Positive Thompson links are arborescent tangles.
problem Understanding the structure of Thompson group elements.
method Analyzing closures of bipartite arborescent tangles.
result Positive Thompson links correspond to arborescent tangles.
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48 results for “Thompson links”
Positive Thompson links are proven for oriented subgroup elements.
problem Proving properties of Thompson links with positive elements.
method Analyzing elements of the oriented subgroup of the Thompson group.
result Positive oriented Thompson links are established.
New method associates annular links to elements of Thompson's group T.
problem Associating annular links to elements of Thompson's group T.
method Edge-signed graphs embedded in annuli and unitary representations of T.
result Tait graphs of annular links can be constructed from elements of T.
This paper explores links from Thompson's group conjugacy classes.
problem Understanding the relationship between Thompson's group conjugacy classes and links.
method Using Jones's construction to link elements of to unoriented links.
result Found sequences of elements from distinct conjugacy classes yielding specific links.
New method counts link components from Thompson group elements.
problem Counting connected components of positive Thompson links.
method Defining Thompson permutations from Thompson group elements.
result Number of orbits equals number of link components.
Half grid diagrams prove every link can be represented by a special type of grid diagram.
problem Representing links using grid diagrams and related invariants.
method Defining half grid diagrams and constructing canonical pairs, proving equivalence to Jones' construction, relating to classical link invariants.
result Established a new method to relate the oriented Thompson index to classical link invariants and provided bounds for knot invariants.
Extends Jones' construction to Thompson's group F and link homology.
problem Link homology and Thompson's group F.
method Asymptotic construction of -tangles in Thompson's group F.
result Asymptotically faithful association of -tangles with elements of Thompson's group F.
Classifies isotopy classes of links from Thompson's group F and its subgroup.
problem Classify isotopy classes of links from Thompson's group F and its subgroup.
method Introduced a method to produce links from elements of Thompson's group F and its subgroup.
result Classified isotopy classes of links from Thompson's group F and its subgroup.
Virtual Thompson's groupmath.GT
New virtual version of Thompson's group created to handle virtual knots.
problem Creating a mathematical framework for virtual knots.
method Defining a virtual version of Thompson's group and proving its properties.
result Any virtual link can be constructed from an element of the new virtual Thompson's group.
New -colorable subgroup derived from Thompson's group.
problem Constructing -colorable knots and links from Thompson's group elements.
method Defining and proving isomorphism of -colorable subgroup.
result The -colorable subgroup is isomorphic to a Brown--Thompson group.
Extends Jones' construction to map Thompson group to pointed links.
problem Constructing a map from Thompson group to pointed links.
method Extended Jones' construction, introduced operations, defined new monoid.
result Surjective map from central monoid to linked monoid.
New methods use Conway tangles to generate knots and links.
problem Constructing knots and links using Thompson's group elements.
method Using Conway rational tangles to find Thompson group elements.
result Methods to construct any product or concatenation of simple tangles.
This paper shows all elements in the 3-colorable subgroup of Thompson's group give 3-colorable links.
problem Exploring the relationship between elements in the 3-colorable subgroup of Thompson's group and 3-colorable links.
method Defined the 3-colorable subgroup and used Jones's method to construct knots and links from elements of Thompson's group.
result All elements in the 3-colorable subgroup give 3-colorable links.
Paper proves almost all stabilizer subgroups of Thompson's group satisfy Alexander's theorem.
problem Alexander's theorem for stabilizer subgroups of Thompson's group.
method Defined a method to construct knots and links from Thompson's group F and proved Alexander's theorem for stabilizer subgroups.
result Almost all stabilizer subgroups under the natural action on the unit interval satisfy Alexander's theorem.
In a "naive" attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups T and F for any subfactor. The Thompson group elements are the "local scale transformations" of the theory. In a simple case the coefficients of the representations are polyn…
We review recent developments in the theory of Thompson group representations related to knot theory.
