Generates minimal hard surface-link diagrams up to ten crossings.
problem Finding minimal diagrams for surface-links.
method Describes a method for generating minimal hard prime surface-link diagrams.
result Generates all minimal hard prime surface-unknot and surface-unlink diagrams up to ten crossings.
Sharp lower bound on fold singularities self-intersections.
problem Finding a lower bound on the number of self-intersections of fold singularities.
method Established a sharp lower bound on the number of self-intersections of the boundary of an immersed surface, then applied this to fold singularities.
result Sharp lower bound on the number of self-intersections of fold singularities.
(NOTE: per referee comments, this article has been split; it is now superseded by "Existence of thread-wire minimizers" and "Near-wire thread-wire minimizers"; please see http://www.bkstephens.net.) Alt's thread problem asks for least-area surfaces bounding a fixed "wire" curve and a movable "thread" curve of length L.…
Proves prime theta-curves for knots on minimal genus surfaces.
problem Prime knots and essential arcs on Seifert surfaces.
method Analyzes prime knot unions with essential arcs on minimal genus surfaces.
result Each prime knot union an essential arc on a minimal genus Seifert surface is a prime theta-curve.
Decomposes prime alternating links into prime tangles.
problem Understanding how to decompose prime alternating links into prime tangles.
method Refined results from Menasco and Thistlethwaite, focusing on alternating property and pseudo-Montesinos links.
result If a prime alternating link can be decomposed into two prime tangles, it must be visible in an alternating link diagram or the link is pseudo-Montesinos.
The main purpose of this note is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The se…
Listed 19,513 prime knots with arc index 12-16.
problem Tabulating prime knots with specific arc indices.
method Provided list of prime knots with minimal grid diagrams.
result 19,513 prime knots with arc index 12-16.
Classifies doodles into prime and super prime types, describing them with doodle codes.
problem Classifying doodles into prime and super prime types.
method Using doodle codes to describe complementary regions and enumerate doodle diagrams.
result Super prime doodles have a Hamiltonian circuit.
Homogeneous braids are visually prime, solving a Cromwell question.
problem Addressing Cromwell's question about visually prime braids.
method Using a criterion for primeness of open books and Murasugi sums.
result Closures of homogeneous braids are visually prime.
Kirby and Lickorish showed that every knot in the 3-sphere is concordant to a prime knot, equivalently, every concordance class contains a prime knot. We prove here that their result can be strengthened: Every knot in the 3-sphere is invertibly concordant to a prime knot. A consequence is that every double concordance …
This is the first in a series of four papers wherein we enumerate all prime alternating knots and links. In this first paper, we introduce four operators on knots and show that, when used according to very simple rules on the prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossing…
New prime amphicheiral knots found with a specific period.
problem Finding amphicheiral knots with a specific free period.
method Constructed new prime amphicheiral knots with free period 2.
result Solved an open question about amphicheiral knots and free periods.
This is the third paper in a series devoted to enumerating the prime alternating knots and links. This paper establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (T …
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.
It is known that the first two-variable Links--Gould quantum link invariant LG≡LG2,1 is more powerful than the HOMFLYPT and Kauffman polynomials, in that it distinguishes all prime knots (including reflections) of up to 10 crossings. Here we report investigations which greatly expand the set of evaluations o…
Alexander polynomial values at prime powers discussed.
problem Calculating Alexander polynomial values at prime powers.
method Finite set cardinality calculation based on Alexander polynomial.
result Finite set cardinality equals Alexander polynomial values at prime powers.
The paper generalizes CR invariants using renormalized characteristic forms.
problem Defining new CR invariants via renormalized characteristic forms.
method Introducing new curvatures for each renormalized characteristic form.
result The new curvatures' integrals match CR invariants constructed by Marugame.
Minimal grid diagrams for 12-crossing prime knots identified.
problem Identifying minimal grid diagrams for prime knots.
method Listed minimal grid diagrams for 12-crossing prime knots.
result Provided a list of minimal grid diagrams for 12-crossing prime knots.
