We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,341 papers · 148 categories
Trend · papers per month
48 results for “dealternation number”
Upper bounds for Khovanov width and dealternation number derived for positive braids.
problem Determining bounds for Khovanov width and dealternation number of positive braid links.
method Braid-theoretic technique combined with Upsilon invariant.
result Asymptotically sharp upper bounds for Khovanov width and dealternation number in terms of crossing number.
Study determines Turaev genus and dealternating number of torus knots up to a small error.
problem Measuring how close torus knots are to alternating links.
method Analyzes Turaev genus and dealternating number of torus knots with up to 6 strands.
result Exact or near-exact values for Turaev genus and bounds for dealternating number of torus knots.
Let D be a link diagram with n crossings, s_A and s_B its extreme states and |s_AD| (resp. |s_BD|) the number of simple closed curves that appear when smoothing D according to s_A (resp. s_B). We give a general formula for the sum |s_AD|+|s_BD| for a k-almost alternating diagram D, for any k, characterising this sum as…
New proof shows knot Floer thickness limits bad domains in diagrams.
problem Understanding knot thickness and dealternating number.
method Modified Stipsicz-Szabo approach using Kauffman states.
result Knot Floer thickness is a lower bound on dealternating number.
The paper calculates a knot invariant for 3-braid knots.
problem Calculating the concordance invariant for 3-braid knots.
method Constructing cobordisms between 3-braid knots and torus knots.
result Explicit formulas and values for the invariant for 3-braid knots.
Constructs links that are both quasi-alternating and almost alternating.
problem Creating links that are both quasi-alternating and almost alternating.
method Using dealternator extension to replace dealternator with rational tangle.
result All not alternating and quasi-alternating Montesinos links can be obtained.
Study eigenvalues of knot covers to bound knot properties.
problem Bounding knot genera, dealternating number, and alternation number.
method Minimal positive eigenvalues of double branched covers over spanning surfaces.
result Eigenvalue bounds provide estimates for knot properties.
We introduce the warping polynomial of an oriented knot diagram. In this paper, we characterize the warping polynomial, and define the span of a knot to be the minimal span of the warping polynomial for all diagrams of the knot. We show that the span of a knot is one if and only if it is non-trivial and alternating, an…
The study shows a limit on cosmetic surgeries for certain knots.
problem Cosmetic surgeries on knots and their finiteness.
method Estimating knot Floer thickness using Turaev genus and dealternation number.
result A knot in does not admit purely cosmetic surgery if its genus is sufficiently large.
Study on knot diagrams showing bridge number can differ from crossing number.
problem Incompatibility between crossing number and bridge number for knot diagrams.
method Defined and compared various bridge number computations for knot diagrams, studied minimizing diagrams, and constructed families of minimal crossing diagrams.
result Found examples where bridge number differs from crossing number, and demonstrated this difference can grow infinitely.
A new knot measure, the untwisting number, equals the algebraic unknotting number.
problem Measuring the minimum number of twists needed to untangle a knot.
method Defined and compared the untwisting number with the unknotting number.
result The algebraic untwisting number equals the algebraic unknotting number.
Straight knotsmath.GT
Straight numbers generalize Meander and OGC numbers for all knots.
problem Defining invariants for all knots based on Meander and OGC numbers.
method Generalized Meander and OGC numbers to all knots and proved their well-definedness.
result Straight numbers and contained straight numbers are well-defined for all knots.
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.
The paper bounds the handle number of sutured manifolds using Morse-Novikov numbers and tunnel numbers.
problem Bounding the handle number of sutured manifolds.
method Developed bounds on the Morse-Novikov number of a link in terms of its tunnel number, and used these to bound the handle number of Heegaard splittings.
result The handle number function is bounded, constant on rays from the origin, and locally maximal.
New measure shows how links can be untangled as twists increase.
problem Understanding how links can be simplified through repeated twists.
method Introduced the stable unknotting number to analyze links in a twist family.
result The stable unknotting number depends only on the winding number of the link, not the wrapping number.
