The Three-Variable Bracket Polynomial for Reduced, Alternating Links
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Ribbon Concordances and Doubly Slice Knots
Ribbon concordances and doubly slice knots Ruppik, Benjamin Matthias Advisors: Arunima Ray & Peter Teichner The knot concordance group Glossary Definition 1. A knot K ⊂ S3 is called (topologically/smoothly) slice if it arises as the equatorial slice of a (locally flat/smooth) embedding of a 2-sphere in S4. – Abelian invariants: Algebraic invari- Proposition. Under the equivalence relation “K ∼ J iff K#(−J) is slice”, the commutative monoid ants extracted from the Alexander module Z 3 −1 A (K) = H1( \ K; [t, t ]) (i.e. from the n isotopy classes of o connected sum S Z K := 1 3 , # := oriented knots S ,→S of knots homology of the universal abelian cover of the becomes a group C := K/ ∼, the knot concordance group. knot complement). The neutral element is given by the (equivalence class) of the unknot, and the inverse of [K] is [rK], i.e. the – Blanchfield linking form: Sesquilin- mirror image of K with opposite orientation. ear form on the Alexander module ( ) sm top Bl: AZ(K) × AZ(K) → Q Z , the definition Remark. We should be careful and distinguish the smooth C and topologically locally flat C version, Z[Z] because there is a huge difference between those: Ker(Csm → Ctop) is infinitely generated! uses that AZ(K) is a Z[Z]-torsion module. – Levine-Tristram-signature: For ω ∈ S1, Some classical structure results: take the signature of the Hermitian matrix sm top ∼ ∞ ∞ ∞ T – There are surjective homomorphisms C → C → AC, where AC = Z ⊕ Z2 ⊕ Z4 is Levine’s algebraic (1 − ω)V + (1 − ω)V , where V is a Seifert concordance group. -
THE JONES SLOPES of a KNOT Contents 1. Introduction 1 1.1. The
THE JONES SLOPES OF A KNOT STAVROS GAROUFALIDIS Abstract. The paper introduces the Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, the knots with at most 9 crossings, the torus knots and the (−2, 3,n) pretzel knots. Contents 1. Introduction 1 1.1. The degree of the Jones polynomial and incompressible surfaces 1 1.2. The degree of the colored Jones function is a quadratic quasi-polynomial 3 1.3. q-holonomic functions and quadratic quasi-polynomials 3 1.4. The Jones slopes and the Jones period of a knot 4 1.5. The symmetrized Jones slopes and the signature of a knot 5 1.6. Plan of the proof 7 2. Future directions 7 3. The Jones slopes and the Jones period of an alternating knot 8 4. Computing the Jones slopes and the Jones period of a knot 10 4.1. Some lemmas on quasi-polynomials 10 4.2. Computing the colored Jones function of a knot 11 4.3. Guessing the colored Jones function of a knot 11 4.4. A summary of non-alternating knots 12 4.5. The 8-crossing non-alternating knots 13 4.6. -
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. -
Arxiv:2008.02889V1 [Math.QA]
NONCOMMUTATIVE NETWORKS ON A CYLINDER S. ARTHAMONOV, N. OVENHOUSE, AND M. SHAPIRO Abstract. In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder Σ, which gives rise to the quasi Poisson bracket of G.Massuyeau and V.Turaev on the group algebra kπ1(Σ,p) of the fundamental group of a surface based at p ∈ ∂Σ. This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space C♮, which is a k-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative r-matrix formalism. This gives a more conceptual proof of the result of [Ove20] that traces of powers of Lax operator form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on C♮. 1. Introduction The current manuscript is obtained as a continuation of papers [Ove20, FK09, DF15, BR11] where the authors develop noncommutative generalizations of discrete completely integrable dynamical systems and [BR18] where a large class of noncommutative cluster algebras was constructed. Cluster algebras were introduced in [FZ02] by S.Fomin and A.Zelevisnky in an effort to describe the (dual) canonical basis of universal enveloping algebra U(b), where b is a Borel subalgebra of a simple complex Lie algebra g. Cluster algebras are commutative rings of a special type, equipped with a distinguished set of generators (cluster variables) subdivided into overlapping subsets (clusters) of the same cardinality subject to certain polynomial relations (cluster transformations). -
Using March Madness in the First Linear Algebra Course
