We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial.
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In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, th…
In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruen…
Conway-Gordon proved that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 mo…
We describe the Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaeger's state model for the HOMFLY polynomi…
The multivariable Conway function is generalized to oriented framed trivalent graphs equipped with additional structure (coloring). This is done via refinements of Reshetikhin-Turaev functors based on irreducible representations of quantized gl(1|1) and sl(2). The corresponding face state sum models for the generalized…
Empirical evidence suggests link polynomials can detect causality in spacetimes.
problem Detect causality in -dimensional globally hyperbolic spacetimes.
method Introduced a new invariant of certain tangles related to the Conway polynomial.
result The Conway polynomial does not detect causality in certain spacetime scenarios.
New bounds for knot complexity based on Jones polynomial coefficients.
problem Finding bounds for the crosscap number of knots and links.
method Using coefficients from the Jones polynomial, we derive two-sided bounds for Conway sums of strongly alternating tangles.
result Neither linear bound generalizes for all knots and links.
For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this formula in terms of the square of the linking number and the second coefficient of t…
Quandle coloring detects causality in spacetime links.
problem Detecting causality in spacetime links using quandle colorings.
method Conjugation quandle of dihedral group D5 combined with Alexander-Conway polynomial.
result The conjugation quandle over D5 distinguishes causally related and unrelated events.
Alexander quandles fail to distinguish certain links, thus not detecting causality.
problem Detecting causality in spacetimes using link polynomials.
method Examined Alexander quandles' ability to distinguish specific links.
result Alexander quandles cannot distinguish the connected sum of two Hopf links and Allen-Swenberg Links.
The paper generalizes linking number properties for complete graphs.
problem Understanding linking numbers in spatial complete graphs.
method Analyzing the sum of square linking numbers and triangle-triangle links.
result Explicit formula for the sum of square linking numbers in large complete graphs.
Symplectic quandles can detect causality in spacetimes, improving on existing methods.
problem Detecting causality in (2+1)-dimensional spacetimes using existing methods.
method Combining Alexander-Conway polynomial with symplectic quandles.
result Symplectic quandles can distinguish between different types of links, suggesting their ability to detect causality.
We show that essential punctured spheres in the complement of links with distance three bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bri…
Polyak showed that any Milnor's -invariant of length 3 can be represented as a combination of Conway polynomials of knots obtained by certain band sum of the link components. On the other hand, Habegger and Lin showed that Milnor invariants are also invariants for string links, called -invariants. We sho…
In~\cite{Kim} the author generalized the Conway algebra and constructed the invariant valued in the generalized Conway algebra defined by applying two skein relations to crossings, which is called a generalized Conway type invariant. The generalized Conway type invariant is a generalization of Homflypt polynomial. In t…
A simplified proof of the Alexander-Conway polynomial exists.
problem Existence of the Alexander-Conway polynomial for links in 3D space.
method Presented an accurate detailed exposition of the proof.
result Existence of the Alexander-Conway polynomial proved.
We consider an algebra of (classical or virtual) tangles over an ordered circuit operad and introduce Conway-type invariants of tangles which respect this algebraic structure. The resulting invariants contain both the coefficients of the Conway polynomial and the Milnor's mu-invariants of string links as partial cases.…
Formulas previously presented for the Casson-Walker invariant are generalized to Lescop's extension. These formulas in terms of linking numbers and surgery coefficients compute the change in Lescop's invariant under crossing changes in a framed link presenting a 3-manifold. This leads us to revisit an old formula for a…
L-space knots lack essential Conway spheres, proven with Floer theory.
problem Essential Conway spheres in L-space knots.
method Floer theoretic invariant for tangles.
result L-space knots have no essential Conway spheres.
The paper characterizes Conway-Coxeter friezes using rational links.
problem Characterizing Conway-Coxeter friezes of zigzag type.
method Characterization via rational links and application to Jones polynomial.
result Jones polynomial can be defined for Conway-Coxeter friezes of zigzag type.
In the present paper, we build a bridge between Conway-Coxeter friezes and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomials attached to rational links by using Conway-Coxeter friezes. As an application one can give a complete invariant on Conway-Coxeter friezes o…
This paper focuses on the graphs in the Petersen family, the set of minor minimal intrinsically linked graphs. We prove there is a relationship between algebraic linking of an embedding and knotting in an embedding. We also present a more explicit relationship for the graph between knotting and linking, whi…
We give a closed formula for the Conway function of a splice in terms of the Conway function of its splice components. As corollaries, we refine and generalize results of Seifert, Torres, and Sumners-Woods.
The Conway knot can't be smoothly tied to any other knot an infinite number of times.
problem Understanding the concordance order of knots.
method Rasmussen invariant, satellite operations, null-homologous twists.
result The Conway knot has infinite concordance order.
We give a geometric construction of the multivariable Conway potential function for colored links. In the case of a single color, it is Kauffman's definition of the Conway polynomial in terms of a Seifert matrix.
The Conway potential function (CPF) for colored links is a convenient version of the multi-variable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway's `smoothing of crossings' is not in the axioms. The proof uses a reduction scheme in a t…
Khovanov homology invariant under Conway mutation.
problem Invariance of Khovanov homology under specific transformations.
method Strong geography restrictions and homological mirror symmetry.
result Classification of components of a Khovanov multicurve invariant.
