Table of symmetric diagrams for knots up to 10 crossings.
problem Finding symmetric diagrams for strongly invertible knots.
method Compilation of symmetric diagrams for knots up to 10 crossings.
result Similarity of transversal diagrams to symmetric union diagrams for strongly invertible knots.
The paper explores when specific knot operations simplify diagrams.
problem Understanding when arc crossing changes simplify knot diagrams.
method Examined two types of arc crossing changes on link diagrams and determined when they are unknotting operations.
result Any two crossing points in an alternating knot diagram are arc crossing change admissible.
New method finds grid diagrams for many fibered knots.
problem Detecting fibered knots using grid diagrams.
method Developed an efficient method to identify grid diagrams with unique maximal Alexander grading states.
result Found suitable grid diagrams for 5385 of 5397 fibered prime knots with crossing number ≤ 13.
Virtual knots and links get multicrossings and petal diagrams.
problem Defining multicrossings for virtual knots and links.
method Extending multicrossings to virtual knots and links, showing petal diagrams exist.
result Petal diagrams exist for all virtual knots.
Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alter…
Study minimum ribbonlength of immersed flat knots and links.
problem Finding the minimum ribbonlength for immersed planar knots and links.
method Embedding into disk diagram space to find length minimizers.
result Computed minimal ribbonlength for some knot and link diagrams.
The article enumerates doubly symmetric diagrams for knots up to 18 crossings.
problem Enumerating doubly symmetric diagrams for knots.
method Developed an enumeration strategy for prime knots given by doubly symmetric diagrams.
result Determined all cases of doubly symmetric diagrams up to 18 crossings.
We prove all knots can be transformed into a trefoil using special diagrams.
problem Transforming any knot into a trefoil using magic tricks.
method Introducing knotholder diagrams to encode transformations.
result All knots can be transformed into a trefoil.
Minimal grid diagrams for 12-crossing prime knots identified.
problem Identifying minimal grid diagrams for prime knots.
method Listed minimal grid diagrams for 12-crossing prime knots.
result Provided a list of minimal grid diagrams for 12-crossing prime knots.
We consider a natural model of random knotting- choose a knot diagram at random from the finite set of diagrams with n crossings. We tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type, allowing us to compute exact probabilities for knots in this model. As expected, most diagrams with 1…
Generalizes meander diagrams to virtual knots and introduces new invariants.
problem Classifying and comparing virtual knots.
method Generalization of meander diagrams to virtual knots, introduction of virtual k-arc crossing numbers. result Classes of meander and semimeander diagrams are universal for virtual knots.
GridPyM handles grid diagrams for knot theory.
problem Handling grid diagrams for knot theory.
method Generates and simplifies grids, models local transformations.
result Models local transformations between grid diagrams.
Study categorizes knots and links as rigid or shaky based on Reidemeister moves.
problem Classifying knots and links as rigid or shaky based on adaptability to Reidemeister moves.
method Categorization of hard diagrams as rigid or shaky, investigation of rigid and shaky hard diagrams for specific knots and links.
result Every link has a rigid hard diagram, and there is an upper limit for the number of crossings in such diagrams.
This paper finds all prime alternating knots with minimal warping degree two.
problem Finding knots with minimal warping degree.
method Examined all prime alternating knots and determined those with minimal warping degree two.
result All prime alternating knots with minimal warping degree two were identified.
New diagonal knots found with non-torus structure.
problem Identifying knots with diagonal grid diagrams.
method Analysis of knots represented by diagonal grid diagrams.
result All diagonal knots are positive, and a new non-torus example is found.
Study essential diagrams of knots in SgimesS1 and their relation to virtual knots.
problem Understanding essential diagrams and their relation to virtual knots in SgimesS1. method Analyzing knots with minimal double lines and embedding virtual knot theory.
result Virtual knot theory is embedded in the theory of knots in SgimesS1. New estimate of semimeander complexity for knots with more than 10 crossings.
problem Estimating the complexity of semimeander diagrams of knots.
method Proved a new upper bound on the number of crossings for semimeander diagrams of knots with more than 10 crossings.
result For knots with more than 10 crossings, semimeander diagrams have no more than 0.31⋅1.558cr(K) crossings. Enhances Vassiliev knot invariants using chord diagrams.
problem Improving Vassiliev knot invariants.
method Generalizes Vassiliev knot invariants to framed chord diagrams.
result Enhances Vassiliev knot invariants.
Problems on region choices for knot and link diagrams solved using Alexander numbering.
problem Existence of solutions for region choice problems on knot and link diagrams.
method Alexander numbering for regions, alternative proofs, necessary and sufficient conditions.
result Existence of solutions for region choice problems on link diagrams.
New method finds knots without low treewidth diagrams.
problem Finding knots without low treewidth diagrams.
method Structural graph theory and knot theory.
result Optimal obstruction for high representativity knots.
In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Fl…
We explore free knot diagrams, which are projections of knots into the plane which don't record over/under data at crossings. We consider the combinatorial question of which free knot diagrams give which knots and with what probability. Every free knot diagram is proven to produce trefoil knots, and certain simple fami…
We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and l…
The paper examines trisection diagrams of spun knots and shows they are standard for certain cases.
problem Whether trisection diagrams induced by the Gluck surgery on specific knots are standard.
method Explicit depiction and analysis of trisection diagrams for spun (2n+1,−2)-torus knots. result Trisection diagrams are standard for spun (2n+1,−2)-torus knots when n=1 and homologically standard for all n. Minimal grid diagrams found for 13-crossing prime knots with 13 arc index.
problem Finding minimal grid diagrams for prime knots with specific crossing and arc indices.
method Used Knotscape to generate spanning trees and obtain minimal arc presentations in grid diagrams.
result 9,988 prime knots with 13 crossings and 13 arc index were identified.
