Brunnian local moves of knots and Vassiliev invariants

Akira Yasuhara

Displaying similar documents to “Brunnian local moves of knots and Vassiliev invariants”

Elementary moves for higher dimensional knots

Dennis Roseman (2004)

Fundamenta Mathematicae

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For smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in $ℝ^{n + 2}$ (or $^{n + 2}$ ), we generalize the notion of knot moves to higher dimensions. This reproves and generalizes the Reidemeister moves of classical knot theory. We show that for any dimension there is a finite set of elementary isotopies, called moves, so that any isotopy is equivalent to a finite sequence of these moves.

Legendrian and transverse twist knots

John B. Etnyre, Lenhard L. Ng, Vera Vértesi (2013)

Journal of the European Mathematical Society

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In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m (5_{2})$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston-Bennequin number of the twist knot $K_{- 2 n}$ with crossing number $2 n + 1$ . In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that $K_{- 2 n}$ has exactly $⌈\frac{n^{2}}{2}⌉$ Legendrian representatives with maximal Thurston–Bennequin...

On the Signatures of Torus Knots

Maciej Borodzik, Krzysztof Oleszkiewicz (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study properties of the signature function of the torus knot $T_{p, q}$ . First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.

Cocycle invariants of codimension 2 embeddings of manifolds

Józef H. Przytycki, Witold Rosicki (2014)

Banach Center Publications

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We consider the classical problem of a position of n-dimensional manifold Mⁿ in $ℝ^{n + 2}$ . We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting $M ⁿ \to ℝ^{n + 2}$ . In particular, we show that for any fixed quandle, quandle coloring, and shadow quandle coloring, of a diagram of Mⁿ embedded in $ℝ^{n + 2}$ we have (n+1)- and (n+2)-(co)cycle invariants (i.e. invariant under Roseman moves).

On malnormal peripheral subgroups of the fundamental group of a $3$ -manifold

Pierre de la Harpe, Claude Weber (2014)

Confluentes Mathematici

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Let $K$ be a non-trivial knot in the $3$ -sphere, $E_{K}$ its exterior, $G_{K} = π_{1} (E_{K})$ its group, and $P_{K} = π_{1} (\partial E_{K}) \subset G_{K}$ its peripheral subgroup. We show that $P_{K}$ is malnormal in $G_{K}$ , namely that $g P_{K} g^{- 1} \cap P_{K} = {e}$ for any $g \in G_{K}$ with $g \notin P_{K}$ , unless $K$ is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in $E_{K}$ attached to $T_{K}$ which are not boundary parallel (Theorem 1 and Corollary 2). More generally, we characterise malnormal peripheral subgroups in the fundamental...

Unconditionality of general Franklin systems in $L^{p} [0, 1]$ , 1 < p < ∞

Gegham G. Gevorkyan, Anna Kamont (2004)

Studia Mathematica

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By a general Franklin system corresponding to a dense sequence = (tₙ, n ≥ 0) of points in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots , that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is that each general Franklin system is an unconditional basis in $L^{p} [0, 1]$ , 1 < p < ∞.

On the Definition of $2$ -Category of $2$ -Knots

V. M. Kharlamov, V. G. Turaev (1993)

Recherche Coopérative sur Programme n°25

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Two cases when $d_{κ}$ and $d *_{κ}$ are equal

Ofer Shafir (2001)

Fundamenta Mathematicae

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We deal with two cardinal invariants and give conditions on their equality using Shelah's pcf theory.

Witten's Conjecture for many four-manifolds of simple type

Paul M. N. Feehan, Thomas G. Leness (2015)

Journal of the European Mathematical Society

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We prove that Witten’s Conjecture [40] on the relationship between the Donaldson and Seiberg-Witten series for a four-manifold of Seiberg-Witten simple type with $b_{1} = 0$ and odd $b_{2}^{+} \geq 3$ follows from our $(3)$ -monopole cobordism formula [6] when the four-manifold has $c_{1}^{2} \geq χ_{h} - 3$ or is abundant.

The Lebesgue constant for the periodic Franklin system

Markus Passenbrunner (2011)

Studia Mathematica

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We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_{j}$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n, ν}$ be the space of piecewise linear continuous functions on the torus with knots $t_{j} : 0 \leq j \leq N - 1$ . Finally, let $P_{n, ν}$ be the orthogonal projection operator from L²([0,1)) onto $V_{n, ν}$ . The main result is $l i m_{n \to \infty, ν = 1} | | P_{n, ν} : L^{\infty} \to L^{\infty} | | = s u p_{n \in ℕ, 0 \leq ν \leq n} | | P_{n, ν} : L^{\infty} \to L^{\infty} | | = 2 + (33 - 18 \sqrt 3) / 13$ . This shows in particular that the Lebesgue constant of the classical...

