DOCSLIB.ORG

Knot Polynomials the Alexander-Conway and the Jones Polynomial

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Knot Polynomials the Alexander-Conway and the Jones Polynomial Knot Polynomials The Alexander-Conway and the Jones polynomial Adrian Dawid May 27, 2020 Institut f¨urMathematik, Humboldt Universit¨at zu Berlin Table of Contents 1. What are Knot Polynomials? 2. The Alexander-Conway Polynomial 3. The Jones Polynomial 1 What are Knot Polynomials? • polynomials are very easy to distinguish • form a ring • \intuitively" have more information than numbers or true-false properties What are Knot Polynomials? Idea: Associate a polynomial with a knot Advantages: 2 • form a ring • \intuitively" have more information than numbers or true-false properties What are Knot Polynomials? Idea: Associate a polynomial with a knot Advantages: • polynomials are very easy to distinguish 2 • \intuitively" have more information than numbers or true-false properties What are Knot Polynomials? Idea: Associate a polynomial with a knot Advantages: • polynomials are very easy to distinguish • form a ring 2 What are Knot Polynomials? Idea: Associate a polynomial with a knot Advantages: • polynomials are very easy to distinguish • form a ring • \intuitively" have more information than numbers or true-false properties 2 • only depends on knot type • easy to compute from diagram • gives us more information than current invariants Goals Must-have properties for a knot polynomial: 3 • easy to compute from diagram • gives us more information than current invariants Goals Must-have properties for a knot polynomial: • only depends on knot type 3 • gives us more information than current invariants Goals Must-have properties for a knot polynomial: • only depends on knot type • easy to compute from diagram 3 Goals Must-have properties for a knot polynomial: • only depends on knot type • easy to compute from diagram • gives us more information than current invariants 3 The Alexander-Conway Polynomial The Alexander-Conway Polynomial Definition A polynomial rL 2 Z[z] is called the Alexander-Conway polynomial of an oriented link L if the following conditions are satisfied: 1. rK = 1 if K is the unknot 2. (Skein-Relation): For all L+; L−; L0 as defined on the next slide: rL+ − rL− = zrL0 4 The Skein Relation In an arbitrary oriented link diagram replace any crossing locally with: (a) L+ (b) L− (c) L0 Skein-Relation: rL+ − rL− = zrL0 5 The Alexander-Conway Polynomial Theorem Let L be the (diagram) of a unlink with n ≥ 2 components, then rL = 0. 6 The Alexander-Conway Polynomial Proof. (Case n=2) View L as L0 in the skein relation. Then L+ and L− are diagrams of the unknot. =) rL− = rL+ = 1. (a) Example of L+ or L− (b) Example of L Thus we have rL+ − rL− = 0 = zrL0 =) rL0 = rL = 0: 7 The Alexander-Conway Polynomial Proof. (Case n ≥ 3) View L as L0 in the skein relation. Then L+ and L− are diagrams of unlinks with n − 1 components. Thus we get rL = 0 via the skein relation and induction from the case n = 2. (b) Example of L (a) Example of L+ or L− 8 The Alexander-Conway Polynomial Lemma Any knot diagram K can be transformed into an unknot with a finite number of crossing changes. Proof. Procedure: Go through the knot and see any crossing as an overcrossing on the first and as an undercrossing the second time. −−−−! 9 The Alexander-Conway Polynomial Lemma Let L be a link so that every component is the unknot then any diagram can be transformed into the unlink with a finite number of crossing changes. Proof. Pick any ordering of the components. Then in any crossing see the higher ranked strand as the \over" strand. −−−−! Figure 5: Example of unlinking. Order:1,2,3. 10 The Alexander-Conway Polynomial Theorem For all oriented links L the Alexander-Conway Polynomial rL is uniquely defined by the skein relation. 11 The Alexander-Conway Polynomial Theorem For all oriented links L the Alexander-Conway Polynomial rL is uniquely defined by the skein relation. Note: We will assume without proof that rL is invariant under Reidemeister moves, i.e. that it is independent of the choice of diagram. 11 The Alexander-Conway Polynomial Proof. By the previous lemma 9 links L2; :::; Lk s.t. Lk is an unlink and L −−−−! L2 −−−−! L3 −−−−! ::: −−−−! Lk are related by single crossing changes. Then rLk = 0. And 8i = 1; :::; k the skein relation implies r − r = ±zr 0 : Li Li+1 Li 0 Where Li has one crossing less than Li . Because two or less crossings imply the unkot, the claim follows by induction. 