Using March Madness in the first Linear Algebra course Steve Hilbert Ithaca College [email protected] Background • National meetings • Tim Chartier • 1 hr talk and special session on rankings • Try something new Why use this application? • This is an example that many students are aware of and some are interested in. • Interests a different subgroup of the class than usual applications • Interests other students (the class can talk about this with their non math friends) • A problem that students have “intuition” about that can be translated into Mathematical ideas • Outside grading system and enforcer of deadlines (Brackets “lock” at set time.) How it fits into Linear Algebra • Lots of “examples” of ranking in linear algebra texts but not many are realistic to students. • This was a good way to introduce and work with matrix algebra. • Using matrix algebra you can easily scale up to work with relatively large systems. Filling out your bracket • You have to pick a winner for each game • You can do this any way you want • Some people use their “ knowledge” • I know Duke is better than Florida, or Syracuse lost a lot of games at the end of the season so they will probably lose early in the tournament • Some people pick their favorite schools, others like the mascots, the uniforms, the team tattoos… Why rank teams? • If two teams are going to play a game ,the team with the higher rank (#1 is higher than #2) should win. • If there are a limited number of openings in a tournament, teams with higher rankings should be chosen over teams with lower rankings. -
On the Number of Unknot Diagrams Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar
On the number of unknot diagrams Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar To cite this version: Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar. On the number of unknot diagrams. SIAM Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2019, 33 (1), pp.306-326. 10.1137/17M115462X. hal-02049077 HAL Id: hal-02049077 https://hal.archives-ouvertes.fr/hal-02049077 Submitted on 26 Feb 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the number of unknot diagrams Carolina Medina1, Jorge L. Ramírez-Alfonsín2,3, and Gelasio Salazar1,3 1Instituto de Física, UASLP. San Luis Potosí, Mexico, 78000. 2Institut Montpelliérain Alexander Grothendieck, Université de Montpellier. Place Eugèene Bataillon, 34095 Montpellier, France. 3Unité Mixte Internationale CNRS-CONACYT-UNAM “Laboratoire Solomon Lefschetz”. Cuernavaca, Mexico. October 17, 2017 Abstract Let D be a knot diagram, and let D denote the set of diagrams that can be obtained from D by crossing exchanges. If D has n crossings, then D consists of 2n diagrams. A folklore argument shows that at least one of these 2n diagrams is unknot, from which it follows that every diagram has finite unknotting number. -
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. -
Arxiv:1501.00726V2 [Math.GT] 19 Sep 2016 3 2 for Each N, Ln Is Assembled from a Tangle S in B , N Copies of a Tangle T in S × I, and the Mirror Image S of S
HIDDEN SYMMETRIES VIA HIDDEN EXTENSIONS ERIC CHESEBRO AND JASON DEBLOIS Abstract. This paper introduces a new approach to finding knots and links with hidden symmetries using \hidden extensions", a class of hidden symme- tries defined here. We exhibit a family of tangle complements in the ball whose boundaries have symmetries with hidden extensions, then we further extend these to hidden symmetries of some hyperbolic link complements. A hidden symmetry of a manifold M is a homeomorphism of finite-degree covers of M that does not descend to an automorphism of M. By deep work of Mar- gulis, hidden symmetries characterize the arithmetic manifolds among all locally symmetric ones: a locally symmetric manifold is arithmetic if and only if it has infinitely many \non-equivalent" hidden symmetries (see [13, Ch. 6]; cf. [9]). Among hyperbolic knot complements in S3 only that of the figure-eight is arith- metic [10], and the only other knot complements known to possess hidden sym- metries are the two \dodecahedral knots" constructed by Aitchison{Rubinstein [1]. Whether there exist others has been an open question for over two decades [9, Question 1]. Its answer has important consequences for commensurability classes of knot complements, see [11] and [2]. The partial answers that we know are all negative. Aside from the figure-eight, there are no knots with hidden symmetries with at most fifteen crossings [6] and no two-bridge knots with hidden symmetries [11]. Macasieb{Mattman showed that no hyperbolic (−2; 3; n) pretzel knot, n 2 Z, has hidden symmetries [8]. Hoffman showed the dodecahedral knots are commensurable with no others [7]. -
