A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Al…
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48 results for “Wada representations”
New method computes knot invariants using free group automorphisms.
problem Computing knot invariants efficiently and accurately.
method Using representations of braid groups by automorphisms of a free group.
result Compared isotopic invariants to Alexander polynomials.
In this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Muras…
Generalized knot groups were introduced independently by Kelly (1991) and Wada (1992). We prove that determines the unoriented knot type and sketch a proof of the same for for .
We give a survey of some recent papers by the authors and Masaaki Wada relating the twisted Alexander polynomial with a partial order on the set of prime knots. We also give examples and pose open problems.
Study Alexander matrices for link quandles and their relation to knot invariants.
problem Understanding Alexander matrices for link quandles and their applications to knot invariants.
method Investigate -twisted Alexander matrices and their connection to quandle cocycle invariants.
result Show that -twisted Alexander invariants of knot quandles are stronger than those of knot groups.
We prove that for n>2 there exists a quandle of cyclic type of size n if and only if n is a power of a prime number. This establishes a conjecture of S. Kamada, H. Tamaru and K. Wada. As a corollary, every finite quandle of cyclic type is an Alexander quandle. We also prove that finite doubly transitive quandles are of…
Generalised knot groups distinguish the square and granny knots (with an appendix by David Savitt)math.GT
Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalised knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and…
Core groupsmath.GT
Core groups are link invariants defined by arc or region presentations.
problem Defining link invariants using different presentations of arcs and regions.
method Introducing core groups as link invariants defined by presentations involving arcs or regions, and extending these to virtual link diagrams.
result Properties of core groups and their extensions to virtual link diagrams are discussed.
In this paper, we first develope the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on . WLG is quite sensitive to NMS flows on . For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tool…
On Generalized Knot Groupsmath.GT
Generalized knot groups G_n(K) were introduced first by Wada and Kelly independently. The classical knot group is the first one G_1(K) in this series of finitely presented groups. For each natural number n, G_1(K) is a subgroup of G_n(K) so the generalized knot groups can be thought of as extensions of the classical kn…
New representation theory for closed geodesic subflows.
problem Classifying representations with good geometric properties.
method Restricting to invariant closed geodesic subflows.
result Equivalent characterizations and properties of new representations.
Proves EGF representations in specific geometric contexts.
problem Understanding representations of groups with hyperbolic properties.
method Analyzes projectively convex cocompact manifolds and convex projective manifolds with generalized cusps.
result Holonomy representations of specific geometric manifolds are EGF representations.
The paper establishes isomorphisms and constructs colored versions of Lawrence representations.
problem Understanding isomorphisms and colored versions of Lawrence representations.
method Explicit isomorphisms and construction of colored versions.
result Matrices for colored versions of BKL and Lawrence representations provided.
Study k-positive surface group representations and their degenerations.
problem Understanding the behavior of surface group representations under degenerations.
method Introduced k-positive representations and studied their degenerations using a limit theorem for positively ratioed representations.
result Degenerations of k-positive representations can lead to limits that are at least (k-3)-positive and irreducible limits are (k-1)-positive.
Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groupsmath.DG
Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.
problem Understanding geometrically finite Fuchsian groups and their representations.
method Theory of Anosov representations, type-preserving deformations, limit maps, relative Anosov and dominated representations.
result Cusped Hitchin representations are Borel Anosov, stable under deformations, and limit maps vary analytically.
New representations defined for groups and graphs, with applications to stable representations.
problem Defining and constructing new types of representations for groups and graphs.
method Introducing -directed Anosov representations and using Fock-Goncharov positivity to construct them.
result Constructs large families of primitive stable representations from to , including non-discrete and non-faithful examples.
Collar lemma proven for certain surface group representations.
problem Proving a collar lemma for specific surface group representations.
method Using partial hyperconvexity properties and Anosov representations.
result 'Positivity properties' hold for partially hyperconvex representations.
Study subgroup actions on mapping class groups using Heisenberg representations.
problem Untwisting representations of mapping class groups on Heisenberg subgroups.
method Restrict and analyze twisted representations of mapping class groups to Heisenberg subgroups.
result Untwisting representations on Torelli group for any Heisenberg representation.
Researchers describe unitary representations of mixed braid groups.
problem Understanding unitary representations of mixed braid groups.
method Explicitly describe unitary representations on cohomology of Abelian branched covers.
result Image of the representation is generated by complex reflections and related to the multivariate Burau representation.