12 Example: The Trefoil Knot Let's compute the Alexander-Conway Polynomial of the trefoil knot: Figure 6: The well-known trefoil knot. 13 0 = T+ (b) T− (c) T0 = T+ (e) T 0 0 0 (d) T− Example: The Trefoil Knot Let's compute the Alexander-Conway Polynomial of the trefoil knot: (a) T 14 Example: The Trefoil Knot Let's compute the Alexander-Conway Polynomial of the trefoil knot: 0 (a) T = T+ (b) T− (c) T0 = T+ (e) T 0 0 0 (d) T− 14 The Trefoil Knot By the skein relation we have r − r = zr = zr 0 : T+ T− T0 T+ |{z} =1 Also r 0 − r 0 = z r : T+ T− T0 |{z} |{z} =0 =1 We then have in total 2 rT = zrT0 + 1 = z(z + 0) + 1 = 1 + z as the Alexander-Conway polynomial of the trefoil. 15 The Hopf Links We can also compute the Alexander-Conway polynomial of a link. (c) H0 (a) H = H+ (b) H− Here we have rH − rH− = zrH0 = z: |{z} =0 16 The Hopf Links Now we change the orientations a bit: (c) H0 (a) H = H− (b) H+ Suddenly we have rH = −zrH0 + rH+ = −z: |{z} =0 17 The Alexander-Conway Polynomial • The Alexander-Conway polynomial can distinguish links with different orientations • Can distingish many knots (e.g. trefoil and unknot) Interesting Question: Can it detect the unknot? 18 The Conway-Knot Sadly, it cannot: Figure 10: The Conway knot has trivial Alexander-Conway polynomial but is not the unknot. 19 But not all hope is lost: Maybe we can define a more powerful knot polynomial. 20 The Jones Polynomial The Bracket-Polynomial The definition of the Jones polynomial given by Jones cannot be stated using our methods. But Kauffman gave an easier definition we can use. Definition The Kauffman bracket h·i is a mapping from the unoriented link −1 diagrams into Z[A ; A] given by the following three axioms: 1. h i = 1 2. h i = Ah i + A−1h i 3. h [ Li = (−A2 − A−2)hLi The second axiom should be interpreted as a local change to a single crossing. 21 The Kauffman Bracket Note: The Kauffman bracket of any link diagram is defined by these axioms because by resolving all crossing we always get the diagram of an unlink and h i = Ah i + A−1h i can then be applied recursively. 22 The Kauffman Bracket Note: The Kauffman bracket of any link diagram is defined by these axioms because by resolving all crossing we always get the diagram of an unlink and h i = Ah i + A−1h i can then be applied recursively. But: We don't know if the Kauffman bracket depends on the choice of link diagram. 22 The Kauffman Bracket And it actually does: Theorem The Kauffman bracket changes under a Ω1-move the following way: h i = −A−3h i h i = −A3h i 23 The Kauffman Bracket Proof. By definition: h i = Ah i + A−1h i It follows: h i = Ah i + A−1h i = (A + (−A2 − A−2)A−1)h i = −A−3h i: 24 The Kauffman Bracket However it looks better for Ω3; Ω2: Theorem The Kauffman bracket does not change under Ω2 and Ω3 moves. 25 The Kauffman Bracket Proof. (For Ω2) By definition: h i = Ah i + A−1h i = −A−2h i + h i + A−2h i = h i 26 The Kauffman Bracket Proof. (For Ω3) By definition: h i = Ah i + A−1h i = Ah i + A−1h i = h i 27 • Goal: Define a correction term for Ω1-move. The Kauffman Bracket • Kauffman bracket is invariant under Ω2; Ω3. =) There is hope. • Change under Ω1 is known exactly. 28 The Kauffman Bracket • Kauffman bracket is invariant under Ω2; Ω3. =) There is hope. • Change under Ω1 is known exactly. • Goal: Define a correction term for Ω1-move. 28 The Writhe Definition Let L be an oriented link diagram then the writhe w(L) is defined as follows: X w(L) = "i i where for the ith crossing "i is (a) " = +1 (b) " = −1 29 The Writhe Lemma Let L be an oriented link diagram then w(L) = w(−L). Proof. (a) " = +1 (b) " = −1 30 The Writhe Lemma The writhe behaves under Reidemeister moves as follows: 1. w( ) = w( ) − 1 2. w( ) = w( ) 3. w( ) = w( ) 31 The Kaufmann Polynomial Using the writhe we can now define a real link invariant. Definition Let L be an oriented link diagram then we define χ(L) = (−A)−3w(L)hLi as the Kaufmann polynomial of L. 32 The Kaufmann Polynomial Theorem The Kaufmann polynomial χ(L) only depends on the link type. 33 The Kaufmann Polynomial Proof. We have Ω2; Ω3-invariance by definition. We also have −3w( ) χ( ) = (−A) h i = (−A)−3w( )+3h i = (−A)−3w( )+3(−A)−3h i = (−A)−3w( )h i Thus the Kauffman polynomial is invariant under Reidemeister moves and thus a link invariant. 34 The Kaufmann Polynomial Definition Let L be an oriented link and DL any diagram of it. Then we call χ(L) = χ(DL) the Kaufmann polynomial of the link L.
