Related papers: Notes on unknotting algorithms using normal surfac…
In this paper, we give the trivializing number of all minimal diagrams of positive 2-bridge knots, and study the relation between the trivializing number and the unknotting number for a part of these knots.
A non-singular connected algebraic curve $A$ in a simply connected algebraic surface $X$ can be knotted so that its homology class and the fundamental group of its complement in $X$ is preserved, provided $A$ is sufficiently complex (not…
We show that colored Khovanov homology detects classes of essential surfaces as a direct analogue of the slope conjectures for the colored Jones polynomial. We do this by identifying certain generators of the colored Khovanov chain complex…
We investigate to what extent renormalization can be understood as an algebraic manipulation on concatenated one-loop integrals. We find that the resulting algebra indicates a useful connection to knot theory.
A knot theoretic algorithm is proposed to model `fragile topology' of quantum physics.
Spectral graph theory is well known and widely used in computer vision. In this paper, we analyze image segmentation algorithms that are based on spectral graph theory, e.g., normalized cut, and show that there is a natural connection…
A knitted surface is a surface with or without closed components smoothly properly embedded in $D^2 \times B^2$, which is a generalization of a braided surface. A knitted surface is called a 2-dimensional knit if its boundary is the closure…
The Kinoshita graph is a particular embedding in the 3-sphere of a graph with three edges, two vertices and no loops. It has the remarkable property that although the removal of any edge results in an unknotted loop, the Kinoshita graph is…
The most tight conformations of prime knots are found with the use of the SONO algorithm. Their curvature and torsion profiles are calculated. Symmetry of the knots is analysed. Connections with the physics of polymers are discussed.
We introduce a technique called graph fission which takes in a graph which potentially contains only one observation per node (whose distribution lies in a known class) and produces two (or more) independent graphs with the same node/edge…
The evaluation of partition functions is a central problem in statistical physics. For lattice systems and other discrete models the partition function may be expressed as the contraction of a tensor network. Unfortunately computing such…
In this paper we use twisted Alexander polynomials to prove that the exterior of a particular graph knot is not fibered. Then we build three 2-component graph links out of this knot, and use similar techniques to discuss their fiberedness.
We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of…
We present a systematic classification of uncolored bonded knots with singularity number at most seven. Bonded knots provide a topological model for closed protein chains with intramolecular bridges, such as disulfide bonds. Following the…
We provide three 3-dimensional characterizations of the Z-slice genus of a knot, the minimal genus of a locally-flat surface in 4-space cobounding the knot whose complement has cyclic fundamental group: in terms of balanced algebraic…
This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove…
While neural networks have achieved vastly enhanced performance over traditional iterative methods in many cases, they are generally empirically designed and the underlying structures are difficult to interpret. The algorithm unrolling…
This paper concerns the H(2)-unknotting numbers of links related to 2-bridge links. It consists of three parts. In the first part, we consider a necessary and sufficient condition for a 2-bridge link to have H(2)-unknotting number one. The…
Two natural generalizations of knot theory are the study of spatial graphs and virtual knots. Our goal is to unify these two approaches into the study of virtual spatial graphs. This paper is a survey, and does not contain any new results.…
We present quantum algorithms for various problems related to graph connectivity. We give simple and query-optimal algorithms for cycle detection and odd-length cycle detection (bipartiteness) using a reduction to st-connectivity.…