Generalised Weber functions

Andreas Enge, François Morain (2014)

Acta Arithmetica

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A generalised Weber function is given by $_{N} (z) = η (z / N) / η (z)$ , where η(z) is the Dedekind function and N is any integer; the original function corresponds to N=2. We classify the cases where some power $_{N}^{e}$ evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating $_{N} (z)$ and j(z). Our ultimate goal is the use of these invariants in...

Welschinger invariants of small non-toric Del Pezzo surfaces

Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin (2013)

Journal of the European Mathematical Society

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We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at $\sim q$ real and $s \leq 1$ pairs of conjugate imaginary points, where $q + 2 s \leq 5$ , and the real quadric blown up at $s \leq 1$ pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil’s recursive formula [22] for Gromov–Witten invariants of these surfaces and generalizes our recursive formula [12] for purely real Welschinger invariants...

Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure

Changxing Miao, Youbin Zhu (2006)

Applicationes Mathematicae

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We discuss invariants and conservation laws for a nonlinear system of Klein-Gordon equations with Hamiltonian structure ⎧ $u_{t t} - Δ u + m ² u = - F ₁ (| u | ², | v | ²) u$ , ⎨ ⎩ $v_{t t} - Δ v + m ² v = - F ₂ (| u | ², | v | ²) v$ for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). Based on Morawetz-type identity, we prove that solutions to the above system decay to zero in local L²-norm, and local energy also decays to zero if the initial energy satisfies $E (u, v, ℝ ⁿ, 0) = 1 / 2 \int_{ℝ ⁿ} (| \nabla u (0) | ² + | u_{t} (0) | ² + m ² | u (0) | ² + | \nabla v (0) | ² + | v_{t} (0) | ² + m ² | v (0) | ² + F (| u (0) | ², | v (0) | ²)) d x < \infty$ , and F₁(|u|²,|v|²)|u|² + F₂(|u|²,|v|²)|v|² - F(|u|²,|v|²) ≥ aF(|u|²,|v|²) ≥ 0, a >...

The Lebesgue constants for the Franklin orthogonal system

Z. Ciesielski, A. Kamont (2004)

Studia Mathematica

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To each set of knots $t_{i} = i / 2 n$ for i = 0,...,2ν and $t_{i} = (i - ν) / n$ for i = 2ν + 1,..., n + ν, with 1 ≤ ν ≤ n, there corresponds the space $_{ν, n}$ of all piecewise linear and continuous functions on I = [0,1] with knots $t_{i}$ and the orthogonal projection $P_{ν, n}$ of L²(I) onto $_{ν, n}$ . The main result is $l i m_{(n - ν) \land ν \to \infty} | | P_{ν, n} | | ₁ = s u p_{ν, n : 1 \leq ν \leq n} | | P_{ν, n} | | ₁ = 2 + (2 - \sqrt 3) ²$ . This shows that the Lebesgue constant for the Franklin orthogonal system is 2 + (2-√3)².

Analytic invariants for the $1 : - 1$ resonance

José Pedro Gaivão (2013)

Annales de l’institut Fourier

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Associated to analytic Hamiltonian vector fields in $ℂ^{4}$ having an equilibrium point satisfying a non semisimple $1 : - 1$ resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.

On the bounding, splitting, and distributivity numbers

Alan S. Dow, Saharon Shelah (2023)

Commentationes Mathematicae Universitatis Carolinae

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The cardinal invariants $𝔥, 𝔟, 𝔰$ of $𝒫 (ω)$ are known to satisfy that $ω_{1} \leq 𝔥 \leq min {𝔟, 𝔰}$ . We prove that all inequalities can be strict. We also introduce a new upper bound for $𝔥$ and show that it can be less than $𝔰$ . The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).

Invariants for the modular cyclic group of prime order via classical invariant theory

David L. Wehlau (2013)

Journal of the European Mathematical Society

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Let $𝔽$ be any field of characteristic $p$ . It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_{1}, V_{2}, ..., V_{p}$ of $C_{p}$ defined over $𝔽$ . Thus if $V$ is any finite dimensional $C_{p}$ -representation there are non-negative integers $0 \leq n_{1}, n_{2}, ..., n_{k} \leq p - 1$ such that $V ≅ \oplus_{i = 1}^{k} V_{n_{i} + 1}$ . It is also well-known there is a unique (up to equivalence) $d + 1$ dimensional irreducible complex representation of $S L_{2} (ℂ)$ given by its action on the space $R_{d}$ of $d$ forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation...