Load more

Recommended publications
  • Complete Invariant Graphs of Alternating Knots
    Complete invariant graphs of alternating knots Christian Soulié First submission: April 2004 (revision 1) Abstract : Chord diagrams and related enlacement graphs of alternating knots are enhanced to obtain complete invariant graphs including chirality detection. Moreover, the equivalence by common enlacement graph is specified and the neighborhood graph is defined for general purpose and for special application to the knots. I - Introduction : Chord diagrams are enhanced to integrate the state sum of all flype moves and then produce an invariant graph for alternating knots. By adding local writhe attribute to these graphs, chiral types of knots are distinguished. The resulting chord-weighted graph is a complete invariant of alternating knots. Condensed chord diagrams and condensed enlacement graphs are introduced and a new type of graph of general purpose is defined : the neighborhood graph. The enlacement graph is enriched by local writhe and chord orientation. Hence this enhanced graph distinguishes mutant alternating knots. As invariant by flype it is also invariant for all alternating knots. The equivalence class of knots with the same enlacement graph is fully specified and extended mutation with flype of tangles is defined. On this way, two enhanced graphs are proposed as complete invariants of alternating knots. I - Introduction II - Definitions and condensed graphs II-1 Knots II-2 Sign of crossing points II-3 Chord diagrams II-4 Enlacement graphs II-5 Condensed graphs III - Realizability and construction III - 1 Realizability
    [Show full text]
  • Oriented Pair (S 3,S1); Two Knots Are Regarded As
    S-EQUIVALENCE OF KNOTS C. KEARTON Abstract. S-equivalence of classical knots is investigated, as well as its rela- tionship with mutation and the unknotting number. Furthermore, we identify the kernel of Bredon’s double suspension map, and give a geometric relation between slice and algebraically slice knots. Finally, we show that every knot is S-equivalent to a prime knot. 1. Introduction An oriented knot k is a smooth (or PL) oriented pair S3,S1; two knots are regarded as the same if there is an orientation preserving diffeomorphism sending one onto the other. An unoriented knot k is defined in the same way, but without regard to the orientation of S1. Every oriented knot is spanned by an oriented surface, a Seifert surface, and this gives rise to a matrix of linking numbers called a Seifert matrix. Any two Seifert matrices of the same knot are S-equivalent: the definition of S-equivalence is given in, for example, [14, 21, 11]. It is the equivalence relation generated by ambient surgery on a Seifert surface of the knot. In [19], two oriented knots are defined to be S-equivalent if their Seifert matrices are S- equivalent, and the following result is proved. Theorem 1. Two oriented knots are S-equivalent if and only if they are related by a sequence of doubled-delta moves shown in Figure 1. .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .. .... .... .... .... .... .... .... .... .... ...
    [Show full text]
  • Mutant Knots and Intersection Graphs 1 Introduction
    Mutant knots and intersection graphs S. V. CHMUTOV S. K. LANDO We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. Conversely, if we have a weight system depending only on the intersection graphs of chord diagrams, then the composition of such a weight system with the Kontsevich invariant determines a knot invariant that does not distinguish mutant knots. Thus, an equivalence between finite order invariants not distinguishing mutants and weight systems depending on intersections graphs only is established. We discuss relationship between our results and certain Lie algebra weight systems. 57M15; 57M25 1 Introduction Below, we use standard notions of the theory of finite order, or Vassiliev, invariants of knots in 3-space; their definitions can be found, for example, in [6] or [14], and we recall them briefly in Section 2. All knots are assumed to be oriented. Two knots are said to be mutant if they differ by a rotation of a tangle with four endpoints about either a vertical axis, or a horizontal axis, or an axis perpendicular to the paper. If necessary, the orientation inside the tangle may be replaced by the opposite one. Here is a famous example of mutant knots, the Conway (11n34) knot C of genus 3, and Kinoshita–Terasaka (11n42) knot KT of genus 2 (see [1]). C = KT = Note that the change of the orientation of a knot can be achieved by a mutation in the complement to a trivial tangle.