A Remarkable 20-Crossing Tangle Shalom Eliahou, Jean Fromentin
A remarkable 20-crossing tangle Shalom Eliahou, Jean Fromentin To cite this version: Shalom Eliahou, Jean Fromentin. A remarkable 20-crossing tangle. 2016. hal-01382778v2 HAL Id: hal-01382778 https://hal.archives-ouvertes.fr/hal-01382778v2 Preprint submitted on 16 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A REMARKABLE 20-CROSSING TANGLE SHALOM ELIAHOU AND JEAN FROMENTIN Abstract. For any positive integer r, we exhibit a nontrivial knot Kr with r− r (20·2 1 +1) crossings whose Jones polynomial V (Kr) is equal to 1 modulo 2 . Our construction rests on a certain 20-crossing tangle T20 which is undetectable by the Kauffman bracket polynomial pair mod 2. 1. Introduction In [6], M. B. Thistlethwaite gave two 2–component links and one 3–component link which are nontrivial and yet have the same Jones polynomial as the corre- sponding unlink U 2 and U 3, respectively. These were the first known examples of nontrivial links undetectable by the Jones polynomial. Shortly thereafter, it was shown in [2] that, for any integer k ≥ 2, there exist infinitely many nontrivial k–component links whose Jones polynomial is equal to that of the k–component unlink U k. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
The Notion Of" Unimaginable Numbers" in Computational Number Theory
Beyond Knuth’s notation for “Unimaginable Numbers” within computational number theory Antonino Leonardis1 - Gianfranco d’Atri2 - Fabio Caldarola3 1 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] 2 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy 3 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] Abstract Literature considers under the name unimaginable numbers any positive in- teger going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathemati- cal definition (it is indeed used in many sources without a proper definition). This simply means that research in this topic must always consider shortened representations, usually involving recursion, to even being able to describe such numbers. One of the most known methodologies to conceive such numbers is using hyper-operations, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. arXiv:1901.05372v2 [cs.LO] 12 Mar 2019 The most important notations to represent such hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work’s introduction. Within this work we will give an axiomatic setup for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorith- mic nature of representations and the increased computation capabilities of 1 computers give the perfect field to develop further the topic, exploring some possibilities to effectively operate with such big numbers. -
MUTATION of KNOTS Figure 1 Figure 2
proceedings of the american mathematical society Volume 105. Number 1, January 1989 MUTATION OF KNOTS C. KEARTON (Communicated by Haynes R. Miller) Abstract. In general, mutation does not preserve the Alexander module or the concordance class of a knot. For a discussion of mutation of classical links, and the invariants which it is known to preserve, the reader is referred to [LM, APR, MT]. Suffice it here to say that mutation of knots preserves the polynomials of Alexander, Jones, and Homfly, and also the signature. Mutation of an oriented link k can be described as follows. Take a diagram of k and a tangle T with two outputs and two inputs, as in Figure 1. Figure 1 Figure 2 Rotate the tangle about the east-west axis to obtain Figure 2, or about the north-south axis to obtain Figure 3, or about the axis perpendicular to the paper to obtain Figure 4. Keep or reverse all the orientations of T as dictated by the rest of k . Each of the links so obtained is a mutant of k. The reverse k' of a link k is obtained by reversing the orientation of each component of k. Let us adopt the convention that a knot is a link of one component, and that k + I denotes the connected sum of two knots k and /. Lemma. For any knot k, the knot k + k' is a mutant of k + k. Received by the editors September 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. ©1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page 206 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MUTATION OF KNOTS 207 Figure 3 Figure 4 Proof.