A prime geodesic theorem for singular geodesics in a locally symmetric space is proved. As an application, an asymptotic formula for units in number fields is given.
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A prime geodesic theorem is proven for singular geodesics in quotients of SL(4). This is a case where regularity assumptions of previous papers fail. As a consequence, the analysis becomes much more involved. For applications in number theory (class number asymptotics) it is, however, necessary to consider this case, t…
The prime geodesic theorem for regular geodesics in a higher rank locally symmetric space is proved. An application to class numbers is given. The proof relies on a Lefschetz formula that is based on work of Andreas Juhl.
It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders which are maximal at a given set of primes containing an even number of elements …
Counting periodic geodesics of bounded length and commutator structure on hyperbolic surfaces.
problem Counting periodic geodesics with specific commutator structure.
method Reduction to counting critical realizations of trivalent graphs.
result Asymptotic count of geodesics with bounded length and commutator structure.
The paper improves bounds on geodesic lengths and their simplicity on hyperbolic surfaces.
problem Estimating the distribution of geodesic lengths on hyperbolic surfaces.
method Analyzing the moduli space of hyperbolic surfaces with the Weil-Petersson metric.
result Most geodesics of certain lengths are simple and non-separating, confirming a conjecture.
We present an enhanced prime decomposition theorem for knots that gives the isotopy classes of composite knots that can be constructed from a given list of prime factors (allowing for the mirroring and orientation reversing for each factor). Underlying the theorem is an algebraic construction that also allows for the c…
We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their multiplier, and also obtain equidistribution of the associated holonomies; both e…
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…
The absolute Galois group of 3-manifolds determines their structure up to homeomorphism.
problem Determining the structure of 3-manifolds using their absolute Galois groups.
method Defined a relative absolute Galois group for 3-manifolds and used Chebotarev density properties and Hilbert ramification theory.
result Two branched covers of the three-sphere over a stably Chebotarev link are homeomorphic if and only if their absolute Galois groups are isomorphic.
We generalise theorems of Cochran-Lickorish and Owens-Strle to the case of links with more than one component. This enables the use of linking forms on double branched covers, Heegaard Floer correction terms, and Donaldson's diagonalisation theorem to complete the table of unlinking numbers for nonsplit prime links wit…
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.
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
Erdős-Kac theorem applied to geodesics on modular surface.
problem Understanding the distribution of geodesics on modular surfaces.
method Analyzing the number of scattering geodesics with a fixed sojourn time.
result Gaussian behavior for the number of scattering geodesics on modular surface.
Short proofs for complex Tverberg theorems using prime powers.
problem Proving Tverberg-type theorems for cell complexes.
method Short proofs using prime powers and continuous maps.
result Pairwise disjoint faces of a complex intersect under continuous maps.
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.
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 use 3-manifold techniques to illuminate the structure of the string link monoid. In particular, we give a prime decomposition theorem for string links on two components as well as give necessary conditions for string links to commute under the stacking operation.
We use Heegaard Floer homology to obtain bounds on unknotting numbers. This is a generalisation of Ozsvath and Szabo's obstruction to unknotting number one. We determine the unknotting numbers of 9_10, 9_13, 9_35, 9_38, 10_53, 10_101 and 10_120; this completes the table of unknotting numbers for prime knots with crossi…
The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles (arXiv:1702.06462), where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic number 6 or less. In this paper, we extend those results to prime knots with mosaic n…
In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact simply-connected manifold with a bumpy, irreversible Finsler metric and $H^*(M;{\b…
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.
Knots 4_1 and 5_1 can be paired to show unknotting number is not additive.
problem Exploring the additivity of the unknotting number for specific knots.
method Pairing knots and analyzing their connected sums.
result Unknotting number is not additive for knots 4_1 and 5_1.
Neural networks generalize on simple data generated by a programming language.
problem Generalization of neural networks on low complexity data.
method Minimum description length (MDL) approach for feedforward neural networks.
result MDL feedforward neural networks generalize with high probability on simple data.
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…
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.