The study of trivializing number for positive knots.
problem Understanding the trivializing number of positive knots.
method Analysis of minimal diagrams and relation study with unknotting number.
result Results on the trivializing number of positive 2-bridge knots.
Wirtinger number equals virtual bridge number for virtual links.
problem Calculating the virtual bridge number of virtual links.
method Algorithmically computing the minimum number of generators of the link group.
result The Wirtinger number equals the virtual bridge number for virtual links.
The study provides bounds and necessary conditions for tunnel and cutting numbers of knots and handlebody-knots.
problem Determining bounds for tunnel and cutting numbers of knots and handlebody-knots.
method Using G-family of quandles colorings and constructing handlebody-knots.
result Lower bounds and necessary conditions for tunnel and cutting numbers of knots and handlebody-knots.
Study unlinking numbers of 10-crossing links using link invariants.
problem Investigate unlinking numbers of 10-crossing links.
method Use various link invariants and explore their behavior with crossing changes.
result Find the unlinking numbers of all but 2 of the 287 prime, non-split links with crossing number 10.
New number bounds knot complexity, including unknotting and crosscap numbers.
problem Bounding knot complexity and understanding knot types.
method Introducing an unknotting-type number to estimate crosscap number.
result Determines set of knots with crosscap number at most two.
New invariant refines Milnor's triple linking number, revealing more information for complex links.
problem Indeterminacy of Milnor's triple linking number in complex link configurations.
method Introduced a new invariant called the total triple linking number, refining Milnor's original.
result The total triple linking number is non-trivial for every -component link, providing more information than classical triple linking numbers.
Paper shows that for torus knots, the pinch number equals the unoriented band unknotting number.
problem Determining the minimum number of band surgeries to unknot torus knots.
method Used the torsion order of unoriented knot Floer homology.
result Pinch number and unoriented band unknotting number coincide for torus knots.
The paper studies tunnel and bridge numbers of composite genus 2 spatial graphs.
problem Understanding the tunnel and bridge numbers of composite genus 2 spatial graphs.
method Analyzes connected sum and trivalent vertex sum operations on genus 2 spatial graphs, proving bounds for tunnel and bridge numbers.
result Sharp bounds for the tunnel number of composite genus 2 spatial graphs, including lower bounds for bridge numbers.
The paper finds bounds and specific tile numbers for knot mosaics.
problem Determining the minimum number of tiles needed to represent knots.
method Analyzing the relationship between tile number and mosaic number of knots.
result Strict bounds and specific tile numbers for various knots are determined.
Delta-unlinking number measures how to unlink algebraically split links.
problem Measuring unlinking complexity of algebraically split links.
method Defining delta-unlinking number as minimum delta-moves to unlink, proving bounds and calculating specific values.
result Precise delta-unlinking numbers for algebraically split prime links up to 9 crossings, and 4-genus values for most.
The paper explores different types of knot unknotting numbers using various local moves.
problem Investigating the unknotting numbers of knots using different local moves.
method Examined ribbon-move and pass-move on 2-knots and 1-knots, and high-dimensional-pass-move on high-dimensional knots.
result Found examples and bounds for various unknotting numbers associated with different local moves.
Study confirms a knot's crosscap number equals its splice-unknotting number for alternating knots.
problem Determining the crosscap number of alternating knots.
method Using a splice-unknotting number defined by Ito-Takimura, and computing through Gauss codes.
result Crosscap numbers of all prime alternating knots up to 13 crossings are computed.
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.
Odd crossing numbers and even rotation numbers for cycles in plane immersions.
problem Analyzing crossing and rotation numbers of cycles in plane immersions of graphs.
method Generic immersions and Legendrian embeddings of graphs, focusing on cycles of specific lengths.
result Sum of rotation numbers of all 5-cycles is even, and sum of crossing numbers is odd.
