The paper studies twisted Alexander polynomials for knot groups in various extensions.
problem Understanding twisted Alexander polynomials in knot groups for different extensions.
method Developed mod p formula for twisted Alexander polynomials and studied central extensions.
result Established formulas for twisted Alexander polynomials in knot groups for various extensions.
Characterizes groups of branched twist-spun knots.
problem Understanding the groups of branched twist-spun knots.
method Characterization through 3-manifold groups and conjectural algebraic approach.
result Each group is the group of at most finitely many branched twist spins.
Paper discusses groups where twisted Alexander polynomials vanish.
problem Understanding groups with vanishing twisted Alexander polynomials.
method Introduced and studied twisted Alexander vanishing groups, constructed knots, and analyzed representations.
result Every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to be zero.
Study of Dehn twists in free groups generates right-angled Artin groups.
problem Understanding dynamics of Dehn twists in free groups.
method Geometry of spheres, tori, and curves in a doubled handlebody; analysis of compatibility conditions.
result Sufficiently large powers of Dehn twists generate right-angled Artin groups.
In this article we define the twisted product of groups as the generalization of the semidirect product of groups. We will find the necessary and sufficient condition in order that the twisted product of groups to be a group. In particular, for two copies of the same group, the twisted product of group by itself throug…
Paper proves any twisted link can be described as a unique twisted braid.
problem Understanding twisted links and braids.
method Proved using Alexander and Markov Theorems for classical braids and links.
result Any twisted link can be described as the closure of a unique twisted braid.
Study finds the order of Dehn twists in various groups.
problem Determining the order of Dehn twists in different topological groups.
method Analyzing the specific groups mentioned in the abstract.
result The order of Dehn twists was calculated in the specified groups.
The paper explores the twisted Rokhlin property in mapping class groups of surfaces.
problem Classifying surfaces whose mapping class groups have the twisted Rokhlin property.
method Generalizing the Rokhlin property to the twisted version, the authors classify surfaces based on their mapping class groups' properties.
result The mapping class groups of connected orientable infinite-type surfaces without boundaries have the twisted Rokhlin property, while those of other surfaces do not.
The paper extends knot theory to twisted virtual braids and links.
problem Generalizing knot theory to include twists.
method Introduced twisted virtual braids and proved theorems for twisted links.
result The Alexander and Markov theorems were extended to twisted links.
Study on realizing subgroup twists in 3-manifolds.
problem Realizing subgroups of twist groups in 3-manifolds.
method Analyzing Nielsen realization problem for Twist(M) subgroups, applying to Burnside problem.
result Nontrivial subgroups of Twist(M) are realized by diffeomorphisms if and only if they are cyclic and M is a connected sum of lens spaces.
This paper characterizes a specific type of twisted Artin groups embedded in knot groups.
problem Embedding twisted right-angled Artin groups in knot groups.
method Defined and characterized twisted right-angled Artin groups through mixed graphs and Klein bottle relations.
result Completely determined which twisted right-angled Artin groups can be embedded in knot groups.
Calculates Dehn twist actions on conformal blocks for modular categories.
problem Order of Dehn twists on spaces of conformal blocks.
method Quantum representations of mapping class groups, ribbon twists, monoidal powers.
result Order of Dehn twists generalizes previous results for small quantum group.
Let Gamma be a group generated by two positive multi-twists. We give some sufficient conditions for Gamma to be free or have no `unexpectedly reducible' elements. For a group Gamma generated by two Dehn twists, we classify the elements in Gamma which are multi-twists. As a consequence we are able to list all the lanter…
Study spherical twists on K3 surfaces, compute their centers.
problem Understanding autoequivalence groups of K3 surfaces.
method Introduced spherical twists, studied their intersection numbers, and classified subgroups.
result Computed the center of autoequivalence groups of K3 surfaces.
New insights into cohomology of closed 1-forms.
problem Understanding twisted cohomology of closed 1-forms.
method Construction of examples and analysis of fundamental group representations.
result Non-trivial twisted cohomology of nowhere-vanishing 1-forms.
We study subgroups of the mapping class group of the torus generated by powers generated by powers of Dehn twists. We give a criterion to show when a collection of powers Dehn twists generates a free group using the ping pong lemma. We show that the subgroup generated by three uniform powers of Dehn twists can be eithe…
We classify groups generated by powers of 2 Dehn twists which are 1) free or 2) have no ``unexpected'' reducible elements. We give some sufficient conditions in the case of groups generated by powers of more than two twists.
The paper calculates Alexander polynomials for knots using finite group representations.
problem Calculating Alexander polynomials for knots using specific group representations.
method Defined twisted Alexander polynomials associated with regular representations of finite groups.
result Several formulas for the twisted Alexander polynomial are provided.
We give a small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists. The difference between the number of the generators and a lower bound of numbers of generators for the twist subgroup by Dehn twists is one. The lower bounds is obtained from an argument of Hiros…
Paper distinguishes 2-knots with circle actions using fundamental groups.
problem Distinguishing 2-knots with circle actions.
method Using fundamental groups of 3-orbifolds of cyclic type.
result Proves that most pairs of 1-knots yield distinct branched twist spins.