Expanding on prime knots with 6 or less mosaic tiles, this paper analyzes those with 7 tiles.
problem Determining the tile number and space-efficiency for prime knots with mosaic number 7.
method Extending the methods of Heap and Knowles (2017) to include prime knots with mosaic number 7.
result Identifying the possible tile numbers and space-efficient layouts for all prime knots with mosaic number 7.
New method proves topological Tverberg problem for all q, not just primes.
problem Establish sufficient conditions for simplicial complexes to map q points to the same point.
method General method that yields results beyond prime powers.
result Proves previously conjectured upper bounds for topological Tverberg problem for all q.
The paper computes CR GJMS operators and CR tractor calculus for pseudo-Einstein forms.
problem Computing CR GJMS operators and CR tractor calculus for pseudo-Einstein forms.
method Algorithm using CR tractors to compute CR GJMS operators, P' operator, and Q' curvature.
result Explicit factorization of CR GJMS operators and P' operator, and constant Q' curvature.
Study of origamis' singularities for groups of prime-power order.
problem Classifying singularities of origamis for groups of prime-power order.
method Geometric and group-theoretic ideas used to classify strata.
result Many groups of prime-power order have only one stratum, but some do not.
Upper bound on Jones polynomials density modulo primes.
problem Density of Jones polynomials modulo prime numbers.
method Derived an upper bound on Jones polynomials density within a large degree range.
result Upper bound on Jones polynomials density modulo primes.
New invariant for prime alternating knots from error-correcting codes
problem Distinguishing prime alternating knots
method Alexander-Briggs code
result New invariant succeeds in separating knots that other invariants fail
We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theor…
The Q-prime curvature is a local invariant of pseudo-Einstein contact forms on integrable strictly pseudoconvex CR manifolds. The transformation law of the Q-prime curvature under scaling is given in terms of a differential operator, called the P-prime operator, acting on the space of CR pluriharmonic functions. …
Computations for prime knots up to 11 crossings.
problem Computing HOMFLY homology for prime knots.
method Direct computations for all prime knots up to 11 crossings.
result HOMFLY homology determined for all prime knots up to 11 crossings.
New tool lassos connects virtual and surface link diagrams, changing primeness rules.
problem Establishing a new correspondence between virtual and surface link diagrams.
method Using a new tool called lassos, establishing a new correspondence between cellular link diagrams on closed surfaces and equivalence classes of virtual link diagrams.
result Traditional primeness for virtual links is stricter than for links in thickened surfaces.
We prove hyperbolicity for links in thickened surfaces.
problem Proving hyperbolicity for links in thickened surfaces.
method Defining fully alternating links and proving their hyperbolicity; extending results to essential surfaces.
result Prime, fully alternating links in SimesI are hyperbolic. Study on Finsler spaces with unique prime geodesics.
problem Estimating the number of orbits of prime closed geodesics in Finsler manifolds.
method Generalizing works on two prime closed geodesics to equivariant situation, studying homogeneous Finsler geometry.
result Closed Finsler manifold with only one orbit of prime closed geodesic is a compact rank-one Riemannian symmetric space when dimension is even or metric is reversible.
Open books link prime positive braids to checkerboard graphs.
problem Determining link types of prime positive braids.
method Associated open books with checkerboard graphs.
result Link type of a prime positive braid closure is determined by the linking graph.
The paper proves a generalized Kauffman-Harary conjecture for prime determinant links.
problem Proving a generalized Kauffman-Harary conjecture for prime determinant links.
method Using Fox colorings and properties of reduced alternating diagrams.
result For every pair of distinct arcs in a prime determinant link, there exists a Fox coloring that distinguishes them.
