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.
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.
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.
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. 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
The study connects knot crossing numbers to surface properties and tunnel numbers.
problem Understanding the relationship between knot crossing numbers and surface properties.
method Combines surface ascending-number estimates, bridge-number estimates, and amalgamation arguments for Heegaard splittings.
result Establishes a linear relationship between the crossing number and the Heegaard deficiency of the surface.
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 shows how many crossings arise in curves on surfaces.
problem Understanding crossings in curves on surfaces.
method Proved a minimum crossing number growth rate for m curves. result Effective bounds with optimal growth rates on every orientable surface.
The paper bounds crossing numbers of dense graphs on surfaces.
problem Estimating the minimum number of edge crossings for dense graphs on surfaces.
method Proved lower and upper bounds on crossing numbers, providing explicit families of surfaces.
result Upper and lower bounds on crossing numbers match up to constant factors.
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.
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.
This paper calculates the non-orientable 4-genus for knots with 10 crossings.
problem Determining the non-orientable 4-genus for knots with a specific number of crossings.
method Calculating the minimal first Betti number of non-orientable surfaces smoothly embedded in a 4-ball with boundary the knot.
result The non-orientable 4-genus for knots with 10 crossings has been calculated.
New invariant distinguishes Legendrian surfaces in 5-manifolds.
problem Distinguishing Legendrian surfaces in closed contact 5-manifolds.
method Introduced a new Legendrian isotopy invariant, MPX(L), and extended it to an absolute invariant. result New invariant distinguishes surfaces not distinguishable by Thurston-Bennequin invariant.
Study knots that divide ribbon knotted surfaces, computing their half ribbon genus and fusion number.
problem Understanding knots that divide ribbon knotted surfaces and their properties.
method Defining half ribbon knots, computing half ribbon genus and fusion number, and comparing with Levine-Tristram signatures.
result Computed half ribbon genus and fusion number for various knots, including new computations of doubly slice genus.
The paper develops a theory of skein adequate links in thickened surfaces and proves Tait conjectures.
problem Establishing Tait conjectures for adequate links in thickened surfaces.
method Applying Kauffman bracket skein algebras to develop a theory of skein adequate links and proving Tait conjectures.
result The crossing number is additive under connected sum for adequate links in thickened surfaces.
We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.
The paper defines new polynomials for links and linkoids.
problem No specific problem stated; focus on new polynomials.
method Defined as sums over states of link or linkoid diagrams with f=n. result Constructed new polynomials for starred links and linkoids.
A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-brid…
Optimizes the crossing number for curve systems on surfaces.
problem Finding a more efficient upper bound for the crossing number of curve systems.
method Relaxing parameter symmetry and solving an entropy balance problem.
result Reduced the constant by about 30% while maintaining the same framework.
We show that a finite numerical boundary slope of an essential surface in the exterior of a Montesinos knot is bounded above and below in terms of the numbers of positive/negative crossings of a specific minimal diagram of the knot.
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, t…
New theory proves infinite homology 3-spheres in homology 4-spheres.
problem Existence of homology 3-spheres in homology 4-spheres.
method Diagrammatics of surface cross sections, Taubes' work.
result Infinite number of homology 3-spheres in homology 4-spheres.
Study non-orientable 4-genus for 11-crossing non-alternating knots.
problem Computing the non-orientable 4-genus for specific knots.
method Survey tools and use various techniques to calculate the invariant.
result Calculate non-orientable 4-genus for 11-crossing non-alternating knots.
New polynomials defined for virtual knots, calculated up to crossing 4.
problem Defining and calculating invariants for virtual knots.
method Intersection number of curves on a closed surface.
result Intersection polynomials calculated up to crossing 4.
Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichmüller space…
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.
Identifies doubly slice genera for 2909 prime knots with up to 12 crossings.
problem Determining the doubly slice genera of prime knots.
method Identifies the minimal genus g for each knot K that divides a surface in S4. result Identified the doubly slice genera for 2909 prime knots with up to 12 crossings.
