. The paper is concluded by a list of unsolved problems. Keywords: Braid, virtual braid, group, presentation, parity. Mathematics Subject Classification 2000: 57M25, 57M27 arXiv:1507.02700v2 [math.GT] 22 Jul 2015 1. Introduction In the present paper we introduce the notion of Z2−braids as well as their gen- eralization: groups of G−braids for any arbitrary group G. They form a natural group{theoretical analog of G−knots, first introduced by the second author in [2]. Those objects naturally generalize the classical (Artin) braids. -
Arxiv:Math/0005108V1 [Math.GT] 11 May 2000 1.1
INVARIANTS OF KNOT DIAGRAMS AND RELATIONS AMONG REIDEMEISTER MOVES OLOF-PETTER OSTLUND¨ Abstract. In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some Ω3-moves that only differ in how the three strands that are involved in the move are ordered on the knot. To transform knot diagrams of isotopic knots into each other one must in general use Ω3-moves of at least two different classes. To show this, knot diagram invariants that jump only under Ω3-moves are introduced. Knot diagrams of isotopic knots can be connected by a sequence of Rei- demeister moves of only six, out of the total of 24, classes. This result can be applied in knot theory to simplify proofs of invariance of diagrammatical knot invariants. In particular, a criterion for a function on Gauss diagrams to define a knot invariant is presented. 1. Knot diagrams and Reidemeister moves. 1.1. Knot diagrams. A knot diagram is a picture of an oriented smooth knot like the figure eight knot diagram in Figure 1. Formally, this is the image of an immersion of S1 in R2, with transversal double points and no points of higher multiplicity, which has been decorated so that we can distinguish: a: An orientation of the strand, and b: an overpassing and an underpassing strand at each double point. Of course, the manner of decoration, such as where an arrow that indicates the orientation is placed, does not matter: Two knot diagrams are the same if they are made from the same image, have the same orientation, and have the same over-undercrossing information at each crossing point. -
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. -
Unknot Diagrams Requiring a Quadratic Number of Reidemeister Moves to Untangle
Discrete Comput Geom (2010) 44: 91–95 DOI 10.1007/s00454-009-9156-4 Unknot Diagrams Requiring a Quadratic Number of Reidemeister Moves to Untangle Joel Hass · Tahl Nowik Received: 20 November 2008 / Revised: 28 February 2009 / Accepted: 1 March 2009 / Published online: 18 March 2009 © The Author(s) 2009. This article is published with open access at Springerlink.com Abstract Given any knot diagram E, we present a sequence of knot diagrams of the same knot type for which the minimum number of Reidemeister moves required to pass to E is quadratic with respect to the number of crossings. These bounds apply both in S2 and in R2. Keywords Reidemister moves · Unknot 1 Introduction In this paper we give a family of unknot diagrams Dn with Dn having 7n − 1 cross- 2 ings and with at least 2n + 3n − 2 Reidemeister moves required to transform Dn to the trivial diagram. These are the first examples for which a nonlinear lower bound has been established. We then use Dn to construct for any knot diagram E with k crossings, a sequence of diagrams En of the same knot type, having k + 7n − 1 cross- ings, and requiring at least 2n2 + 3n − 2 Reidemeister moves to transform to E. A knot in R3 or S3 is commonly represented by a knot diagram, a generic projec- tion of the knot to a plane or 2-sphere. A diagram is an immersed oriented planar or spherical curve with finitely many double points, called crossings. Each crossing is marked to indicate a strand, called the overcrossing, that lies above the second strand, The research of J. -
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. -
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. -
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. -
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−). -
On the Reidemeister Moves of a Classical Knot
proceedings of the american mathematical society Volume 89. Number 4. December 1983 ON THE REIDEMEISTER MOVES OF A CLASSICAL KNOT BRUCE TRACE Abstract. We show that under certain reasonable criteria one of the Reidemeister moves can be elminated. It is a well-known fact in classical knot theory that the following three moves, known as the Reidemeister moves, generate all the possible projections of a knot. To be more precise, given two oriented knots, A", and K2, in R3, where K¡ is in general position with respect to the natural projection m: R3 -» R2, having the property that there is an isotopy carrying A', to K2 and preserving the orientations of the knots, then there is a sequence of the Reidemeister moves taking A^, to K2 and preserving the orientations of the knots. Note that the last two Reidemeister moves preserve two properties of the knots projection. (1) U (2) \_/ (3) / / \ Figure 1. The Reidemeister moves The first is the winding number, the second is the framing associated to the knot's projection. Upon defining these concepts we will show that given two projections of an oriented knot having the same winding numbers and associated framings then only the last two Reidemeister moves are required in passing from either of the projections to the other. Received by the editors January 24, 1983. Presented to the AMS at the Norman, Oklahoma, meeting, March 1983. 1980 Mathematics Subject Classification. Primary 57C99, 57065. 'Research supported in part by NSF Grant MCS-810-3387. ©1983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page 722 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE REIDEMEISTER MOVES OF A CLASSICAL KNOT 723 Suppose /: S ' -» R3 is a smooth embedding.