Symmetric Union Research Articles

Ribbon knots with different symmetric union presentations

Ribbon knots with different symmetric union presentations

A short proof of a theorem of Tanaka on composite knots with symmetric union presentations

We present a short proof of a theorem of Tanaka that if a composite ribbon knot admits a symmetric union presentation with one twist region, then it has a non-trivial knot and its mirror image as connected summands.

Symmetric union presentations for 2-bridge ribbon knots

We prove that all 2-bridge ribbon knots are symmetric unions.

On equivalence of symmetric union presentations for ribbon knots

In this paper, we show that there exists a knot which has infinitely many non-equivalent symmetric union presentations with the same partial knots. We also define an invariant of a symmetric union presentation and give alternative proofs to results of Eisermann and Lamm and a result of Collari and Lisca by using methods in 3-dimensional topology.

On composite knots with symmetric union presentations

A symmetric union in the 3-space [Formula: see text] is obtained from a knot in [Formula: see text] and its mirror image, which is symmetric with respect to a 2-plane in [Formula: see text], by connecting them with some vertical 2-string tangles with twists and a horizontal 2-string tangle with no twists along the plane. We introduce the minimal twisting number of a symmetric union and show that if a knot [Formula: see text] is a composite symmetric union with minimal twisting number one, then [Formula: see text] has a non-trivial knot and its mirror image as connected summands. As an application, we study the minimal twisting number of a symmetric union and show that there exist infinitely many symmetric unions with minimal twisting number two.

The Search for Nonsymmetric Ribbon Knots

We present the results of Axel Seeliger’s tabulation of symmetric union presentations for ribbon knots with crossing numbers 11 and 12 and exhibit possible examples for ribbon knots which are not representable as symmetric unions. In addition, we give a complete atlas of band diagrams for prime ribbon knots with 11 and 12 crossings.

Symmetric Union Diagrams and Refined Spin Models

AbstractAn open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence among symmetric union diagrams and showing that non-equivalent diagrams can be detected using a refined version of the Jones polynomial. We prove that every topological spin model gives rise to many effective invariants of symmetric equivalence, which can be used to distinguish infinitely many Reidemeister equivalent but symmetrically non-equivalent symmetric union diagrams. We also show that such invariants are not equivalent to the refined Jones polynomial and we use them to provide a partial answer to a question left open by Eisermann and Lamm.

Intersecting Families in Symmetric Unions of Direct Products of Set Families

Let $X_{1}, \ldots,X_m$ be $m$ pairwise disjoint sets with the same size $n$, and let $\{k_1,\ldots,k_m\}$ be a multiset of positive integers such that $k_i\leq n/2$. Let $X=X_{1}\cup\cdots\cup X_{m}$ and $k=k_1+\cdots+k_m$. Given a $k$-set $A\subseteq X$, let $I(A)$ be the $m$-element multiset $\{|A \cap X_{1}|,\ldots,|A\cap X_m|\}$. Set $\Omega=\{ A\in \binom {X}{k}\colon\,{$I(A)=\k_1,łdots,k_m\$}\}$. In this paper, we prove that if $\mathcal{F}$ is an intersecting family in $\Omega$, then $|\mathcal F|\leq \frac{k}{nm}p(k_1,\ldots,k_m)\prod_{i=1}^m\binom{n}{k_i}$, where $p(k_1,\ldots,k_m)$ equals the number of permutations of $k_1,\ldots,k_m$. Furthermore, equality holds if and only if $\mathcal{F}=\left\{ A\in \Omega:a\in A\right\}$ for some $a\in X$, except in the case $k_1=\cdots=k_m=n/2$.

The macro-event property and the layered structure of the clause

We ask whether there is a “macro-event phrase,” a uniform level of syntax at which complex scenarios may be described as single events under the Macro-Event Property (MEP). The MEP is a form-meaning mapping property that constrains the compatibility of event descriptions with time-positional modifiers. An examination of English infinitival complements, Ewe serial verb constructions, and Japanese converb constructions suggests that the putative crosslinguistic “macro-event phrase” is the verbal core of the Layered Structure of the Clause theory of Role and Reference Grammar. Across languages, simple cores necessarily have the MEP, whereas complex cores have it if and only if they are integrated by ‘cosubordinate’ nexus, defined as a symmetric union of two cores that together behave like a single core. We furthermore argue that this connection between core cosubordinations and the MEP may help explain why cosubordinate cores seem to always share an argument through control.

