A mathematical link between knots and primes formalized.
problem Formalizing the relationship between knots and primes.
method Constructing a functor from 3D manifolds to algebraic number fields using cluster C*-algebras.
result The functor realizes all axioms of the arithmetic topology conjectured in the 1960s.
We developed an efficient algorithm to factorize knots.
problem Computing the prime factorization of knots efficiently.
method Introduced an edge-ideal triangulation to represent knots and developed an algorithm using Regina.
result Our algorithm works well for knots up to 19 crossings and provides new complexity results.
Study introduces dynamical ideals for non-commutative rings and classifies knots and links.
problem Classifying surface knots and links in smooth 4-manifolds.
method Introduced dynamical analog of prime ideals for non-commutative rings and proved a factorization theorem.
result Classified surface knots and links in smooth 4-manifolds.
The Frohman Kania-Bartoszynska ideal is an invariant associated to a 3-manifold with boundary and a prime p >3. We give some estimates of this ideal. We also calculate this invariant for some 3-manifolds constructed by doing surgery on a knot in the complement of another knot.
Proves ideal triangulations of hyperbolic alternating links are non-degenerate.
problem Proving ideal triangulations of hyperbolic alternating links are non-degenerate.
method Using non-positively curved cubings of prime alternating link exteriors.
result Ideal triangulations are non-degenerate, guaranteeing hyperbolicity equations have solutions.
Geometric proof shows primes of form 3k+1 are norms of Eisenstein integers.
problem Geometric proof of primes of form 3k+1 being norms of Eisenstein integers.
method Geometric proof using Penner's λ-length and norms of Eisenstein integers.
result Every prime p of the form 3k+1 is the norm of an Eisenstein integer. We find bases for naturally defined lattices over certain rings of integers in the SU(2)-TQFT-theory modules of surfaces. We consider the TQFT where the Kauffman's A variable is a root of unity of order four times an odd prime. As an application, we show that the Frohman Kania-Bartoszynska ideal invariant for 3-manifol…
The paper shows that arithmetic hyperbolic 3-orbifolds have many non-commensurable pairs with similar geodesic spectra.
problem Understanding the relationship between geodesic length spectra and commensurability of arithmetic hyperbolic 3-orbifolds.
method Using a bounded gaps result for prime ideals in number fields, the paper constructs infinitely many non-commensurable pairs of arithmetic hyperbolic 3-orbifolds with similar geodesic spectra.
result Arithmetic hyperbolic 3-orbifolds can have many non-commensurable pairs with similar geodesic spectra.
New invariant from links to polyhedra volumes.
problem Computing hyperbolic volumes of link complements.
method Geometric, topological, and combinatorial methods to decompose link complements into ideal polyhedra.
result A new geometric link invariant, the right-angled volume, is a lower bound for hyperbolic volume.
The first purpose of this paper is to point out a curious result announced by Macaulay on the Hilbert function of a differential module in his famous book The Algebraic Theory of Modular Systems published in 1916. Indeed, on page 78/79 of this book, Macaulay is saying the following: " A polynomial ideal $\mathfrak{a} \…
Researchers find higher symmetries in symplectic Dirac operator.
problem Understanding symmetries in symplectic Dirac operator.
method Constructing higher symmetry algebra of symplectic Dirac operator Daise0.22ex/s. result Higher symmetry algebra structure corresponds to a completely prime primitive ideal.
This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle c…
Formulates Hilbert reciprocity law on 3-manifolds.
problem Developing arithmetic topology on 3-manifolds.
method Formulated an analogue of Hilbert reciprocity law using intersection forms and Kummer extensions.
result Cyclic covers of links are analogues of Kummer extensions.
Prime Match protects client stock trades from market price manipulation.
problem Protecting client stock trades from market price manipulation.
method Prime Match uses a two-round secure linear comparison protocol to match orders without revealing information.
result Prime Match reduces market impact and maintains client privacy.
We study the action of the Veech group of square-tiled surfaces of genus two on homology. This action defines the homology Veech group which is a subgroup of SL2(OD) where OD is a quadratic order of square discriminant. Extending a result of Weitze-Schmithüsen we show that also the h…
Natural coordinates for SL3-webs on surfaces are shown to be consistent under triangulation changes.
problem Natural coordinates for SL3-webs on surfaces.
method Showed natural coordinates are consistent under triangulation changes using cluster transformations.
result Natural coordinates for SL3-webs on surfaces are consistent under triangulation changes.
Locates entanglement in curves using knot intensity distribution.
problem Finding robust methods for locating entanglement in embedded curves.
method Introducing knot intensity distribution as a local quantifier for entanglement contribution.
result Intensity distributions identify regions in knots accommodating topological changes.
New links are shown to be stably generic based on Chebotarev law.
problem Understanding the relationship between knot analogues and prime ideals.
method Analyzing sequences of knots and their decompositions in 3-manifolds.
result Chebotarev knots form stably generic links.
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…
Uniform mixing proved for hyperbolic manifolds using congruence covers.
problem Proving uniform exponential mixing for hyperbolic manifolds.
method Using Dolgopyat's method and spectral bounds for congruence transfer operators.
result Uniform exponential mixing for congruence covers of hyperbolic manifolds.
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.
New properties established for SO(3) quantum representations, showing density and surjectivity.
problem Properties of SO(3) quantum representations of mapping class groups.
method Analyzing roots of unity and maximal ideals of Z[ζ_p] to establish properties.
result SO(3) quantum representations have dense image and are surjective modulo unramified maximal ideals.
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…
If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its twist number t(D), and show that the volume lies between v_3(t(D) - 2)/2 and v_3(16t(D) - 16), where v_3 is the volume of a…
Study shows Steinberg representation's multiplicity in cohomology of congruence subgroups.
problem Analyzing multiplicity of Steinberg representation in cohomology of congruence subgroups.
method Computation of cohomology of S-arithmetic groups outside a linear range of degrees. result Multiplicity of Steinberg representation is 1 in top-degree cohomology.
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.
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.
New prime knots found with high-genus surfaces.
problem Finding knots with high-genus surfaces.
method Analyzing prime knot complements for meridional essential surfaces.
result Existence of infinitely many prime knots with high-genus surfaces.
The paper proves a free product decomposition for congruence subgroups with constraints.
problem Proving a free product decomposition for congruence subgroups with specific constraints.
method Using the convex hull of the extended Farey sequence and properties of the hyperbolic plane.
result Established bounds and characterizations for minimum denominators in cusp sets.
In this note, we embed the set of all Fricke characters of a free group F -- the set of all characters of representations of F into SL(2,C) -- as an irreducible affine variety V in complex affine space of dimension 2^n-1. Using the Horowitz generating set as the indeterminates, we show that the ideal I of all polynomia…
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.
We develop new algebraic methods refining the Witt group of linking forms and Ranicki's torsion algebraic L-groups into double Witt groups and double L-groups. At each prime ideal of the underlying ring, our double Witt groups capture infinitely many more integral signatures of the linking form than the single Witt gro…
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.
The paper introduces lattice homology for integrally closed submodules and applies it to geometric invariants.
problem Computing numerical invariants of geometric objects.
method Introduces lattice homology for integrally closed submodules and applies it to geometric invariants.
result Well-defined lattice homology associated to quotient modules of type M/N.