Study counts geodesic surfaces in knot complements, finding unique ones for small knots.
problem Counting totally geodesic surfaces in knot complements.
method Adapting boundary slope and intersection techniques, extending obstructions.
result Uniqueness of geodesic surfaces for specific knots, no geodesic surfaces for 47 knots.
A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length o…
Let L be a prime alternating link with n crossings. We show that for each fixed g, the number of genus g incompressible surfaces in the complement of L is bounded by a polynomial in n. Previous bounds were exponential in n.
This paper shows how to create surface-links with many triple points.
problem Creating surface-links with a large number of triple points.
method Analogous to knot diagrams, the paper uses broken sheet diagrams to project surface-links and analyze their triple points.
result There are non-split surface-links with arbitrarily many triple points.
Polynomial-time algorithm finds short non-orientable loops intersecting graph edges up to 30 times.
problem Finding short non-orientable loops intersecting graph edges efficiently.
method Combining computational biology techniques with recent graph theory results.
result Existence of short canonical non-orientable systems of loops.
Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface in…
This paper compiles and calculates triple point numbers for surface-links in Yoshikawa's table.
problem Determining the triple point number of surface-links in Yoshikawa's table.
method Using broken sheet diagrams, the paper compiles known triple point numbers and calculates or bounds the remaining ones.
result Compilation and calculation of triple point numbers for surface-links in Yoshikawa's table.
New polynomial invariants defined for long virtual knots.
problem Defining and studying polynomial invariants for long virtual knots.
method Intersection numbers of cycles on a closed surface, considering crossing order.
result Intersection polynomials are finite-type invariants of degree two under crossing changes, but not under virtualizations.
Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a circle pattern on the closed surface. In particular, there is an ass…
The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable 4-genus for all knots with crossing number 8 or 9. As applications we prove a conject…
We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative …
A new method for Gaussian Processes handles mixed continuous and categorical inputs.
problem Modeling cross-correlations between continuous and categorical data.
method Low-Rank Correlation (LRC) method for Gaussian Processes with flexible rank approximation.
result LRC outperforms existing methods in estimating cross-correlations and predicting response surfaces.
Study Mazur doubles of knots and their relation to the Slope Conjecture.
problem Proving the Strong Slope Conjecture for Mazur doubles of knots.
method Analyzing Mazur doubles under certain hypotheses and using the Slope Conjecture.
result Mazur doubles of knots satisfy the Strong Slope Conjecture under certain conditions.
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.
The paper studies families of curves on surfaces that realize all types of pants decompositions.
problem Finding the minimal size of families of curves on surfaces that realize all types of pants decompositions.
method Investigates exponential and superlinear bounds for surfaces without punctures, and provides bounds for surfaces with punctures.
result Provides bounds for the minimal size of families of curves on surfaces with and without punctures.
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.
We prove that if M is a CW-complex and M1 is its 1-skeleton then the crossed module Π2(M,M1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D2,S1). From this it follows that, if G is a finite crossed module and M is finite, then th…
Let L be a non-split prime alternating link with n>0 crossings. We show that for each fixed g, the number of genus-g Seifert surfaces for L is bounded by an explicitly given polynomial in n. The result also holds for all spanning surfaces of fixed Euler characteristic. Previously known bounds were exponenti…
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…
Study counts specific surfaces in Montesinos knots with 4 rational tangles.
problem Investigating closed essential surfaces in Montesinos knots with 4 rational tangles.
method Analyzing the number of closed, connected, essential, orientable surfaces of fixed genus in knot complements.
result Exactly 12 genus 2 surfaces and 8φ(g - 1) surfaces of genus greater than 2 are found, independent of knot crossings.
Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. These flat plumbing basket surface can be presented by a sequential presentation known as flat plumbing basket code first found by Furihata, Hirasawa and Kobayashi. The minimum number of flat plumbings to obtain…