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…
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48 results for “Crossing”
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…
We introduce a local move on a link diagram named a region freeze crossing change which is close to a region crossing change, but not the same. We study similarity and difference between region crossing change and region freeze crossing change.
Multi-crossing Braidsmath.GT
Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A multi-crossing is a crossing where more than two strands meet at a single point, such that each strand bisects the crossing. In this paper we generalize ideas in traditional braid theory to multi-…
Classifies foliations on CROSSes.
problem Classifying Riemannian foliations on CROSSes.
method Classification based on manifold homeomorphism.
result Classification of Riemannian foliations on CROSSes.
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:…
Alexander polynomial condition blocks crossing changes in some knots.
problem Cosmetic crossing changes in knot diagrams.
method Alexander polynomial obstruction for -space knots.
result Proved the conjecture for a five-parameter family of pretzel knots.
Proves minimal crossing diagrams for specific spatial graphs.
problem Proving minimal crossing diagrams for spatial graphs.
method Analyzing adequate diagrams and replacing vertices and edges.
result All 1-vertex spatial graphs with adequate diagrams have minimal crossing number.
Colored Unlinkingmath.GT
In links with two components there are three different types of crossings: self-crossings in the first component, self crossings in the second component, and crossings between components. In this paper we examine the minimum number of crossing changes needed to unlink without changing the crossings between components. …
Self-crossing geodesicsmath.DG
Self-crossing geodesics on convex surfaces are studied.
problem Understanding patterns of geodesics crossing themselves.
method Analyzing closed geodesics on convex surfaces.
result Self-crossing geodesics exist on convex surfaces.
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…
There are non-vanishing price responses across different stocks in correlated financial markets. We further study this issue by performing different averages, which identify active and passive cross-responses. The two average cross-responses show different characteristic dependences on the time lag. The passive cross-r…
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…
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…
Proves special alternating knots can't have cosmetic crossings.
problem Cosmetic crossing changes in special alternating knots.
method Combines Lidman-Moore and Scharlemann-Thompson results.
result Special alternating knots do not admit non-nugatory crossing changes.
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 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 .
Nonabelian bundle 2-gerbesmath.DG
We define 2-crossed module bundle 2-gerbes related to general Lie 2-crossed modules and discuss their properties. A 2-crossed module bundle 2-gerbe over a manifold is defined in terms of a so called 2-crossed module bundle gerbe, which is a crossed module bundle gerbe equipped with an extra sructure. It is shown that s…
New method detects intrinsic cross-correlations in non-stationary time series affected by common factors.
problem Bias in cross-correlation analysis due to common external factors.
method Multifractal temporally weighted detrended partial cross-correlation analysis (MF-TWDPCCA).
result MF-TWDPCCA accurately detects intrinsic cross-correlations between non-stationary time series.
The study finds significant power-law cross correlations in Bitcoin's return-volatility dynamics.
problem Investigating asymmetry in Bitcoin's return-volatility relationships.
method Analysis of daily and high-frequency Bitcoin data to identify cross correlations.
result Power-law cross correlations between returns and future volatilities are observed, indicating long-range dependencies.
In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for empirical risk minimizers. In the general setting, we prove sanity-check bounds in the spirit of \cite{KR99} \textquotedblleft\textit{bounds showing that the worst-case error of this estimate is not m…
Two crossing homomorphisms on braid groups are shown to be equivalent.
problem Comparing two definitions of crossing homomorphisms on braid groups.
method Diagrammatic and algebraic definitions of crossing homomorphisms compared and computed for simple braids.
result Diagrammatic and algebraic crossing homomorphisms are equivalent.
Pseudo links have two crossing types: classical crossings and indeterminate crossings. They were first introduced by Ryo Hanaki as a possible tool for analyzing images produced by electron microscopy of DNA. A normalized bracket polynomial is defined for pseudo links and then used to construct and obstruction to cosmet…
It is classically known that generic smooth maps of R^2 into R^3 admit only cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap (u,uv,v^2) has non-trivial isometric deformations with infinite dimensional freedom. …
Quantum machine learning uses quantum cross entropy to minimize loss, but measurement loss affects this process.
problem Quantum machine learning's loss minimization through cross entropy is affected by measurement outcomes.
method Defined quantum cross entropy, proved its lower bounds, and investigated its relation to quantum fidelity and likelihood.
result Quantum cross entropy is lower-bounded by negative log-likelihood when derived from quantum data, but measurement outcomes can cause loss.
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.
Graphs represent knot adjacency for n crossings.
problem Understanding adjacency relationships between knots.
method Defined a new graph to represent -adjacency.
result Proved several results about the new graph .
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.
Folding maps on a cross-capmath.DG
Study symmetry of cross-cap surfaces with folding maps.
problem Reflectional symmetry of cross-cap surfaces.
method Characterization of singularities in folding maps.
result Characterized generic singularities on cross-cap.
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.
Paper classifies link diagrams on nonorientable surfaces using region crossing changes.
problem Classifying link diagrams on nonorientable surfaces.
method Classification through region crossing changes.
result Classification of link diagrams on nonorientable surfaces.
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.
New index principle shows indistinguishability of certain knot crossings.
problem Identifying indistinguishable crossings in knot diagrams.
method Developed a universal index function on knot diagrams that respects Reidemeister moves and sign of crossings.
result Crossings of the same sign in a classical knot diagram cannot be distinguished by any inherent property.
Region crossing change is a local operation on link diagrams. The behavior of region crossing change on is well understood. In this paper, we study the behavior of (modified) region crossing change on higher genus surfaces.
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.
Proposes MFPC for cross-manifold clustering.
problem Cross-manifold clustering challenges traditional methods.
method Multiple Flat Projections Clustering (MFPC).
result MFPC distinguishes cross-manifold clusters from projections.
We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic …
A crossing in a knot is nugatory if changing the crossing does not change the knot type. Using an invariant of certain types of closed 3-braid diagrams, we show that if a closed 3-braid contains a nugatory crossing then its braid index is one or two. This proves a special case of a conjecture on nugatory crossings due …
Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.
problem Unknottability of spatial graphs by region crossing changes.
method Region crossing changes to switch over/under relations within regions of spatial graph diagrams.
result Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.
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.
Enumerated all genus two handlebody-knots with seven crossings.
problem Counting genus two handlebody-knots with specific crossings.
method Extending an existing table of genus two handlebody-knots.
result Enumerated all genus two handlebody-knots with seven crossings.
Computed the 4-genus for all 12-crossing prime knots.
problem Calculating the 4-genus for all prime knots with 12 or fewer crossings.
method Computed the smooth 4-genera of knots with 12 crossings.
result Completed the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.
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.
Symmetries of cross capsmath.DG
Paper classifies symmetries of cross caps using invariants.
problem Classifying symmetries of cross caps.
method Used Bruce-West's normal form and associated functions to create invariants.
result Classified possible symmetries on cross caps.
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.
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 …
Calculates twist in Teichmüller space using cross ratios.
problem Calculating the Fenchel-Nielsen twist in Teichmüller space.
method Using cross ratio coordinates.
result Compact calculation of twist in Teichmüller space.
In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for stable predictors in the context of risk assessment. The notion of stability has been first introduced by \cite{DEWA79} and extended by \cite{KEA95}, \cite{BE01} and \cite{KUNIY02} to characterize cla…