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.
Minimal coloring number found for Z-colorable links.
problem Finding the minimum number of colors needed for Z-colorings of Z-colorable links.
method Defined Z-coloring as a generalization of Fox coloring for links with zero determinants. Provided sufficient conditions for non-splittable Z-colorable links to have the least minimal coloring number.
result Sufficient conditions for non-splittable Z-colorable links to have the least minimal coloring number.
This paper proves the minimal coloring number for a specific type of link is exactly 4.
problem Determining the minimal coloring number for a specific type of link.
method Investigated Z-colorable links and used their properties to prove the minimal coloring number is 4. result The minimal coloring number of any non-splittable Z-colorable link is exactly 4. 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…
This paper shows the minimal coloring number for certain Z-colorable links is four.
problem Determining the minimal number of colors for Z-colorings of links. method Analyzing diagrams of Z-colorable links and constructing specific diagrams to find the minimal coloring number. result The minimal coloring number for non-splittable Z-colorable links is four. Study finds minimal coloring numbers for torus links using rack colorings.
problem Determining the minimal number of colors for torus link diagrams.
method Using rack colorings on link diagrams to classify minimal colorings.
result Complete classifications of Z-colorings by four colors. New knots found with Seifert genus not matching minimal genus Seifert surfaces.
problem Discrepancy between Seifert genus and minimal genus Seifert surfaces.
method Constructed knots with specific genus and handle numbers to demonstrate the discrepancy.
result Found knots where Seifert genus is not realized by minimal genus Seifert surfaces.
The paper explores minimal coloring numbers for Z-colorable links.
problem Finding the minimum number of colors needed for Z-colorings on minimal diagrams of Z-colorable links. method Investigates minimal diagrams and Z-colorings for Z-colorable links. result For any positive integer N, there exists a minimal diagram of a Z-colorable link with at least N colors in any Z-coloring. Improved lower bound for geodesics on manifolds.
problem Finding a lower bound for the number of minimal geodesics on Riemannian manifolds.
method Refined Bangert's method using the stable norm unit ball on the first homology.
result Quadratic lower bound for the number of minimal geodesics.
We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson equation. We obtain a classification theorem for bi-umbilical hypersurfaces of t…
The study finds large Betti numbers in minimal hypersurfaces with positive Ricci curvature.
problem Minimal hypersurfaces with large Betti numbers in manifolds with positive Ricci curvature.
method Constructing sequences of manifolds with embedded minimal hypersurfaces.
result Minimal hypersurfaces have unbounded first Betti numbers.
Study estimates index of minimal hypersurfaces using Betti numbers.
problem Estimating the index of unstable minimal hypersurfaces.
method Extends previous method using first Betti number.
result Morse index is bounded by first Betti number.
We investigate the minimal number of links and knots in complete partite graphs. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links for K4,4,1…
Relations will be described between the quandle cocycle invariant and the minimal number of colors used for non-trivial Fox colorings of knots and links. In particular, a lower bound for the minimal number is given in terms of the quandle cocycle invariant.
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.
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.
Study minimal annuli in a slab, estimating their area.
problem Estimating the area of minimal annuli in a slab.
method Organized minimal annuli based on winding number, deduced convexity of length function, compared to catenoid waist area.
result Deduced convexity of length function and estimated area of minimal annuli.
Paper finds minimal number of curves in surface systems.
problem Determining minimal number of curves in surface systems.
method Analyzes oriented surfaces of genus g for positive integers k and g.
result Exact minimal number of curves in filling k-systems.
Study bounds index of minimal hypersurfaces in curved spaces.
problem Bounding the index of minimal hypersurfaces.
method Proved linear index bound using first Betti number and curvature.
result Index is bounded below by a linear function of first Betti 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.
We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in R2. We tackle this problem by the so-called…
The study finds an upper limit for the number of minimal origami pairs on a surface.
problem Counting the minimal origami pairs on a surface of genus g.
method Algorithm to count minimal origami pairs and using Ménage Problem to establish an upper bound.
result Established a new upper bound for the count of minimal origami pairs.
This paper disproves a conjecture about knot projections under specific homotopy conditions.
problem Reidemeister moves of types 1 and 3 are insufficient to describe all homotopies of circle immersions.
method Constructs counterexamples with minimal crossing numbers of 15 and higher, extending previous results.
result Obtains the first counterexample with a minimal crossing number of 15, extending to higher odd numbers.
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.