This paper addresses law invariant coherent risk measures and their Kusuoka representations. By elaborating the existence of a minimal representation we show that every Kusuoka representation can be reduced to its minimal representation. Uniqueness -- in a sense specified in the paper -- of the risk measure's Kusuoka r…
A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this pape…
Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain …
In this paper, we introduce a study of prolongations of representations of Lie groups. We obtain a faithful (one-to-one) representation of TG where G is a finite-dimensional Lie group and TG is the tangent bundle of G, by using (not necessarily faithful) representations of G. We show that tangent functions of Lie group…
Polynomial representations found in surface braid and mapping class groups.
problem Homological representations of surface braid and mapping class groups.
method Study of homological representation functors and short exact sequences.
result Many homological representation functors are polynomial.
Paper addresses the disparity between sampled and mean representations in disentangled learning.
problem Disparity between sampled and mean representations in disentangled learning.
method Proposes a method to eliminate the disparity by proving and utilizing the relationship between total correlation of sampled and mean representations for multivariate normal distributions.
result Demonstrates that a factorized mean representation can have lower total correlation than the sampled representation.
New -positive representations of surface groups discovered.
problem Generalizing Lusztig's total positivity to surface groups.
method Introducing -positivity and proving properties of -positive representations.
result Discrete and faithful -positive representations exist and form open sets in representation varieties.
New representations for surface groups expand known Anosov classes.
problem Understanding new types of representations for surface groups.
method Introducing and studying simple Anosov representations of closed hyperbolic surface groups.
result Simple Anosov representations strictly contain Anosov representations.
New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in to .
result Cusped Borel Anosov representations with specific properties are Hitchin representations.
The paper formalizes criteria for non-spurious and disentangled representations using causal methods.
problem Formalizing criteria for non-spurious and disentangled representations in representation learning.
method Causal perspective, counterfactual quantities, observable consequences of causal assertions.
result Computable metrics for assessing representation learning based on observed data.
Characterizes Anosov reducible representations in terms of eigenvalues.
problem Understanding Anosov representations in reducible settings.
method Characterizes Anosov representations using eigenvalue magnitudes of irreducible block factors.
result Connected components of character varieties do not contain reducible representations for many non-elementary hyperbolic groups.
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermit…
Study local structure of knot group representations into SL(n,C).
problem Understanding the local structure of knot group representations.
method Analysis of tangent cone and use of Luna's slice theorem.
result Local structure of representation variety at diagonal representations.
We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.
We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that t…
We characterize groups admitting Anosov representations into , projective Anosov representations into , and Borel Anosov representations into . More generally, we obtain bounds on the cohomological dimension of groups admitting -Anosov r…
New method finds Fuchsian representations dominating others in surface group representations.
problem Finding Fuchsian representations that dominate others in surface group representations.
method Straightening the pleated plane and applying strip deformations.
result There exists a Fuchsian representation that strictly dominates a given non-Fuchsian representation.
The paper studies conjugating complex representations into real ones.
problem Understanding representations of surface groups into complex Lie groups.
method Analyzes representations of finitely generated groups into and determines conjugacy conditions.
result Identifies representations in the larger variety that are conjugate in to a representation in .
Survey on stated skein algebras and their representations.
problem None explicitly stated in the abstract.
method None explicitly stated in the abstract.
result None explicitly stated in the abstract.
Develops theory of relatively Anosov representations using flow examples.
problem Understanding relatively Anosov representations.
method Uses a contracting flow on a bundle to define Anosov representations and builds examples.
result Builds families of examples of relatively Anosov representations.
This paper explores the complexity of learning representations in contextual linear bandits.
problem Understanding the complexity of representation learning in contextual linear bandits.
method Systematic approach to representation learning in contextual linear bandits, focusing on instance-dependent perspective.
result Representation learning is fundamentally more complex than linear bandits, with some cases being arbitrarily harder.
New CR representations are found and shown to be redundant.
problem Identifying and classifying CR representations of 3-manifolds.
method Experimental computation of limit sets and exact computations of triangle groups.
result Many CR representations are redundant and conjugate.
In this paper, I introduce weak representations of a Lie groupoid . I also show that there is an equivalence of categories between the categories of 2-term representations up to homotopy and weak representations of . Furthermore, I show that any VB-groupoid is isomorphic to an action groupoid associated to a weak…
We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants of different types. We also study properties of the augmentation representation…
The paper proposes a learning-theoretic perspective on representation alignment.
problem Understanding how AI models' representations become aligned as they scale.
method Reviewing and connecting different notions of alignment, focusing on stitching.
result Relating properties of stitching to kernel alignment of representations.
This paper improves few-shot learning by reducing sample complexity using representation learning.
problem Reducing sample complexity for target tasks with limited data.
method Representation learning to pool all source task samples for target task learning.
result Representation learning can achieve substantial sample size reduction, bypassing the $Ω( rac{1}{T})$ barrier.
This work improves understanding of reinforcement learning state representations.
problem Lack of precise characterization of how and when state representations generalize.
method Developed a bound on the generalization error based on effective dimension.
result Bound quantifies the tension between generalization and approximation.