The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number were principally no more than linear in the number of classic…
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48 results for “crossing number”
Study shows crossing numbers of cable knots are larger than previously thought.
problem Determining the crossing numbers of cable knots.
method Using colored Jones knot polynomials and degree analysis.
result Crossing numbers of -cables of adequate knots are larger than .
Spatial embeddings of planar graphs can have higher unknotting numbers than crossing numbers.
problem Understanding the relationship between unknotting numbers and crossing numbers of spatial embeddings of planar graphs.
method Analyzing specific examples of planar graphs and their spatial embeddings to find counterexamples.
result There exist planar graphs and their spatial embeddings where the unknotting number is greater than half the crossing number.
It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams o…
In this paper we investigate the unlinking numbers of 10-crossing links. We make use of various link invariants and explore their behaviour when crossings are changed. The methods we describe have been used previously to compute unlinking numbers of links with crossing number at most 9. Ultimately, we find the unlinkin…
A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing…
Study minimizes crossing points of up to 12 curves on a genus 2 surface.
problem Minimizing intersection points of curves on a surface.
method Analyzes systems of up to 12 simple closed curves on a genus 2 surface to find the minimum crossing number.
result Determines the minimal crossing number of up to 12 curves on a genus 2 surface and proves the minimization systems are unique.
The study calculates the average genus of 2-bridge knots based on their crossing numbers.
problem Determining the average genus of 2-bridge knots with a given crossing number.
method Analytical approach focusing on the properties of 2-bridge knots.
result Obtained the oblique asymptote of the average genus as crossing numbers increase.
Quantifies the crossing number of knots based on genus and braid index.
problem Estimating the crossing number of knots given their genus and braid index.
method Quantitative Birman-Menasco finiteness theorem applied to crossing numbers.
result Estimates the crossing number of knots in terms of genus and braid index.
New bounds on odd multicrossing numbers of knots and links are established.
problem Determining bounds on the number of crossings in knots and links.
method Proved inequalities involving the -crossing number, 3-genus, and number of components of a link.
result Established new bounds on the odd crossing numbers of torus knots and links.
The paper studies how the crossing number of graphs changes with a specific transformation called ΔY-move.
problem Investigating how the crossing number of graphs changes under the ΔY-move transformation.
method Analyzing the behavior of crossing number under the ΔY-move transformation on complete graphs.
result For any natural number k, there exists a sequence of ΔY-moves that decreases the crossing number of a complete graph.
An -crossing is a point in the projection of a knot where strands cross so that each strand bisects the crossing. An übercrossing projection has a single -crossing and a petal projection has a single -crossing such that there are no loops nested within others. The übercrossing number, , is the…
Determines the crossing number of polynomial curve systems on surfaces.
problem Calculating the crossing number of polynomial curve systems on surfaces.
method Determines the crossing number in terms of the genus for polynomial curve systems.
result High precision determination of crossing number for polynomial curve systems.
The paper defines and calculates an upper bound for the equivariant crossing number of two-bridge knots.
problem Finding the minimum number of crossings in symmetric diagrams for two-bridge knots.
method Defining and calculating for two-bridge knots by restricting diagrams to two types.
result An algorithm to determine for any two-bridge knot and results up to 14 crossings.
Minimal crossing number found in arithmetic curve systems.
problem Finding the minimal crossing number in arithmetic curve systems.
method Analyzing systoles of hyperbolic surfaces associated with congruence lattices in SL2(Z).
result Minimal crossing number is asymptotically achieved.
Study on knot properties, showing relation between unknotting and crossing numbers.
problem Relations between unknotting and crossing numbers of spatial embeddings.
method Analyzes handcuff-graphs and theta curves, extends known results to handlebody-knots.
result Characterizes handlebody-knots satisfying the equality between unknotting and crossing numbers.
A new knot invariant measures crossings in three orthogonal directions.
problem Defining a new knot invariant for certain knot diagrams.
method Defining the simultaneous crossing number for knots with doubly transvergent diagrams.
result The limit of the ratio of the new invariant to the usual crossing number is at most 8.
Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and th…
2-twist trefoil has 6 crossings, proving non-trivial knotted surface.
problem Computing the crossing number of non-trivial knotted surfaces.
method Bridge trisection and tri-plane diagrams to minimize crossings.
result 2-twist trefoil has crossing number 6.
Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Re…
We show that the crossing number of a satellite knot is at least 10^{-13} times the crossing number of its companion knot.
The paper tabulates and computes the number of alternating pretzel links up to a given crossing number.
problem Computing the total number of alternating pretzel links for a given crossing number.
method Derived a closed formula to compute the total number of alternating pretzel links, , for any given crossing number .
result The number of alternating pretzel links grows exponentially with the crossing number.
Characterizes adequate links using Jones polynomial and crossing number.
problem Characterizing adequate links.
method Using Jones polynomial and crossing number, proving links are adequate.
result Links with specific polynomial properties are adequate.
Jones polynomial bounds and crossing numbers of knots.
problem Bounding the degree of colored Jones polynomial in terms of crossing number.
method Sharpness of bounds determined for adequate knots; application to satellite knots.
result Sharpness of bounds for adequate knots; determination of crossing numbers for specific knots.
The paper improves bounds on knot crossings and tabulates minimal diagrams.
problem Improving bounds on knot crossings and tabulating minimal diagrams.
method Analyzing triple-crossing and delta-crossing numbers, proving tangle existence, generating tables.
result Improved bounds on knot crossings and tabulated minimal diagrams for prime knots up to delta-crossing number 4.