Robot untangles knots by walking and switching crossings.
problem Untangling knots using a robot with limited memory.
method The robot walks a knot diagram, switching crossings, and combinatorially proves knot transformations.
result Minimal moves to transform knots into an unknot are bounded by (7C+1)C.
The warping matrix has been defined for knot projections and knot diagrams by using warping degrees. In particular, the warping matrix of a knot diagram represents the knot diagram uniquely. In this paper we show that the rank of the warping matrix is one greater than the crossing number. We also discuss the linearly i…
New formula for knot invariants simplifies calculations and counts.
problem Calculating knot invariants for various diagrams.
method Localized configuration space integral with a new Gauss form.
result Yields arrow diagram expressions and new lower bounds.
A knot is fertile if it can generate all smaller knots through a specific diagram modification.
problem Understanding the conditions under which a knot can generate all smaller knots.
method Using obstructions and diagram modifications to analyze knot fertility.
result Finiteness of (c(K)+f,c(K)+p)-fertile knots for all f,p. Standard trisection diagrams found for a specific type of knot.
problem Finding standard trisection diagrams for Gluck twists.
method Using a specific method to obtain trisection diagrams for spun (p+1,p)-torus knots. result Standard trisection diagrams were found for the spun (p+1,p)-torus knots. Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in R4; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we e…
This paper studies virtual knots using mosaic diagrams.
problem Understanding virtual knots through mosaic diagrams.
method Developed moves to preserve knot type and showed all virtual knots can be represented.
result Any virtual knot can be represented as a virtual mosaic.
We give a simple example showing that a knot or link diagram that lies in the Z2 lattice is not necessarily the projection of a lattice stick knot or link in the Z3 lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the Z2 lat…
We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.
Given a knot diagram D, we construct a semi-threading circle for it which can be an axis of D as a closed braid depending on knot diagrams. In particular, we consider semi-threading circles for minimal diagrams of a knot with respect to overpasses which give us some information related to the braid index. By this n…
The JSJ decomposition helps classify genus two handlebody-knots.
problem Classifying genus two handlebody-knots.
method JSJ decomposition and annulus classification applied to handlebody-knot exteriors.
result Genus two handlebody-knots classified using annulus diagrams.
This paper tabulates prime knot projections up to eight double points.
problem Tabulating prime knot projections and their mirror images up to a certain number of double points.
method Systematic flypes and enumeration of tangles with at most four double points, using arrow diagrams.
result Complete table of prime knot projections with their mirror images up to eight double points.
Minimal grid diagrams found for 13-crossing prime knots.
problem Finding the simplest grid diagrams for prime knots with 13 crossings.
method Converted prime alternating knots to grid diagrams, focusing on minimal configurations.
result 4878 prime alternating knots with 13 crossings have been represented by grid diagrams with 15 vertical segments.
We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for building knot diagrams out of full twists on two strings rather than out of crossings. We describe an equivalence relation on clasp diagrams which…
Algorithm finds knot diagrams from exterior triangulations.
problem Finding knot diagrams from their exteriors.
method First practical algorithm for finding diagrams from triangulations of exteriors.
result First diagrams for 23 link exteriors with over 2,500 crossings.
Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number c grows exponentially with c, and to date computer-assisted proofs can only classify diagrams up to around twenty crossings. Instead, we consider diagrams enumerated by bridge number, following the lead of S…
Region crossing change is a local transformation on a knot or link diagram. We show that a region crossing change on a knot diagram is an unknotting operation, and we define the region unknotting numbers for a knot diagram and a knot.
Minimal grid diagrams for 15,735 knots with 14 crossings and arc index 14.
problem Representing prime knots with 14 crossings and specific arc indices using grid diagrams.
method Enumerated all prime knots with 14 crossings, categorized by arc index, and found minimal grid diagrams for those with arc index 14.
result 8,027 knots with arc index 13 and 15,735 knots with arc index 14 were represented by minimal grid diagrams.
New method computes knot Floer homology for satellite knots.
problem Computing knot Floer homology for satellite knots.
method Using immersed Heegaard diagrams and bordered Floer homology.
result Computation of knot Floer homology for satellite knots streamlined.
We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and pulling them uniformly from among sets of a given number of vertices n, as first established in recent work with Cantarella and Mastin. The knot diagram model is an exciting new model which captures b…
Generalizes region select game to k-colored knot diagrams.
problem Play game on knot diagrams with multiple colors.
method Generalize region select game to k-colored knot diagrams. result Generalization of the region select game to k-colored knot diagrams. Study identifies prime strongly positive amphicheiral knots with double symmetry.
problem Characterizing prime strongly positive amphicheiral knots with specific symmetries.
method Examined knots up to 16 crossings, identified prime knots with double symmetry, and presented almost doubly symmetric diagrams.
result Found the first prime strongly positive amphicheiral knot not slice.
In this article, we define an independence system for a classical knot diagram and prove that the independence system is a knot invariant for alternating knots. We also discuss the exchange property for minimal unknotting sets. Finally, we show that there are knot diagrams where the independence system is a matroid and…