On Fibonacci Knotsmath.GT
We show that the Conway polynomials of Fibonacci links are Fibonacci polynomials modulo 2. We deduce that, when $ n \not\equiv 0 \Mod 4$ and the Fibonacci knot $ \cF_j^{(n)} $ is not a Lissajous knot.
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48 results for “Fibonacci”
Study hyperbolic geometry to find Fibonacci numbers.
problem Investigate exponential growth and isoperimetric inequality in hyperbolic geometry.
method Combinatorial approximation of hyperbolic plane, calculations.
result Surprising link to Fibonacci numbers.
Fibonacci Ensembles use Fibonacci weights to improve ensemble learning, inspired by natural growth patterns.
problem Improving ensemble learning methods to enhance model performance and interpretability.
method Introduces Fibonacci weights and a recursive ensemble dynamic to reduce variance and enrich representational depth.
result Fibonacci weighting can match or improve upon uniform averaging in ensemble learning experiments.
New proof shows how to fill a square with Fibonacci curve.
problem How to fill a square with a space-filling curve.
method Used Cartesian product of Fibonacci substitution with itself.
result Different proof of space-filling curve construction.
The paper computes groups and modules for wheel graphs using Fibonacci and Chebyshev polynomials.
problem Computing groups and modules for wheel graphs.
method Utilized Fibonacci and Chebyshev polynomials to compute the Reduced Fox Coloring Group and Alexander-Burau-Fox Module.
result Computed groups and modules for wheel graphs using Fibonacci and Chebyshev polynomials.
This paper studies recursive ensembles driven by Fibonacci updates, improving learning dynamics.
problem Improving learning dynamics in recursive ensemble learning.
method Develops second-order recursive architectures with Fibonacci-type update flows.
result Establishes global convergence conditions and generalization bounds for recursive ensembles.
Lee's work on twisted torus knots with Fibonacci parameters is extended to Horadam parameters.
problem Classifying twisted torus knots with Horadam parameters.
method Using recursive Horadam parameters to generalize Lee's work on Fibonacci parameters.
result Families of twisted torus knots with Horadam parameters are provided.
The paper encloses computation of simple centralizer of simple braids and their connection with Fibonacci numbers. Planarity of some commuting graphs is also discussed in the last section.
Odd Fibonacci groups cannot form hyperbolic 3-orbifolds.
problem Characterizing the geometric properties of Fractional Fibonacci groups.
method Analyzing the fundamental groups of orientable hyperbolic 3-orbifolds and using properties of Fibonacci groups.
result For odd , Fractional Fibonacci groups cannot be fundamental groups of orientable hyperbolic 3-orbifolds of finite volume.
We study the cyclic presentations with relators of the form and the groups they define. These "groups of Fibonacci type" were introduced by Johnson and Mawdesley and they generalize the Fibonacci groups and the Sieradski groups . With the exception of two groups, we classify wh…
The study finds new infinite dilogarithm identities related to number sequences and continued fractions.
problem Finding new infinite dilogarithm identities.
method Demonstrating families of identities associated with specific number sequences and continued fractions.
result New infinite dilogarithm identities related to Fibonacci, Lucas numbers, convergents of even period continued fractions, and recurrence relations.
Quantum representations of mapping class groups are locally rigid at prime levels.
problem Locally rigid properties of quantum representations of mapping class groups.
method Proving local rigidity for Fibonacci representations of mapping class groups at prime levels.
result Local rigidity of Fibonacci representations of mapping class groups at prime levels.
Two new feature selection algorithms improve on RFE.
problem Optimal feature selection for faster and more accurate models.
method Fibonacci and k-Subsecting Recursive Feature Elimination.
result Faster feature selection with comparable predictive performance.
Study connects lens spaces' fundamental group to their symplectic fillings' second Betti numbers.
problem Relationship between lens spaces' fundamental group and symplectic fillings' second Betti numbers.
method Exploration of minimal symplectic fillings of lens spaces.
result Unified and generalized results on lens spaces' fundamental group and symplectic fillings' second Betti numbers.