Connected sum affects crossing numbers of flat virtual knots.
problem Understanding how connected sum impacts the crossing numbers of flat virtual knots.
method Analyzing minimal crossing diagrams and using super-additivity properties.
result Crossing number of flat virtual knots is super-additive under connected sum.
A note on knot fertilitymath.GT
Even knots with more than 30 crossings are not fertile.
problem Understanding fertility in knots with specific crossing numbers.
method Analyzing minimum crossing number diagrams and changing over-under information.
result Even knots with more than 30 crossings cannot be obtained from a minimum crossing number diagram by changing over-under information.
We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing …
The study calculates the average genus of rational knots and links.
problem Finding the average genus of rational knots and links.
method Enumerating and calculating the number of rational knots and links with a given crossing number.
result A precise formula for the average minimal genus of rational knots and links.
Triple-crossing number bound for knots and links, especially torus knots.
problem Finding bounds for triple-crossing numbers of knots and links.
method Using the genus of a knot or link, we derive bounds for the triple-crossing number.
result Triple-crossing number of torus knots and many other knots is at least twice their genus.
A multi-crossing (or n-crossing) is a singular point in a projection at which n strands cross so that each strand bisects the crossing. We generalize the classic result of Kauffman, Murasugi, and Thistlethwaite, which gives the upper bound on the span of the bracket polynomial of K as 4c_2(K), to the n-crossing number:…
The paper classifies links based on signature and crossing number properties.
problem Understanding the relationship between signature and crossing number of knots and links.
method Refined theorems and comprehensive classification of links with specific properties.
result Identification of all links where signature + crossing number = 2, closures of positive 3-braids.
Lower bound on crossing number of knots and links on a 2-sided surface in a 3-manifold
problem Crossing number of knots and links on a 2-sided surface in a 3-manifold
method Rank of fundamental groups
result Lower bound on crossing number
New moves prove crossing number sum for knots.
problem Proving the additivity of crossing number for knots.
method Defined restricted Reidemeister moves to prove additivity.
result Crossing number of combined knots is at least sum of individual knots.
Shows large unknotting number for simple knots.
problem Finding minimum crossing changes for unknotting.
method Positive-to-negative crossing changes without increasing genus.
result Genus non-increasing totally positive unknotting number can be large.
A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c_3(…
Study of generalized knots and links, proving inequality involving crossing number and braid index.
problem Proving an inequality involving the minimal crossing number and braid index for generalized knots and links.
method Introducing generalized crossings and moves, proving inequality for generalized knots and links.
result Proved inequality involving total crossing number and braid index for generalized knots and links.
The stable crossing number of a twist family of knots and the satellite crossing number conjecturemath.GT
The paper studies knots formed by twisting a circle around a base knot and conjectures a linear growth in crossing numbers.
problem Understanding the growth rate of crossing numbers in twist families of knots.
method Introduced the stable crossing number and used geometric wrapping and algebraic winding to establish conjectures.
result The crossing number of grows like as for coherent twist families.
The paper establishes a relation between knotoid crossing number and height.
problem Establishing a relation between knotoid crossing number and height.
method Using Reidemeister moves and isotopies, the paper proves a relation between the crossing number and height of a knotoid.
result The crossing number of a knotoid is at least twice its height.
This paper classifies a specific weave type by their crossing number.
problem Classifying doubly periodic untwisted (p,q)-weaves.
method Classification by crossing number, introducing crossing matrix for equivalence.
result Classification of untwisted (p,q)-weaves by their crossing number.
New methods find minimal crossing numbers for surfaces in .
problem Determining minimal crossing numbers for surfaces in .
method Developed new tri-plane diagrams to analyze surfaces.
result Proved minimal crossing numbers for nonorientable unknotted surfaces in .
New strict inequalities for knot crossing numbers proved.
problem Establishing strict inequalities between different crossing numbers of knots.
method Proving and generalizing inequalities between -crossing numbers for various knots.
result Optimal strict inequality for many knots, with optimality proven.
The purpose of this article is to give a preliminary clarification on the relation between crossing number and crossing change. With a main focus on the span of X polynomial, we prove that, as our theorem claims, the crossing number of the link after crossing change can be estimated when certain conditions are met. At …
Improved bounds for knot crossings in different mosaic patterns.
problem Finding tighter bounds for knot crossings in rectangular and hexagonal mosaics.
method Extended Howard and Kobin's proof to hexagonal mosaics and shortened the rectangular proof.
result New bounds for hexagonal mosaics with improved efficiency in rectangular mosaics.
Positive braids minimize knot untangling steps.
problem Finding the minimum number of steps to untangle knots.
method Analyzing positive braids and their knot closures, comparing ascending number to unknotting number.
result Ascending number equals unknotting number for knots from positive braids.
Optimal curves minimize crossings on surfaces.
problem Minimizing crossings on congruence surfaces.
method Examining systoles and their intersections.
result Modular systoles have minimal crossing numbers.
Enumerates knots up to five crossings and describes moves between them.
problem Counting and classifying knots up to a specific number of crossings.
method Generated tables of minimal diagrams and derived moves between knots.
result Conjecture about a lower bound for the triple-crossing number based on Alexander polynomial.
In this paper we compute the sharp lower bounds for the crossing number of -string -loop essential tangles. For essential tangles with only string components, we characterise the ones with the minimum crossing number for a given number of components, both when the tangle has knotted strings or only unknotted stri…