Lower bound found for Perron-Frobenius degrees of certain complex numbers.
problem Finding lower bounds for the Perron-Frobenius degree of complex numbers.
method Using Doug Lind's idea, proving results for both cubic and biPerron numbers.
result Arbitrary large Perron-Frobenius degrees for certain complex numbers.
This paper calculates stick numbers for rail arcs and knot classes.
problem Calculating the minimum number of sticks needed for rail arcs and knot classes.
method Rail isotopies, ambient isotopies, winding number invariant, and lattice stick number.
result Calculates stick numbers for rail arcs and knot classes with crossing number at most 9.
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…
Study computability of real numbers from group properties.
problem Computability of real numbers from group properties.
method Analyzing -Betti numbers and -torsion of groups.
result Real numbers as -Betti numbers or -torsion are computable.
We define the basket number, the flat plumbing number and the flat plumbing basket number of a link. Then we provide some upperbounds for these plumbing numbers by using Seifert's algorithm. We study the relation between these plumbing numbers and the genera of links.
We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number, which is the minimal number of crossing changes to a slice knot; and the concordanc…
The clock number of a knotmath.GT
This paper is about the clock number of a knot. First we define the clock number by using states of a knot defined by Kauffman. Next we show that if K is a prime knot, its clock number is greater than or equal to its crossing number. Finally we prove that its clock number is equal to its crossing number if and only if …
For a knot the cube number is a knot invariant defined to be the smallest for which there is a cube diagram of size for . There is also a Legendrian version of this invariant called the \emph{Legendrian cube number}. We will show that the Legendrian cube number distinguishes the Legendrian left hand toru…
This paper finds all prime knots with mosaic number 6 and their minimal space-efficient mosaics.
problem Finding minimal space-efficient mosaics for prime knots with a specific mosaic number.
method Examined prime knots with mosaic number 6, determined their minimal space-efficient mosaics, and calculated their tile numbers.
result A complete list of prime knots with mosaic number 6 and their minimal space-efficient mosaics were found.
Study Euler and Betti numbers of homology groups for a specific type of superalgebra.
problem Calculating Euler and Betti numbers for homology groups of pre Lie superalgebras.
method Introduced double weighted chain spaces to analyze pre Lie superalgebras of multi-vector fields with polynomial coefficients. Calculated Euler and Betti numbers for these homology groups.
result Derived formulas for Euler and Betti numbers of homology groups of pre Lie superalgebras.
New insights into knot fusion numbers via cabling.
problem Understanding fusion numbers of ribbon knots and their behavior under cabling.
method Utilizing knot Floer homology and cabling formulas to analyze fusion numbers.
result The fusion number and strong homotopy fusion number of (p,1)-cable knots are preserved.
Survey on two-numbers and their applications in mathematics.
problem Understanding the geometric properties of connected Riemannian manifolds.
method Review of existing literature and open problems.
result Two-numbers are closely related to various mathematical areas.
New q-numbers restore knot skein relations.
problem Restoring knot skein relations.
method Introducing deformed fermionic q-numbers for different skein relations.
result Restored Alexander, Jones, and HOMFLY skein relations.
New bounds and examples for sphere unknotting numbers.
problem Comparing unknotting numbers for 2-spheres in 4-space.
method Algebraic and geometric techniques.
result Stabilization number is bounded above by one more than Casson-Whitney number.
Links with minimum tunnel number have one less component than their number of parts.
problem Finding the minimum tunnel number for complex links.
method Combinatorial argument in link diagrams.
result Links with minimum tunnel number have one less component than their number of parts.
A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three only multiples of the top Chern number, which is the Euler characteristic, are invarian…
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…
The paper calculates the trivializing number for all minimal diagrams of positive 2-bridge knots.
problem Understanding the structure and properties of positive knots.
method Analysis of minimal diagrams and calculation of trivializing number.
result The trivializing number for all minimal diagrams of positive 2-bridge knots is determined.