We show that the Frölicher spectral sequence of a complex parallelizable solvmanifold is degenerate at E2-term. For a semi-direct product $G=\C^{n}\ltimes_φN$ of Lie-groups with lattice Γ=Γ′⋉Γ′′ such that N is a nilpotent Lie-group with a left-invariant complex structure and φ is …
Odd primes act on alternating knots via flypes.
problem Understanding odd order group actions on alternating knots.
method Investigation of flypes between reduced alternating diagrams.
result Odd prime actions on alternating knots are isotopic through a single flype.
Formula for CR curvature in 3D geometry derived.
problem Integral formula for Q′-curvature in 3D CR geometry. method Derivation involving CR Laplacian and Paneitz operator relationships.
result Integral formula for total Q′-curvature. Study L^2-Betti numbers in prime characteristic for a conjecture about 2-complex towers.
problem Verify a conjecture about 2-complex towers using L^2-Betti numbers.
method Systematically study L^2-Betti numbers in zero and prime characteristic.
result Apply L^2-Betti numbers to verify a conjecture about 2-complex towers.
In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an…
The paper finds at least four prime closed characteristics on star-shaped hypersurfaces in 8D space.
problem Finding prime closed characteristics on compact star-shaped hypersurfaces in 8D space.
method Proved existence of at least four prime closed characteristics for non-degenerate C3 compact star-shaped hypersurfaces in R8 without prime closed characteristics of Maslov-type index -1. result Existence of at least four prime closed characteristics on compact star-shaped hypersurfaces in R8. Study counts geodesics on hyperbolic 3-manifolds, proving prime theorems.
problem Counting primitive closed geodesics on compact hyperbolic 3-manifolds.
method Proves prime geodesic theorems with symmetric error terms in length and holonomy.
result Effective equidistribution of holonomy and symmetric error terms.
Decomposes string links in a surface into prime components.
problem Decomposing string links in a surface into prime components.
method Proves a prime decomposition theorem for string links in a thickened surface.
result Any non-braid string link can be uniquely decomposed into prime string links up to braid equivalence.
Hyperbolic knots are not common among prime knots.
problem Proportion of hyperbolic knots among prime knots does not reach 100% as knot complexity increases.
method Analyzing the proportion of hyperbolic knots and satellites among prime knots of increasing complexity.
result The proportion of hyperbolic knots does not converge to 100% as knot complexity increases.
Prime homology detects split links in prime characteristic.
problem Detecting split links in prime characteristic.
method Uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients.
result Reduced sl(P) link homology detects split links in Z/P. pPCA speeds up PCA by priming initial estimates for faster, more accurate results.
problem Improving the speed and accuracy of principal component analysis (PCA).
method pPCA is a two-step algorithm: first, an approximate-PCA method primes the data, then exact PCA is applied in the span of the initial estimate.
result pPCA improves accuracy significantly with a small computational cost, outperforming other methods across various datasets.
Proves rigidity of SU(2) and SO(3) quantum representations at prime levels.
problem Quantum representations of mapping class groups at prime levels.
method Ocneanu rigidity of modular categories and harmonic representatives in Hodge theory.
result Rigidity of SU(2) and SO(3) quantum representations at all prime levels for closed surfaces of genus at least 7.
Researchers create higher-dimensional I′-curvatures and find counterexamples to the Hirachi conjecture.
problem The Hirachi conjecture in higher CR dimensions.
method Constructing higher-dimensional I′-curvatures and analyzing their properties under contact form changes. result Total integrals of I′-curvatures depend on the choice of contact form, providing counterexamples to the Hirachi conjecture. This is the second of a part series devoted to enumerating prime alternating knots and links. In Part I, we introduced four operators on knots and showed that if these operators are applied to the set of all prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossings is obtained. In …
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.
This paper finds all prime alternating knots with minimal warping degree two.
problem Finding knots with minimal warping degree.
method Examined all prime alternating knots and determined those with minimal warping degree two.
result All prime alternating knots with minimal warping degree two were identified.