New methods find minimal crossing numbers for surfaces in S4.
problem Determining minimal crossing numbers for surfaces in S4. method Developed new tri-plane diagrams to analyze surfaces.
result Proved minimal crossing numbers for nonorientable unknotted surfaces in S4. Minimal crossing virtual links have minimal supporting genus.
problem Understanding the relationship between the number of crossings and the genus of virtual links.
method Developed a new parity theory for virtual links to prove minimal crossing implies minimal genus.
result Minimal crossing virtual links have minimal supporting genus.
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.
Minimal singular fibers in genus-g Lefschetz fibrations over torus.
problem Determining the minimum number of singular fibers in Lefschetz fibrations.
method Analyzing genus-g Lefschetz fibrations over the torus.
result The minimal number of singular fibers N(g,1) is at least 3. Paper proves timelike minimal surfaces with any number of ends exist.
problem Existence of timelike minimal surfaces with arbitrary ends.
method Analyzes asymptotic behavior and topology of constructed surfaces.
result Timelike minimal surfaces with arbitrary ends exist.
We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in R^N. The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, …
Bounds on minimal surfaces' ends and stability index.
problem Understanding the topology and stability of minimal surfaces.
method Proving bounds on the number of ends and stability index of minimal surfaces.
result Minimal surfaces of genus g have a finite number of ends and a bounded stability index. Lower bounds for minimal hypersurfaces in flat tori.
problem Finding lower bounds for the Morse index of minimal hypersurfaces.
method Generalizing earlier work by Ros, proving an affine lower bound in terms of first Betti number.
result Proved an affine lower bound for the Morse index of closed minimal hypersurfaces inside a flat torus.
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…
For every odd integer c≥21, we raise an example of a prime component-preservingly amphicheiral link with the minimal crossing number c. The link has two components, and consists of an unknot and a knot which is (−)-amphicheiral with odd minimal crossing number. We call the latter knot a {\it Stoimenow knot}. W…
Study f-minimal hypersurfaces in weighted manifolds, finding index bounds.
problem Understanding the index and first Betti number of f-minimal hypersurfaces.
method Generalized L. Ambrozio, A. Carlotto, and B. Sharp's method to study Morse index.
result Linear lower bound on Morse index via first Betti number.
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…
We show that the minimal number of colors for all effective n-colorings of a link with non-zero determinant is at least 1+log2n.
We find the minimal number of self-intersections of the boundary of a surface of genus g generically immersed in the plane.
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.
Minimal braids found for quasipositive links, solving a Rudolph question.
problem Finding minimal braids for quasipositive links.
method Using quasipositive minimal braids to establish bounds and solve problems.
result Minimal braid representatives for quasipositive links exist and have specific properties.
In this paper I give estimates for the minimal crossing number, leading to a short proof that the crossing number is additive for torus links. These estimates are applied to several classes of links. Finally, I prove a part of a conjecture relating the HOMFLY polynomial and the Kauffman polynomial.
We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.
We find the minimal number of links in an embedding of any complete k-partite graph on 7 vertices (including K7, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete k-partite graphs on 8 vertices. We also look at larger complete bip…
Study on the index and Betti number of f-minimal hypersurfaces and self-shrinkers.
problem Estimating the Morse index of self-shrinkers and f-minimal hypersurfaces.
method Analyzing the relationship between the index and the first Betti number of compact hypersurfaces, and using the dimension of the space of weighted square summable f-harmonic 1-forms in the non-compact case.
result Lower bounds on the index in terms of the first Betti number and the dimension of the space of weighted square summable f-harmonic 1-forms.
Proves large cobordism genus for winding number patterns.
problem Proving large cobordism genus for winding number patterns.
method Using knots and cobordism theory to show existence of large genus.
result Minimal genus of cobordism between winding number patterns can be arbitrarily large.
Minimal link diagrams are unique and have a fixed number of crossings.
problem Characterizing minimal link diagrams under Reidemeister moves.
method Analyzing minimal link diagrams through Reidemeister moves I and II.
result The uniqueness of minimal diagrams and their fixed number of crossings.
We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of complete minimal surfaces that includes algebraic minimal surfaces. It also provides a …
Minimal singular fibers found in nonorientable Lefschetz fibrations.
problem Finding the minimal number of singular fibers in nonorientable Lefschetz fibrations.
method Analyzing admissible nonorientable genus g Lefschetz fibrations over orientable surfaces.
result Existence of one singular fiber if and only if g ≥ 4 and h ≥ 1.