    [Show full text]
  • A Symmetry Motivated Link Table
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 August 2018 doi:10.20944/preprints201808.0265.v1 Peer-reviewed version available at Symmetry 2018, 10, 604; doi:10.3390/sym10110604 Article A Symmetry Motivated Link Table Shawn Witte1, Michelle Flanner2 and Mariel Vazquez1,2 1 UC Davis Mathematics 2 UC Davis Microbiology and Molecular Genetics * Correspondence: [email protected] Abstract: Proper identification of oriented knots and 2-component links requires a precise link 1 nomenclature. Motivated by questions arising in DNA topology, this study aims to produce a 2 nomenclature unambiguous with respect to link symmetries. For knots, this involves distinguishing 3 a knot type from its mirror image. In the case of 2-component links, there are up to sixteen possible 4 symmetry types for each topology. The study revisits the methods previously used to disambiguate 5 chiral knots and extends them to oriented 2-component links with up to nine crossings. Monte Carlo 6 simulations are used to report on writhe, a geometric indicator of chirality. There are ninety-two 7 prime 2-component links with up to nine crossings. Guided by geometrical data, linking number and 8 the symmetry groups of 2-component links, a canonical link diagram for each link type is proposed. 9 2 2 2 2 2 2 All diagrams but six were unambiguously chosen (815, 95, 934, 935, 939, and 941). We include complete 10 tables for prime knots with up to ten crossings and prime links with up to nine crossings. We also 11 prove a result on the behavior of the writhe under local lattice moves.
    [Show full text]
  • 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.
    [Show full text]
  • MUTATIONS of LINKS in GENUS 2 HANDLEBODIES 1. Introduction
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 1, January 1999, Pages 309{314 S 0002-9939(99)04871-6 MUTATIONS OF LINKS IN GENUS 2 HANDLEBODIES D. COOPER AND W. B. R. LICKORISH (Communicated by Ronald A. Fintushel) Abstract. A short proof is given to show that a link in the 3-sphere and any link related to it by genus 2 mutation have the same Alexander polynomial. This verifies a deduction from the solution to the Melvin-Morton conjecture. The proof here extends to show that the link signatures are likewise the same and that these results extend to links in a homology 3-sphere. 1. Introduction Suppose L is an oriented link in a genus 2 handlebody H that is contained, in some arbitrary (complicated) way, in S3.Letρbe the involution of H depicted abstractly in Figure 1 as a π-rotation about the axis shown. The pair of links L and ρL is said to be related by a genus 2 mutation. The first purpose of this note is to prove, by means of long established techniques of classical knot theory, that L and ρL always have the same Alexander polynomial. As described briefly below, this actual result for knots can also be deduced from the recent solution to a conjecture, of P. M. Melvin and H. R. Morton, that posed a problem in the realm of Vassiliev invariants. It is impressive that this simple result, readily expressible in the language of the classical knot theory that predates the Jones polynomial, should have emerged from the technicalities of Vassiliev invariants.
    [Show full text]
  • 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.
    [Show full text]
  • The Jones Polynomial Through Linear Algebra
    The Jones polynomial through linear algebra Iain Moffatt University of South Alabama Workshop in Knot Theory Waterloo, 24th September 2011 I. Moffatt (South Alabama) UW 2011 1 / 39 What and why What we’ll see The construction of link invariants through R-matrices. (c.f. Reshetikhin-Turaev invariants, quantum invariants) Why this? Can do some serious math using material from Linear Algebra 1. Illustrates how math works in the wild: start with a problem you want to solve; figure out an easier problem that you can solve; build up from this to solve your original problem. See the interplay between algebra, combinatorics and topology! It’s my favourite bit of math! I. Moffatt (South Alabama) UW 2011 2 / 39 What we’re trying to do A knot is a circle, S1, sitting in 3-space R3. A link is a number of disjoint circles in 3-space R3. Knots and links are considered up to isotopy. This means you can “move then round in space, but you can’t cut or glue them”. I. Moffatt (South Alabama) UW 2011 3 / 39 What we’re trying to do A knot is a circle, S1, sitting in 3-space R3. A link is a number of disjoint circles in 3-space R3. Knots and links are considered up to isotopy. This means you can “move then round in space, but you can’t cut or glue them”. The fundamental problem in knot theory is to determine whether or not two links are isotopic. =? =? I. Moffatt (South Alabama) UW 2011 3 / 39 To do this we need knot invariants: F : Links (a set) such (Isotopy) ! that F(L) = F(L0) = L = L0, Aim: construct6 link invariants) 6 using linear algebra.