Topological quantum computation with Fibonacci anyons relies on the possibility of efficiently generating unitary transformations upon pseudoparticles braiding. The crucial fact that such set of braids has a dense image in the unitary operations space is well known; in addition, the Solovay-Kitaev algorithm allows to a…
Fibonacci anyons are attractive for use in topological quantum computation because any unitary transformation of their state space can be approximated arbitrarily accurately by braiding. However there is no known braid that entangles two qubits without leaving the space spanned by the two qubits. In other words, there …
We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numb…
We establish two-sided bounds for the complexity of two infinite series of closed orientable 3-dimensional hyperbolic manifolds, the Lobell manifolds and the Fibonacci manifolds.
Bitcoin: A life in crisesq-fin.ST
Study examines Bitcoin's price history and identifies recurring events.
problem Understanding Bitcoin's price fluctuations and recurring events.
method Analyzed BTC price time-series (2010-2021), identified recurring events, and approximated price evolution using a Fibonacci sequence.
result BTC price history shows recurring events with similar duration and can be approximated using a Fibonacci sequence.
Matrix formulas for super Teichmüller spaces generalize previous work and yield super λ-lengths.
problem Calculating super λ-lengths on bordered surfaces with marked points.
method Using holonomy matrices of elements in the supergroup OSp(1|2) to compute super λ-lengths in decorated super Teichmüller spaces.
result Matrix formulas for arcs on bordered surfaces yield super λ-lengths in Penner-Zeitlin's decorated super Teichmüller space.
Introduction 1. The two-eigenvalue problem 2. Hecke algebra representations of braid groups 3. Duality of Jones-Wenzl representations 4. Closed images of Jones-Wenzl sectors 5. Distribution of evaluations of Jones polynomials 6. Fibonacci representations
The paper calculates the number of oriented rational links with a given deficiency.
problem Counting oriented rational links with a specific deficiency.
method Derived precise formulas for the number of oriented rational links with crossing number n and deficiency d.
result Precise formulas for the number of oriented rational links with crossing number n and deficiency d.
New rank 3 distributions with exponentially growing symmetries.
problem Constructing distributions with very large symmetries.
method Using rational normal curves and symplectic flags.
result Symmetry groups grow exponentially in dimension.
The paper proves properties of quantum representations and their Toledo invariants.
problem Proving properties of quantum representations and their Toledo invariants.
method Computing Toledo invariants for specific quantum representations and extending the concept to a series of cohomological invariants.
result The proof of properties of quantum representations and their Toledo invariants, including the computation of the -matrix at first order.
Characterizes unknotted curves on Seifert surfaces of twist knots.
problem Identifying unknotted curves on Seifert surfaces of twist knots.
method Analyzing homologically essential simple closed curves on Seifert surfaces of genus one knots.
result Characterizes unknotted curves on Seifert surfaces of twist knots, including infinitely many for the figure eight knot and one for Whitehead doubles.
3D RadViz improves 3D data visualization of multidimensional datasets.
problem Tackles the challenge of visualizing multidimensional datasets in 3D.
method Develops RadViz3D, a 3D radial visualization tool with uniform anchor points.
result Improves the display of multidimensional datasets, especially for uncorrelated variables.
Turning the skein relation for HOMFLY into a Fibonacci recurrence, we prove that there are only three rational specializations of HOMFLY polynomial: Alexander-Conway, Jones, and a new one. Using the recurrence relation, we find general and relative expansion formulae and rational generating functions for Alexander-Conw…
The paper uses Seshadri constants to construct symplectic ellipsoid embeddings.
problem Constructing symplectic embeddings of ellipsoids.
method Exploring weighted blow-ups and Seshadri constants to demonstrate symplectic embeddings.
result Illustrates constructions of ellipsoid fillings and embeddings.
It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite fami…
The language of maximal lexicographic representatives of elements in the positive braid monoid with generators is a regular language. We describe with great detail the smallest Finite State Automaton accepting such language, and study the proportion of elements of length whose maximal lexicographic repres…
We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched covers of S^3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable.…
New risk models use chaotic attractors to predict extreme events.
problem Predicting Black Swan events in financial markets.
method Combining heavy-tailed priors with chaotic dynamics (Lorenz and Rossler systems).
result Models generate volatility clustering, fat tails, and extreme events.