Upper bounds for surface-links in the Yoshikawa table are estimated.
problem Estimating Kirby-Thompson invariants of surface-links.
method Using tri-plane diagrams and L-, L*-invariants.
result Upper bounds for surface-links in the Yoshikawa table are obtained.
Study of new link types and their invariants, extending previous results.
problem Properties of polynomial invariants and signatures of weakly successively almost positive links.
method Analysis of minimal genus and fibering properties, extending known theorems.
result Extension of Scharlemann-Thompson's theorem to weakly successively almost positive links.
In [Jo14] and [Jo18] Vaughan Jones introduced a construction which yields oriented knots and links from elements of the oriented Thompson group . In this paper we prove, by analogy with Alexander's classical theorem establishing that every knot or link can be represented as a closed braid, that given an orient…
New bounds found for complexity of spun knots.
problem Measuring complexity of spun knots.
method Using bridge trisections and distances in the pants complex.
result Bound the Kirby-Thompson invariant of spun knots.
We show how to construct unitary representations of the oriented Thompson group from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of .
The Jones polynomial and functions of positive type on the oriented Jones-Thompson groups and math.GR
The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups and of the Thompson groups and , respectively, together with a procedure that associates an oriented link diagram to any element o…
Wireless communication systems operate in complex time-varying environments. Therefore, selecting the optimal configuration parameters in these systems is a challenging problem. For wireless links, \emph{rate selection} is used to select the optimal data transmission rate that maximizes the link throughput subject to a…
A Finite Time Analysis of Thompson Sampling for Bayesian Optimization with Preferential Feedbackstat.ML
Proposes a TS approach for Bayesian optimization with preferential feedback.
problem Optimizing with preference feedback in complex applications.
method Uses Thompson Sampling with a dueling kernel and anchor invariance.
result Performance matches standard TS for scalar feedback in finite time.
We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In addition, we describe how the potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of t…
The paper verifies stable handleslide triviality of some R-links and shows many are stably equivalent.
problem Stable handleslide triviality of R-links as potential counterexamples to the generalized property R conjecture.
method Implemented an algorithm to construct all R-links explicitly and verified their stable handleslide triviality.
result Many R-links are stably handleslide equivalent.
Bayesian algorithms minimize cumulative regret in decentralized multi-agent bandits.
problem Minimizing cumulative regret in a decentralized multi-agent multi-armed bandit problem.
method Proposed decentralized Bayesian multi-armed bandit framework, including Thompson Sampling and Bayes-UCB algorithms.
result Regret scales logarithmically with constants matching those of an optimal centralized agent.
We address online combinatorial optimization when the player has a prior over the adversary's sequence of losses. In this framework, Russo and Van Roy proposed an information-theoretic analysis of Thompson Sampling based on the information ratio, resulting in optimal worst-case regret bounds. In this paper we introduce…
The paper sets lower bounds for a Kirby-Thompson invariant of 4-manifolds.
problem Determining the Kirby-Thompson invariant of specific 4-manifolds.
method Using trisections, the paper establishes lower bounds and calculates the invariant for specific examples.
result The paper calculates the Kirby-Thompson invariant of the spin of and shows the existence of 4-manifolds with arbitrarily large invariants.
Earlier work with Robert Gompf and Abigail Thompson classified, via a natural slope indexed by the rationals, all two-component links which contain the square knot and from which can be obtained by surgery. It was argued that a certain family of such links probably contradic…
Abby Thompson proved that if a link is in thin position but not in bridge position then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that the if a link is in thin…
Thompson sampling provides a solution to bandit problems in which new observations are allocated to arms with the posterior probability that an arm is optimal. While sometimes easy to implement and asymptotically optimal, Thompson sampling can be computationally demanding in large scale bandit problems, and its perform…
Jones constructs knots from Thompson group elements.
problem No specific problem stated; focuses on constructions.
method Constructions of knots from Thompson group elements.
result Introduced two types of knots: oriented and unoriented.