Cosmetic surgery in L-spaces and nugatory crossings

The cosmetic crossing conjecture (also known as the “nugatory crossing conjecture”) asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most nine crossings and provide new examples of knots, including pretzel knots, non-arborescent knots and symmetric unions for which the conjecture holds.

Classical homological invariants are not determined by knot Floer homology and Khovanov homology

We illustrate that there are knots for which Heegaard knot Floer homology and Khovanov homology are identical but the Alexander module and torsion invariants differ. The examples are certain symmetric unions. We also give examples of similar flavor, concerning the Kauffman and Q-polynomials in place of the classical homological invariants. This shows there are nonmutant knots with the same knot Floer and Khovanov homology.

The structure of graphs with circular flow number 5 or more, and the complexity of their recognition problem

For some time the Petersen graph has been the only known Snark with circular flow number $5$ (or more, as long as the assertion of Tutte's $5$-flow Conjecture is in doubt). Although infinitely many such snarks were presented eight years ago by Macajova and Raspaud, the variety of known methods to construct them and the structure of the obtained graphs were still rather limited. We start this article with an analysis of sets of flow values, which can be transferred through flow networks with the flow on each edge restricted to the open interval $(1,4)$ modulo $5$. All these sets are symmetric unions of open integer intervals in the ring $\mathbb{R}/5\mathbb{Z}$. We use the results to design an arsenal of methods for constructing snarks $S$ with circular flow number $\phi_c(S)\ge 5$. As one indication to the diversity and density of the obtained family of graphs, we show that it is sufficiently rich so that the corresponding recognition problem is NP-complete.

THE JONES POLYNOMIAL OF KNOTS WITH SYMMETRIC UNION PRESENTATIONS

A symmetric union is a diagram of a knot, obtained from diagrams of a knot in the 3-space and its mirror image, which are symmetric with respect to an axis in the 2-plane, by connecting them with 2-tangles with twists along the axis and 2-tangles with no twists. This paper presents an invariant of knots with symmetric union presentations, which is called the minimal twisting number, and the minimal twisting number of <TEX>$10_{42}$</TEX> is shown to be two. This paper also presents a sufficient condition for non-amphicheirality of a knot with a certain symmetric union presentation.

Symmetric ribbon disks

We study the ribbon disks that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number rS(K) is introduced and compared with the classical ribbon number r(K). We show that the difference rS(K) - r(K) can be arbitrarily large by constructing an infinite family of ribbon knots Kn such that r(Kn) = 2 and rS(Kn) &gt; n. The proof is based on a particularly simple description of symmetric unions in terms of certain band diagrams which leads to an upper bound for the Heegaard genus of their branched double covers.

A refined Jones polynomial for symmetric unions

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams we develop a two-variable refinement $W_D(s,t)$ of the Jones polynomial that is invariant under symmetric Reidemeister moves. Here the two variables $s$ and $t$ are associated to the two types of crossings, respectively on and off the symmetry axis. From sample calculations we deduce that a ribbon knot can have essentially distinct symmetric union presentations even if the partial knots are the same. If $D$ is a symmetric union diagram representing a ribbon knot $K$, then the polynomial $W_D(s,t)$ nicely reflects the geometric properties of $K$. In particular it elucidates the connection between the Jones polynomials of $K$ and its partial knots $K_\pm$: we obtain $W_D(t,t) = V_K(t)$ and $W_D(-1,t) = V_{K_-}(t) \cdot V_{K_+}(t)$, which has the form of a symmetric product $f(t) \cdot f(t^{-1})$ reminiscent of the Alexander polynomial of ribbon knots.

RIBBON 2-KNOTS WITH SYMMETRIC UNION PRESENTATIONS

A symmetric union of a knot is a generalization of the operation of the connected sum of a knot and its mirror image. In this paper, we show that every ribbon 2-knot of 1-fusion has a ribbon presentation associated with a symmetric union which has either the unknot or a 2-bridge knot as a partial knot.

EQUIVALENCE OF SYMMETRIC UNION DIAGRAMS

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is easy to see that every symmetric union represents a ribbon knot, but the converse is still an open problem. Besides existence it is natural to consider the question of uniqueness. In order to attack this question we extend the usual Reidemeister moves to a family of moves respecting the symmetry, and consider the symmetric equivalence thus generated. This notion being in place, we discuss several situations in which a knot can have essentially distinct symmetric union representations. We exhibit an infinite family of ribbon two-bridge knots each of which allows two different symmetric union representations.

Symmetric unions and ribbon knots

Symmetric unions and ribbon knots