Suppose C is a singular curve in CP^2 and it is topologically an embedded surface of genus g; such curves are called cuspidal. The singularities of C are cones on knots K_i. We apply Heegaard Floer theory to find new constraints on the sets of knots {K_i} that can arise as the links of singularities of cuspidal curves.…
There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…
The paper explores mapping class group quotients by Dehn twists and their representations.
problem Finite quotients and representations of mapping class groups by powers of Dehn twists.
method Construction of finite quotients using representations with Zariski dense images into semisimple Lie groups, and Long and Moody's method.
result The Fibonacci TQFT representation is a specialization of the Jones representation in genus 2.
New 3-manifold spines with unique Whitehead graphs identified.
problem Identifying new 3-manifold spines with specific Whitehead graph types.
method Cyclic presentations and conditions on parameters to show spines of 3-manifolds.
result Examples of 3-manifolds with Whitehead graphs of specific types.
We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of dihedral quandles satisfies H_3^R(R_p) = Z \oplus Z_p for p odd prime. We also show that H_n^R(R_p) contains Z_p for n>2. Furthermore, we show that the torsion of H_n^R(R_3) is annihilated by 3. We also prove that the quandle homol…
The twin group is a right angled Coxeter group generated by involutions and the pure twin group is the kernel of the natural surjection from onto the symmetric group on symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conj…
This is the third of three papers that refine and extend portions of our earlier preprint, "The depth of a knot tunnel." Together, they rework the entire preprint. In this paper, we use the theory of tunnel number 1 knots that we introduced in "The tree of knot tunnels" to strengthen the Tunnel Leveling Theorem of H. G…
Braiding Majorana Fermionscond-mat.str-el
In this paper we study a Clifford algebra generalization of the quaternions and its relationship with braid group representations related to Majorana fermions. The Fibonacci model for topological quantum computing is based on the fusion rules for a Majorana fermion. Majorana fermions can be seen not only in the structu…
One of the apparent advantages of quantum computers over their classical counterparts is their ability to efficiently contract tensor networks. In this article, we study some implications of this fact in the case of topological tensor networks. The graph underlying these networks is given by the triangulation of a mani…
We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and their complex dimensions. Moreover, we present an explicit volume formula for the…
The depth of a knot tunnelmath.GT
The theory of tunnel number 1 knots detailed in our previous paper, The tree of knot tunnels, provides a non-negative integer invariant called the depth of the tunnel. We give various results related to the depth invariant. Noting that it equals the minimum number of Goda-Scharlemann-Thompson tunnel moves needed to con…
A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrari…
Parrondo's effects with aperiodic protocolsphysics.soc-ph
Study shows aperiodic sequences enhance Parrondo's effect, with Thue-Morse outperforming others.
problem Enhancing Parrondo's effect through strategic switching protocols.
method Investigated Fibonacci, Thue-Morse, and Rudin-Shapiro sequences; analyzed capital correlation and persistence.
result Thue-Morse sequence outperforms other aperiodic sequences and benchmark games in capital gain.
This paper develops a new theory for ensemble learning beyond variance reduction.
problem Ensemble learning's effectiveness for stable estimators is not fully explained by variance reduction.
method Develops a general weighting theory for ensemble learning, formalizing ensembles as linear operators and introducing geometric and spectral constraints.
result Structured weights can outperform uniform averaging by reshaping approximation geometry and redistributing spectral complexity.
Paper bridges matching rules and height functions in aperiodic tilings.
problem Relationship between matching rules and height functions in aperiodic tilings.
method Cochain-first framework to establish equivalence between matching rules, Ammann bar continuity, cycle closure of 1-cochains, and height-function existence.
result Unified framework for aperiodic tilings including Penrose and canonical projection tilings.
FibQuant improves KV-cache compression for long-context inference.
problem Memory traffic bottleneck in long-context inference due to KV cache growth.
method Introduces FibQuant, a universal vector quantizer that combines Beta-quantile radii, Fibonacci/Roberts-Kronecker directions, and Lloyd-Max refinement.
result FibQuant achieves high compression rates with minimal loss in attention cosine similarity.