Related papers: Conway's doughnuts
An oriented link L in a 3-sphere S in complex 2-space is a C-boundary if it bounds a piece of algebraic curve in the 4-ball bounded by S. Using Kronheimer and Mrowka's proof of the Thom Conjecture, we construct many oriented knots which are…
There is a map, defined and studied by Jones, from Thompson's group $F$ to knots. Jones proved that every knot is in the image of this map -- that is, that every knot can be seen as the "knot closure" of a Thompson group element. We…
By the theorem of Mantel $[5]$ it is known that a graph with $n$ vertices and $\lfloor \frac{n^{2}}{4} \rfloor+1$ edges must contain a triangle. A theorem of Erd\H{o}s gives a strengthening: there are not only one, but at least…
We prove a variant of the Sylvester-Gallai theorem for cubics (algebraic curves of degree three): If a finite set of sufficiently many points in $\mathbb{R}^2$ is not contained in a cubic, then there is a cubic that contains exactly nine of…
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…
The Holant theorem is a powerful tool for studying the computational complexity of counting problems in the Holant framework. Due to the great expressiveness of the Holant framework, a converse to the Holant theorem would itself be a very…
If all but two vertices of a triangulated sphere have degrees divisible by $k$, then the exceptional vertices are not adjacent. This theorem is proved for $k=2$ with the help of the coloring monodromy. For $k = 3, 4, 5$ colorings by the…
We give necessary conditions for a polynomial to be the Conway polynomial of a two-bridge link. As a consequence, we obtain simple proofs of the classical theorems of Murasugi and Hartley. We give a modulo 2 congruence for links, which…
A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. It is known that infinitely many $d$-regular nut graphs exist for $3 \leq d \leq 12$…
In its Euclidean form, the Dense Neighborhood Lemma (DNL) asserts that if $V$ is a finite set of points of $\mathbb{R}^N$ such that for each $v \in V$ the ball $B(v,1)$ intersects $V$ on at least $\delta |V|$ points, then for every…
In this article, we discuss whether a single congruent number $t$ can have two (or more) distinct triangles with the same hypotenuse. We also describe and carry out computational experimentation providing evidence that this does not occur.
A graph $G$ with vertex set $\{v_1,v_2,\ldots,v_n\}$ is an intersection graph of segments if there are segments $s_1,\ldots,s_n$ in the plane such that $s_i$ and $s_j$ have a common point if and only if $\{v_i,v_j\}$ is an edge of~$G$. In…
We suggest a method of solving the problem of existence of a triangle with prescribed two bisectors and one third element which can be taken as one of the angles, the sides, the heights or the medians, or the third bisector.
Let $K$ be a link of Conway's normal form $C(m)$, $m \geq 0$, or $C(m,n)$ with $mn\textgreater{}0$, and let $D$ be a trigonal diagram of $K.$ We show that it is possible to transform $D$ into an alternating trigonal diagram, so that all…
The theorem of three circles in real algebraic geometry guarantees the termination and correctness of an algorithm of isolating real roots of a univariate polynomial. The main idea of its proof is to consider polynomials whose roots belong…
We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology…
We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets $A$, $B$, and $C$ of points in the plane, that satisfy a…
In this paper, we study the self delta-equivalence of pretzel links. If the number of components is 2, then we know the complete invariants in terms of the Conway polynomial for classification. We calculate the values. For pretzel links…
The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to…
A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected $2$-complex every link graph of which is 3-connected…