Related papers: Almost positive links have negative signature
We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen, Beliakova and Wehrli's $s$-invariant. We also study almost positive links, in particular, determine the $s$-invariants of almost…
A signed graph $(G,\sigma)$ is a graph $G$ with a signature $\sigma$ labeling each edge with a positive or negative sign. Two signatures of $G$ are switching equivalent if one is obtained from the other by changing the signs of all edges in…
A bridgeless cubic graph $G$ is said to have a 2-bisection if there exists a 2-vertex-colouring of $G$ (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an…
It is known that there are only finitely many knots with super bridge index 3. Jin and Jeon have provided a list of possible such candidates. However, they conjectured that the only knots with super bridge index 3 are trefoil and the figure…
Binary representations of the trefoil and other knots of up to ten crossings in the simple cubic lattice were created. The BiEntropy of each knot was computed using a variety of binary encodings and compared against controls. This showed…
R. F. Williams showed that all knots in the Lorenz template are prime. His proof included the cases where any number of positive twists were added to one of the template's branches. However, Williams does give an example of a composite knot…
We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify…
Graph connectivity is a fundamental combinatorial optimization problem that arises in many practical applications, where usually a spanning subgraph of a network is used for its operation. However, in the real world, links may fail…
We will develop various methods, some are of geometric nature and some are of algebraic nature, to detect the various achiralities of knots and links in $S^3$. For example, we show that the twisted Whitehead double of a knot is achiral if…
Clustering a signed graph means partitioning the vertices into sets ("clusters") so that every positive edge, and no negative edge, is within a cluster. Clustering is not always possible; the obstruction is circles with exactly one negative…
Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter…
We prove that the detection rate of n-crossing alternating links by many standard link invariants decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular…
It is well-known that a knot is Fox $n$-colorable for a prime $n$ if and only if the knot group admits a surjective homomorphism to the dihedral group of degree $n$. However, this is not the case for links with two or more components. In…
We give a complete characterization of the topological slice status of odd 3-strand pretzel knots, proving that an odd 3-strand pretzel knot is topologically slice if and only if either it is ribbon or has trivial Alexander polynomial. (By…
A projective link is a smooth closed 1-submanifold of the real projective space of dimension three. A projective link is said to be affine if it is isotopic to a link, which does not intersect some projective plane. The main result: a…
We present three "hard" diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in $\mathbb{S}^2$. Both examples are constructed by applying…
We consider signed networks in which connections or edges can be either positive (friendship, trust, alliance) or negative (dislike, distrust, conflict). Early literature in graph theory theorized that such networks should display…
Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded…
We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual "smoothing of a crossing" move. As by-products we have characterizations of these…
We give examples showing that Kidwell's inequality for the maximal degree of the Brandt-Lickorish-Millett-Ho polynomial is in general not sharp.