    [Show full text]
  • Knot Theory and the Alexander Polynomial
    Knot Theory and the Alexander Polynomial Reagin Taylor McNeill Submitted to the Department of Mathematics of Smith College in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Elizabeth Denne, Faculty Advisor April 15, 2008 i Acknowledgments First and foremost I would like to thank Elizabeth Denne for her guidance through this project. Her endless help and high expectations brought this project to where it stands. I would Like to thank David Cohen for his support thoughout this project and through- out my mathematical career. His humor, skepticism and advice is surely worth the $.25 fee. I would also like to thank my professors, peers, housemates, and friends, particularly Kelsey Hattam and Katy Gerecht, for supporting me throughout the year, and especially for tolerating my temporary insanity during the final weeks of writing. Contents 1 Introduction 1 2 Defining Knots and Links 3 2.1 KnotDiagramsandKnotEquivalence . ... 3 2.2 Links, Orientation, and Connected Sum . ..... 8 3 Seifert Surfaces and Knot Genus 12 3.1 SeifertSurfaces ................................. 12 3.2 Surgery ...................................... 14 3.3 Knot Genus and Factorization . 16 3.4 Linkingnumber.................................. 17 3.5 Homology ..................................... 19 3.6 TheSeifertMatrix ................................ 21 3.7 TheAlexanderPolynomial. 27 4 Resolving Trees 31 4.1 Resolving Trees and the Conway Polynomial . ..... 31 4.2 TheAlexanderPolynomial. 34 5 Algebraic and Topological Tools 36 5.1 FreeGroupsandQuotients . 36 5.2 TheFundamentalGroup. .. .. .. .. .. .. .. .. 40 ii iii 6 Knot Groups 49 6.1 TwoPresentations ................................ 49 6.2 The Fundamental Group of the Knot Complement . 54 7 The Fox Calculus and Alexander Ideals 59 7.1 TheFreeCalculus ...............................
    [Show full text]
  • An Odd Analog of Plamenevskaya's Invariant of Transverse
    AN ODD ANALOG OF PLAMENEVSKAYA'S INVARIANT OF TRANSVERSE KNOTS by GABRIEL MONTES DE OCA A DISSERTATION Presented to the Department of Mathematics and the Graduate School of the University of Oregon in partial fulfillment of the requirements for the degree of Doctor of Philosophy September 2020 DISSERTATION APPROVAL PAGE Student: Gabriel Montes de Oca Title: An Odd Analog of Plamenevskaya's Invariant of Transverse Knots This dissertation has been accepted and approved in partial fulfillment of the requirements for the Doctor of Philosophy degree in the Department of Mathematics by: Robert Lipshitz Chair Daniel Dugger Core Member Nicholas Proudfoot Core Member Micah Warren Core Member James Brau Institutional Representative and Kate Mondloch Interim Vice Provost and Dean of the Graduate School Original approval signatures are on file with the University of Oregon Graduate School. Degree awarded September 2020 ii c 2020 Gabriel Montes de Oca iii DISSERTATION ABSTRACT Gabriel Montes de Oca Doctor of Philosophy Department of Mathematics September 2020 Title: An Odd Analog of Plamenevskaya's Invariant of Transverse Knots Plamenevskaya defined an invariant of transverse links as a distinguished class in the even Khovanov homology of a link. We define an analog of Plamenevskaya's invariant in the odd Khovanov homology of Ozsv´ath,Rasmussen, and Szab´o. We show that the analog is also an invariant of transverse links and has similar properties to Plamenevskaya's invariant. We also show that the analog invariant can be identified with an equivalent invariant in the reduced odd Khovanov homology. We demonstrate computations of the invariant on various transverse knot pairs with the same topological knot type and self-linking number.