Lisa Piccirillo solved the mystery of the Conway knot's sliceness.
problem Determining if the Conway knot is slice.
method The main idea of the proof is given in the title.
result The Conway knot is not slice.
In this chapter (Chapter III) we introduce the concept of Conway algebras (the notion related to entropic magmas) and describe invariants of links yielded by (partial) Conway algebras (including the Homflypt polynomial and signatures). We present, in detail, a proof (following the original Przytycki-Traczyk 1984 proof)…
Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and the minimal number of conways. The continued fraction associated to rational kno…
Paper defines half-Conway polynomial and computes it for knots up to 12 crossings.
problem Computing and characterizing half-Conway polynomials of knots.
method Normalized Conway polynomial, equivariant skein relation, diagrammatic interpretation.
result First examples of non-slice strongly negative amphichiral knots with determinant one.
The purely cosmetic surgery conjecture is true for the Kinoshita-Terasaka and Conway knot familiesmath.GT
Two knot families meet cosmetic surgery conjecture.
problem Cosmetic surgery conjecture in knot theory.
method Analyzing Kinoshita-Terasaka and Conway knot families.
result All nontrivial members of both knot families satisfy the conjecture.
Extends Benard-Conway invariant to all two-component links.
problem Counting irreducible SU(2) representations for two-component links.
method Counting irreducible SU(2) representations with fixed meridional traces.
result Invariant equals symmetrized multivariable link signature for (2, 2n)-torus links.
Simplified proof for a theorem about graphs.
problem Proving a generalized Conway--Gordon--Sachs theorem for complete graphs.
method Provided a shorter proof over integers.
result A simpler proof of the generalized theorem.
Study proves nontrivial knots can't undergo cosmetic surgeries.
problem Cosmetic surgeries on nontrivial knots.
method Calculated finite type invariant of order 3.
result Nontrivial knots do not admit chirally cosmetic surgeries.
We give a counterexample to the Kawauchi conjecture on the Conway polynomial of achiral knots which asserts that the Conway polynomial of an achiral knot satisfies the splitting property for a polynomial with integer coefficients. We show that the Bonahon-Siebenmann decomposition of an ac…
New theorem shows every integer can be represented by knot summation.
problem Understanding the summation of knot coefficients.
method Analyzing spatial complete graphs and Hamiltonian knots.
result Every integer can be represented by knot summation.
New method characterizes thin links via Conway spheres and tangle decompositions.
problem Characterize thin links without relying on specific knot invariants.
method Developed a relative version of thinness for tangles and used it to characterize thinness via tangle decompositions along Conway spheres.
result Characterized thin links via Conway spheres and tangle decompositions.
A string link S can be closed in a canonical way to produce an ordinary closed link L. We also consider a twisted closing which produces a knot K. We give a formula for the Conway polynomial of L as a product of the Conway polynomial of K times a power series whose coefficients are given as explicit functions of the Mi…
The purpose of this paper is to present a certain combinatorial method of constructing invariants of isotopy classes of oriented tame links. This arises as a generalization of the known polynomial invariants of Conway and Jones. These invariants have one striking common feature. If L+, L- and L0 are diagrams of oriente…
Conway's drum quiltsmath.GT
John Conway created pairs of domains that sound the same for a special kind of music.
problem Creating domains that sound the same for a special kind of music.
method Using his theory of quilts, Conway developed pairs of glueing diagrams.
result Conway's pairs of domains are isospectral for the Laplace operator.
Formula found for a specific knot's A-polynomial.
problem Computing the A-polynomial of a specific knot.
method Explicit formula derived for the knot with Conway's notation C(2n, 4).
result The A-polynomial contains exactly the same irreducible factors as the one defined in~\cite{CCGLS1}.
We exhibit relations between van Kampen-Flores, Conway-Gordon-Sachs and Radon theorems, by presenting direct proofs of some implications between them. The key idea is an interesting relation between the van Kampen and the Conway-Gordon-Sachs numbers for restrictions of a map of -simplex to to the $…
We show two results about the Conway potential function which is known as the normalized multivariable Alexander polynomial. We first show that the Conway potential function introduced by Kauffman in "Formal Knot Theory" is indeed a link invariant. Next we show that Kauffman's potential function equals Hartley's potent…
Given an oriented link in the 3-sphere, the Euler characteristic of its link Floer homology is known to coincide with its multivariate Alexander polynomial, an invariant only defined up to a sign and powers of the variables. In this paper, we get rid of this ambiguity by proving that this Euler characteristic is equal …
Taffy, Trees, and Tanglesmath.HO
Study connects taffy pulling, fractions, and rational tangles.
problem Understanding the relationship between taffy pulling, fractions, and rational tangles.
method Developed a taffy analogue for Conway's characterization of rational tangles and gave a geometric connection.
result Direct geometric connection between rational tangles and taffy pulls.
We give a Conway-Gordon type formula for invariants of knots and links in a spatial complete four-partite graph in terms of the square of the linking number and the second coefficient of the Conway polynomial. As an application, we show that every rectilinear spatial contains a nontrivial Ha…