$Q$ -curvature and spectral invariants

Branson, Thomas

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On isomorphic classification of tensor products $E_{\infty} (a) \otimes ̂ E_{\infty}^{'} (b)$

Goncharov A., Zahariuta V., Terzioğlu Tosun

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Abstract New linear topological invariants are introduced and utilized to give an isomorphic classification of tensor products of the type $E_{\infty} (a) \otimes ̂ E_{\infty}^{'} (b)$ , where $E_{\infty} (a)$ is a power series space of infinite type. These invariants are modifications of those suggested earlier by Zahariuta. In particular, some new results are obtained for spaces of infinitely differentiable functions with values in a locally convex space X. These spaces coincide, up to isomorphism, with spaces L(s’,X) of all continuous linear...

Every braid admits a short sigma-definite expression

Jean Fromentin (2011)

Journal of the European Mathematical Society

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A result by Dehornoy (1992) says that every nontrivial braid admits a $σ$ -definite expression, defined as a braid word in which the generator $σ_{i}$ with maximal index $i$ appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a $σ$ -definite word expression that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid...

Asymptotic Vassiliev invariants for vector fields

Sebastian Baader, Julien Marché (2012)

Bulletin de la Société Mathématique de France

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We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of $ℝ^{3}$ . More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field.

La Queste del Saint $G R_{a}$ (AL): a Computational Approach to Local Algebra

Teo Mora (1989)

Publications mathématiques et informatique de Rennes

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Labeled floor diagrams for plane curves

Sergey Fomin, Grigory Mikhalkin (2010)

Journal of the European Mathematical Society

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Floor diagrams are a class of weighted oriented graphs introduced by E. Brugallé and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) Gromov–Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In a number of cases, these descriptions can be used to obtain explicit (direct or recursive) formulas for the corresponding enumerative invariants. In particular, we use this approach to enumerate...

Isomorphisms of Cartesian Products of ℓ-Power Series Spaces

E. Karapınar, M. Yurdakul, V. Zahariuta (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let ℓ be a Banach sequence space with a monotone norm $∥ \cdot ∥_{ℓ}$ , in which the canonical system $(e_{i})$ is a normalized symmetric basis. We give a complete isomorphic classification of Cartesian products $E_{0}^{ℓ} (a) \times E_{\infty}^{ℓ} (b)$ where $E_{0}^{ℓ} (a) = K^{ℓ} (e x p (- p^{- 1} a_{i}))$ and $E_{\infty}^{ℓ} (b) = K^{ℓ} (e x p (p a_{i}))$ are finite and infinite ℓ-power series spaces, respectively. This classification is the generalization of the results by Chalov et al. [Studia Math. 137 (1999)] and Djakov et al. [Michigan Math. J. 43 (1996)] by using the method of compound linear topological invariants developed by...

Combinatorics of dense subsets of the rationals

B. Balcar, F. Hernández-Hernández, M. Hrušák (2004)

Fundamenta Mathematicae

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We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants $_{ℚ}$ , $_{ℚ}$ , $_{ℚ}$ , $_{ℚ}$ , $_{ℚ}$ , $_{ℚ}$ describing properties of Dense(ℚ). These invariants satisfy $_{ℚ}$ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ $. W e c o m p a r e t h e m w i t h t h e i r a n a l o g u e s i n t h e w e l l s t u d i e d B o o l e a n a l g e b r a (ω) / f i n . W e s h o w t h a t$ ℚ = p $,$ ℚ = t $a n d$ ℚ = i $, w h e r e a s$ ℚ > h $a n d$ ℚ > r $a r e b o t h s h o w n t o b e r e l a t i v e l y c o n s i s t e n t w i t h Z F C . W e a l s o i n v e s t i g a t e c o m b i n a t o r i c s o f t h e i d e a l n w d o f n o w h e r e d e n s e s u b s e t s o f, ℚ . I n p a r t i c u l a r, w e s h o w t h a t$ non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

$H^{p}$ -spaces of conjugate systems on local fields

Jia Chao (1974)

Studia Mathematica

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Asymptotic Expansions of Witten-Reshetikhin-Turaev Invariants for Some Simple $3$ -Manifolds

R. J. Lawrence (1995)

Recherche Coopérative sur Programme n°25

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