Algorithm finds knot friends in 3-manifolds.
problem Identifying knots that share similar 3-manifold complements.
method Developed an algorithm using SnapPy and Regina.
result Constructed a census of simple knots with friends.
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30 results for “Snappy”
A counterexample disproves the unknotting number conjecture for a specific knot combination.
problem Disproving the unknotting number conjecture for a particular knot combination.
method Used Snappy to discover and verify a counterexample.
result A counterexample showing disproves the conjecture.
In Dunfield's catalog of the hyperbolic manifolds in the SnapPy census which are complements of L-space knots in , we determine that have tunnel number while the remaining all have tunnel number . Notably, these manifolds contain asymmetric L-space knot complements. Furthermore, using SnapPy a…
Study on hidden symmetries in Dehn fillings of tetrahedral links.
problem Existence of infinite families of knot complements with hidden symmetries.
method Analysis of horoball packings and SnapPy algorithm.
result No infinite families of knot complements with hidden symmetries exist for given tetrahedral links and related manifolds.
Experiments with the Censusmath.GT
In this paper we study the manifolds in the census of "small" 3-manifolds as available in SnapPy. We compare our results with the statistics of random 3-manifolds obtained using the Dunfield Thurston and Rivin models.
The paper verifies no cosmetic surgeries on knots and 3-manifolds using hyperbolic geometry.
problem Checking cosmetic surgeries on knots and 3-manifolds.
method Knot invariants and hyperbolic geometry.
result Verification of no cosmetic surgeries on knots and 3-manifolds.
Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.
problem Characterizing manifolds at infinity of complex hyperbolic orbifolds.
method Spherical CR uniformization and Dehn surgery.
result Specific 3-manifolds at infinity of complex hyperbolic triangle groups are identified.
Machine learning identifies 3-manifold triangulations using isomorphism signatures.
problem Differentiating and classifying 3-manifolds and their Dehn surgeries.
method Training machine learning models on isomorphism signatures derived from 3-manifold triangulations and Pachner graphs.
result Gradient saliency analysis reveals key parts of the language-like encoding scheme.
The loop invariants of Dimofte-Garoufalidis is a formal power series with arithmetically interesting coefficients that conjecturally appears in the asymptotics of the Kashaev invariant of a knot to all orders in . We develop methods implemented in SnapPy that compute the first 6 coefficients of the formal power se…
Quasi-alternating surgeriesmath.GT
The study examines quasi-alternating surgeries on knots and their properties.
problem Understanding quasi-alternating surgeries and their relation to L-space knots.
method Exploration of quasi-alternating surgeries, classification of knots, and analysis of slopes.
result All SnapPy census L-space knots except two admit quasi-alternating surgeries.
The article confirms two quasi-alternating surgeries for 9 asymmetric L-space knots.
problem Understanding quasi-alternating surgeries on asymmetric L-space knots.
method Using the Montesinos trick to confirm known surgeries.
result Confirmation of two quasi-alternating surgeries for each of 9 asymmetric L-space knots.
This paper poses some basic questions about instances (hard to find) of a special problem in 3-manifold topology. "Important though the general concepts and propositions may be with the modern industrious passion for axiomatizing and generalizing has presented us ... nevertheless I am convinced that the special problem…
New infinite family of knots found with unique properties.
problem Finding asymmetric L-space knots with specific properties.
method Generalizing known asymmetric L-space knot t12533 to form an infinite family.
result First infinite family of asymmetric hyperbolic L-space knots of braid index 4.
Paper finds first infinite family of hyperbolic knots with specific properties.
problem Identifying new hyperbolic knots with specific properties.
method Examined SnapPy census and used knot theory to find new infinite family.
result First infinite family of strongly invertible hyperbolic L-space knots with braid index four and tunnel number two.
This paper generalizes octahedral decomposition to links in thickened surfaces.
problem Understanding the geometry of links in thickened surfaces.
method Octahedral decomposition of links in thickened surfaces.
result Nonpositive curvature of the complement and essential-ness of edges proved.
We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial tetrahedral tessellations with at most 25 (orientable case) and 21 (non-orienta…
We describe a model of random links based on random 4-valent maps, which can be sampled due to the work of Schaeffer. We will look at the relationship between the combinatorial information in the diagram and the hyperbolic volume. Specifically, we show that for random alternating diagrams, the expected hyperbolic volum…
Authors find no positive spun triangulations for certain hyperbolic 3-manifolds.
problem Finding positive spun triangulations for hyperbolic 3-manifolds.
method Using Choi's result, they provide examples of closed hyperbolic 3-manifolds and geodesics without positive spun ideal triangulations.
result They provide evidence for the conjecture that Vol3 has no positive spun ideal triangulation for any choice of geodesic.
We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of hyperbolic Platonic manifolds and all of their Platonic tessellations. For the octahedra…
Fixed-parameter tractable computation of Reshetikhin--Turaev knot polynomials via tensor networksmath.GT
New algorithm speeds up knot polynomial calculations.
problem Computing Reshetikhin--Turaev knot polynomials efficiently.
method Fixed-parameter tractable computation via tensor networks.
result Knot polynomial computations are fixed-parameter tractable.
Uniformizes CR structure on a specific 3-manifold.
problem Prove discrete and faithful representation of a complex hyperbolic triangle group.
method Analyzes subgroup properties and applies to CR structure.
result Even subgroup represents a uniformizable spherical CR structure.
We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with triv…
We find braid positive presentations for most L-space knots, except one, and explore related knot properties.
problem Characterizing L-space knots and their braid positivity.
method Examined SnapPy census knots, used HOMFLY polynomial analysis, and investigated Alexander polynomials.
result One L-space knot is not braid positive, and a 1-parameter family of hyperbolic L-space knots might not be.
A single qubit may be represented on the Bloch sphere or similarly on the -sphere . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of -manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-value…
This paper describes a Dehn surgery approach to generating asymmetric hyperbolic manifolds with two distinct lens space fillings. Such manifolds were first identified in work of Dunfield-Hoffman-Licata as the result of a computer search of the SnapPy census, but the current work establishes a topological framework for …
The study calculates and analyzes alternating surgeries for various knots.
problem Identifying and understanding alternating surgeries on knots.
method Algorithmic computation and structural analysis of alternating surgery slopes.
result The set of alternating surgery slopes is algorithmically computable and exhibits interesting phenomena.
3-manifolds are CR uniformized on spheres, proving a conjecture.
problem Uniformizing 3-manifolds with cusps using CR methods.
method Spherical CR uniformization of complex hyperbolic triangle groups.
result Magic 3-manifolds and other cusped 3-manifolds are CR uniformizable.
The study characterizes slopes for hyperbolic knots and Whitehead doubles.
problem Identifying characterizing slopes for hyperbolic knots and Whitehead doubles.
method Combining JSJ decompositions, geodesic lengths, and volume inequalities.
result Explicit conditions for characterizing slopes and examples of non-characterizing slopes.
Computer program finds FAMED triangulations for thousands of knots.
problem Proving the Andersen-Kashaev volume conjecture for many knots.
method Straightforward computer implementation in Regina and Snappy.
result Andersen-Kashaev conjecture proven for over 42,000 knots.
The gluing equations of a cusped hyperbolic 3-manifold are a system of polynomial equations in the shapes of an ideal triangulation $\calT$ of that describe the complete hyperbolic structure of and its deformations. Given a Neumann-Zagier datum (comprising the shapes together with the gluing equations in a …