Optimizes Thompson sampling policies using policy gradient methods.
problem Improving Thompson sampling in bandit problems.
method Applies policy gradient algorithms to optimize Thompson sampling policies.
result Direct policy search on Thompson sampling improves performance.
Virtual Covers of Links IImath.GT
A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links with fibered. These are concordances that restrict to fibered concordances on the first …
This paper proves a conjecture about trisections with a specific length.
problem Proving a conjecture about trisections with a specific length.
method Examining trisections with Kirby-Thompson length 2 and proving the conjecture.
result Proves the conjecture about length 2 trisection being a 4-manifold with length 0.
New method targets vaccines for new variants using Thompson sampling.
problem Deciding how to prioritize vaccines for new variants with delayed feedback.
method Partial likelihood Thompson sampling, updating beliefs with partial likelihood.
result Method effectively prioritizes vaccines based on real-world data.
The tree of knot tunnelsmath.GT
We present a new theory which describes the collection of all tunnels of tunnel number 1 knots in the 3-sphere (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus-2 handlebody and associated structures. It shows that each knot tunnel is obtained from the tu…
Thompson sampling has impressive empirical performance for many multi-armed bandit problems. But current algorithms for Thompson sampling only work for the case of conjugate priors since these algorithms require to infer the posterior, which is often computationally intractable when the prior is not conjugate. In this …
The study provides polynomial bounds for essential surfaces in various 3-manifolds.
problem Bounding the number of isotopy classes of embedded essential surfaces in 3-manifolds.
method Restricting to alternating link complements in 3-sphere, then extending results to other classes of cusped 3-manifolds.
result Explicit polynomial bounds on all embedded essential surfaces in 3-manifolds.
Improved Thompson Sampling reduces regret in contextual bandits and reinforcement learning.
problem Thompson Sampling's exploration is insufficient in some contexts.
method Developed Feel-Good Thompson Sampling to address exploration issues.
result Feel-Good Thompson Sampling reduces regret compared to standard Thompson Sampling.
New Thompson sampling algorithm for stochastic partial monitoring achieves logarithmic regret.
problem Limited feedback in sequential learning problems.
method Developed a novel Thompson-sampling-based algorithm to sample from the posterior distribution exactly.
result Achieved logarithmic regret bound of O(log T) for a linearized variant of the problem.
Brin-Thompson groups have new properties for n>=2.
problem Characterize subgroups of Brin-Thompson groups.
method Analyzing torsion and subgroup structure.
result Brin-Thompson groups for contain continuum many copies of .
New family of braided Thompson groups introduced using recursive braids.
problem Dehornoy-Brin braided Thompson group generalization.
method Using recursive braids and strand diagrams to define new groups.
result New groups are finitely generated if is finitely generated.
Extends Thompson sampling for RL with fewer episodes.
problem Limited episodes in RL settings.
method Batch Bayesian optimization over episodes to learn action bias terms.
result Significantly outperforms standard Thompson sampling.
Thompson sampling, a Bayesian method for balancing exploration and exploitation in bandit problems, has theoretical guarantees and exhibits strong empirical performance in many domains. Traditional Thompson sampling, however, assumes perfect compliance, where an agent's chosen action is treated as the implemented actio…
Study of quasimorphisms and bounded cohomology in braided Thompson groups.
problem Investigate quasimorphisms and bounded cohomology in braided versions of Thompson groups.
method Analyze quasimorphisms and bounded cohomology of various braided Thompson groups.
result Found infinite-dimensional spaces of quasimorphisms in some braided Thompson groups and trivial second bounded cohomology in others.
Thompson sampling used for linear bandits with normal-gamma priors.
problem Optimizing decisions in uncertain environments with linear dependencies and unknown parameters.
method Bayesian Thompson sampling with multivariate normal-gamma priors.
result Derivation of a Bayesian regret bound for the approach.