The study proves all prime knots up to 10 crossings have superbridge index ≤ 5.
problem Determining the maximum superbridge index for prime knots up to 10 crossings.
method New upper bounds on stick numbers and equilateral stick numbers for specific knots, leading to conclusions about superbridge index.
result All prime knots through 10 crossings have a superbridge index ≤ 5.
Study of manifolds with prime cyclic group actions and curvature properties.
problem Curvature properties of manifolds with -actions.
method Analysis of -actions on positively curved manifolds, use of error-correcting codes.
result Improved symmetry-rank bounds for -manifolds with -actions, especially for small primes.
In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By intro…
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…
A Generalization of the Prime Geodesic Theorem to Counting Conjugacy Classes of Free Subgroupsmath.GR
The classical prime geodesic theorem (PGT) gives an asymptotic formula (as tends to infinity) for the number of closed geodesics with length at most on a hyperbolic manifold . Closed geodesics correspond to conjugacy classes of where is a lattice in . The theorem can be rephrased in…
Minimal grid diagrams found for 13-crossing prime knots with 13 arc index.
problem Finding minimal grid diagrams for prime knots with specific crossing and arc indices.
method Used Knotscape to generate spanning trees and obtain minimal arc presentations in grid diagrams.
result 9,988 prime knots with 13 crossings and 13 arc index were identified.
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 for each knot that divides a surface in .
result Identified the doubly slice genera for 2909 prime knots with up to 12 crossings.
This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of view.
We obtain asymptotic counting results with error terms for complex orthospectrum for Schottky groups and orbit counting function for quadratic polynomials. Moreover, we prove equidistribution of holonomy associated to these dynamical systems. Our results are obtained by considering generalized -functions coming from…
The main purpose of this note is the study of the total space of a holomorphic Lie algebroid . 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…
This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant. Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2…
We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one.
This paper is devoted to prove the existence of -periodic alternating projections of prime alternating -periodic knots. The main tool is the Menasco-Thistlethwaite's Flyping theorem. Let be an oriented prime alternating knot that is -periodic with , i.e. admits a symmetry that is a rotation of…
We show that a rank two finite group G admits a finite G-CW-complex X homotopy equivalent to a sphere, with rank one prime power isotropy, if and only if G does not p'-involve Qd(p) for any odd prime p. This follows from a more general theorem which allows us to construct a finite G-CW-complex by gluing together a give…
New proof of link factorization theorem, avoiding case exhaustion.
problem Proving every non-split link can be uniquely factored into prime links.
method Shorter proof using string links and ambient isotopy.
result Demonstrates existence of string links with no local knots.
Characterizes slopes for Markov ordering on prime pairs.
problem Investigating the Markov ordering on relatively prime integer pairs.
method Employing the stable norm on modular torus homology.
result Characterizes slopes for monotonicity of Markov ordering.
New knot polynomials yield simple results modulo primes.
problem Understanding knot polynomials modulo primes.
method Constructing knots with specific properties.
result All polynomials modulo with bounded -span are realizable by knots with bounded braid index.
We consider the question of existence of ramified covers over P_1 matching certain prescribed ramification conditions. This problem has already been faced in a number of papers, but we discuss alternative approaches for an existence proof, involving elliptic curves and universal ramified covers with signature. We also …
New findings on prime theta-curves with simple tangles.
problem Understanding prime theta-curves with specific unknotting numbers.
method Analyzing composite theta-curves and their components.
result Composite theta-curves with unknotting number one are prime.
Ito-Takimura recently defined a splice-unknotting number for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same …
The paper proves that rational concordance of double twist knots is reciprocal.
problem Determining when double twist knots are rationally slice.
method Using Donaldson's diagonalization theorem, the paper constructs rational ball obstructions for non-rational sliceness.
result Infinitely many prime power-fold cyclic branched covers do not bound a rational ball, proving the converse of rational concordance.
The study proves CR structures on specific three-manifolds are equivalent to standard structures.
problem Proving CR structures on three-manifolds are equivalent to standard structures.
method Analyzing Yamabe constant and total -curvature to deduce CR equivalence.
result Closed CR three-manifolds with certain curvature properties are equivalent to standard structures.