    [Show full text]
  • Bracket Calculations -- Pdf Download
    Knot Theory Notes - Detecting Links With The Jones Polynonmial by Louis H. Kau®man 1 Some Elementary Calculations The formula for the bracket model of the Jones polynomial can be indicated as follows: The letter chi, Â, denotes a crossing in a link diagram. The barred letter denotes the mirror image of this ¯rst crossing. A crossing in a diagram for the knot or link is expanded into two possible states by either smoothing (reconnecting) the crossing horizontally, , or 2 2 vertically ><. Any closed loop (without crossings) in the plane has value ± = A ³A¡ : ¡ ¡ 1  = A + A¡ >< ³ 1  = A¡ + A >< : ³ One useful consequence of these formulas is the following switching formula 1 2 2 A A¡  = (A A¡ ) : ¡ ¡ ³ Note that in these conventions the A-smoothing of  is ; while the A-smoothing of  is >< : Properly interpreted, the switching formula abo³ve says that you can switch a crossing and smooth it either way and obtain a three diagram relation. This is useful since some computations will simplify quite quickly with the proper choices of switching and smoothing. Remember that it is necessary to keep track of the diagrams up to regular isotopy (the equivalence relation generated by the second and third Reidemeister moves). Here is an example. View Figure 1. K U U' Figure 1 { Trefoil and Two Relatives You see in Figure 1, a trefoil diagram K, an unknot diagram U and another unknot diagram U 0: Applying the switching formula, we have 1 2 2 A¡ K AU = (A¡ A )U 0 ¡ ¡ 3 3 2 6 and U = A and U 0 = ( A¡ ) = A¡ : Thus ¡ ¡ 1 3 2 2 6 A¡ K A( A ) = (A¡ A )A¡ : ¡ ¡ ¡ Hence 1 4 8 4 A¡ K = A + A¡ A¡ : ¡ ¡ Thus 5 3 7 K = A A¡ + A¡ : ¡ ¡ This is the bracket polynomial of the trefoil diagram K: We have used to same symbol for the diagram and for its polynomial.
    [Show full text]
  • On Cosmetic Surgery on Knots K
    Cosmetic surgery on knots On cosmetic surgery on knots K. Ichihara Introduction 3-manifold Dehn surgery Cosmetic Surgery Kazuhiro Ichihara Conjecture Known Facts Examples Nihon University Criteria College of Humanities and Sciences Results (1) Jones polynomial 2-bridge knot Hanselman's result Finite type invariants Based on Joint work with Recent progress Tetsuya Ito (Kyoto Univ.), In Dae Jong (Kindai Univ.), Results (2) Genus one alternating knots Thomas Mattman (CSU, Chico), Toshio Saito (Joetsu Univ. of Edu.), Cosmetic Banding and Zhongtao Wu (CUHK) How To Check? The 67th Topology Symposium - Online, MSJ. Oct 13, 2020. 1 / 36 Cosmetic surgery Papers on knots I (with Toshio Saito) K. Ichihara Cosmetic surgery and the SL(2; C) Casson invariant for 2-bridge knots. Introduction 3-manifold Hiroshima Math. J. 48 (2018), no. 1, 21-37. Dehn surgery Cosmetic Surgery I (with In Dae Jong (Appendix by Hidetoshi Masai)) Conjecture Known Facts Cosmetic banding on knots and links. Examples Osaka J. Math. 55 (2018), no. 4, 731-745. Criteria Results (1) I (with Zhongtao Wu) Jones polynomial 2-bridge knot A note on Jones polynomial and cosmetic surgery. Hanselman's result Finite type invariants Comm. Anal. Geom. 27 (2019), no.5, 1087{1104. Recent progress I (with Toshio Saito and Tetsuya Ito) Results (2) Genus one alternating Chirally cosmetic surgeries and Casson invariants. knots Cosmetic Banding To appear in Tokyo J. Math. How To Check? I (with In Dae Jong, Thomas W. Mattman, Toshio Saito) Two-bridge knots admit no purely cosmetic surgeries. To appear in Algebr. Geom. Topol. 2 / 36 Table of contents Introduction Jones polynomial 3-manifold 2-bridge knot Dehn surgery Hanselman's result Finite type invariants Cosmetic Surgery Conjecture Recent progress Known Facts Results (2) [Chirally cosmetic surgery] Examples Alternating knot of genus one Criteria Cosmetic Banding Results (1) [Purely cosmetic surgery] How To Check? Cosmetic surgery Classification of 3-manifolds on knots K.
    [Show full text]