We revoke the problem of drawing graphs in the plane so that only certain specified pairs of edges are allowed to cross. We overview some previous results and open problems, namely the connection to intersection graphs of curves in the plane. We complement these by stating a new conjecture and showing that its proof would solve the problem of algorithmic decidability of recognition of string gr...

In this paper, we construct a new topological quantum field theory of cohomological type and show that its partition function is a crossing number.

One of the most basic questions in knot theory remains unresolved: is crossing number additive under connected sum? In other words, does the equality c(K1♯K2) = c(K1) + c(K2) always hold, where c(K) denotes the crossing number of a knot K and K1♯K2 is the connected sum of two (oriented) knots K1 and K2? The inequality c(K1♯K2) ≤ c(K1) + c(K2) is trivial, but very little more is known in general...

The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions,...

We prove that it is NP-hard to determine whether the crossing number of an input graph is even or odd.

A knot in S is doubly slice if it is the cross-section of an unknotted two-sphere in S. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We extend that work through 12 crossings, resolving all but four cases among the 2,977 prime knots in that range. The techniques involved in this analysis include considerations of the Ale...

A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The pairwise crossing number (resp. the odd-crossing number) of G is the minimum number of pairs of ...

A center-crossing recurrent neural network is one in which the null-(hyper)surfaces of each neuron intersect at their exact centers of symmetry, ensuring that each neuron's activation function is centered over the range of net inputs that it receives. We demonstrate that relative to a random initial population, seeding the initial population of an evolutionary search with center-crossing networ...

For any simple graph G = (V, E), we can define four types of crossing number: crossing number, rectilinear crossing number, odd-crossing number, and pairwise crossing number. We discuss the relationship of them.