The paper studies submonoids of singular twisted virtual braids and their properties.
problem Examining submonoids within singular twisted virtual braid monoid.
method Established epimorphisms, identified generators and defining relations, constructed other epimorphisms.
result Identified generators and defining relations for STVPn and other submonoids. Formula for Alexander polynomial of twisted torus knots derived.
problem Calculating Alexander polynomial for a specific class of knots.
method Knot group presentation combined with Fox's calculus.
result Explicit formula for Alexander polynomial of twisted torus knots.
By using a notion of a geometric Dehn twist in ♯k(S2×S1), we prove that when projections of two Z-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the Z-splittings generate a free group of ra…
Accessible groups with infinitely many ends have infinitely many twisted conjugacy classes.
problem Characterizing groups with infinitely many ends and their conjugacy classes.
method Analyzing accessible groups and relatively hyperbolic groups to deduce properties.
result Groups with infinitely many ends have infinitely many twisted conjugacy classes.
The paper explores representations of specific knot groups and their properties.
problem Investigating representations of branched twist spins with a non-trivial center of order 2.
method Analyzes mSL2(Z3)-representations and dihedral group representations of branched twist spins. result Provides sufficient conditions for the existence of mSL2(Z3)-representations and determines the number of dihedral group representations. Researchers study Prym representations for handlebody groups, focusing on cyclic cases.
problem Understanding Prym representations for handlebody groups and their images.
method Restrict Prym representations to handlebody groups and twist groups, focusing on cyclic cases.
result Determined the image of Prym representations in the cyclic case for handlebody groups.
Generators found for automorphisms of special groups.
problem Characterizing automorphisms of special groups.
method Using a novel hierarchical shortening argument, the paper provides generators for automorphisms of special groups.
result Generators for automorphisms of special groups are found, including Dehn twists and pseudo-twists.
Study of decorated surfaces with vortices and their group structures.
problem Understanding group structures of decorated surfaces with vortices.
method Proved isomorphism between cluster braid group, braid twist group, and fundamental group of moduli space.
result Finite presentations of isomorphic groups were given.
Study shows exotic Dehn twists on certain 3-sphere fillings.
problem Extending group actions from boundaries to interiors of 4-manifolds.
method Analyzing Dehn twists on Seifert homology spheres and their fillings.
result Dehn twists on certain fillings are infinite order exotic.
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 f-twisted Alexander matrices and their connection to quandle cocycle invariants. result Show that f-twisted Alexander invariants of knot quandles are stronger than those of knot groups. Paper defines doodles on closed surfaces, unifying classical and virtual theories.
problem Classifying doodles on closed surfaces, especially non-orientable ones.
method Introducing twisted virtual doodles, defining twin groups, and proving Alexander- and Markov-type theorems.
result Unified theory of doodles, showing trivial center and residually finite properties.
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.
Paper uses Long-Moody construction for new braid group representations.
problem Constructing new representations of braid groups.
method Applies Fox derivation to matrix presentation of Long-Moody construction.
result Shows relation to twisted Alexander invariants.
Study on a knot invariant's vanishing order.
problem Understanding the vanishing order of twisted Alexander polynomials.
method Defined and explored properties of the twisted Alexander vanishing order.
result Listed twisted Alexander vanishing groups of order less than 201.
New 2-knots found with same knot group but different quandles.
problem Identifying 2-knots with identical knot groups but distinct quandles.
method Analyzing knot quandles of twist spins.
result First example of 2-knots with same knot group but different quandles.
We introduce a notion of a Fox pairing in a group algebra and use Fox pairings to define automorphisms of the Malcev completions of groups. These automorphisms generalize to the algebraic setting the action of the Dehn twists in the group algebras of the fundamental groups of surfaces. This work is inspired by the Kawa…
If phi: G-->G' is a surjective homomorphism, we prove that the twisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G'. As an application, we show non-existence of surjective homomorphism between certain knot groups.
Twists agrarian and ℓ2-Betti numbers for locally indicable groups.
problem Understanding ℓ2-Betti numbers of locally indicable groups. method Using generalised agrarian invariants and twisted Alexander-Thurston norms.
result Twisted ℓ2-Betti numbers are equal to usual ℓ2-Betti numbers rescaled by the dimension of the twisting representation. Study shows non-spin 4-manifolds where smooth Nielsen realization fails.
problem Existence of non-spin 4-manifolds with non-realizable mapping class groups.
method Investigation of multi-twists, projective twists, and multi-reflections.
result Examples of non-spin 4-manifolds with non-realizable mapping class groups.
Paper examines Dehn twists on non-orientable surfaces and their limitations.
problem Limitations of generating Dehn twists on non-orientable surfaces.
method Analyzes the level 2 mapping class group of non-orientable surfaces and their subgroups.
result Dehn twist subgroup of M2(Ng) cannot be generated by squares of Dehn twists about non-separating curves. We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove t…
It is well known that any knot group is torsion-free, but it may admit a generalized torsion element. We show that the knot group of any negative twist knot admits a generalized torsion element. This is a generalization of the same claim for the knot 52, which is the (−2)-twist knot, by Naylor and Rolfsen.
New relation found in 4D symplectic mapping class group.
problem Relation between Dehn twists in symplectic 4-manifolds.
method Holomorphic curve techniques, symplectic isotopy problem solution.
result Relation between two products of Dehn twists.
In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory o…
Defines a new knot invariant and studies its properties.
problem Understanding the structure of knot invariants.
method Defining and analyzing the twisted Alexander vanishing order.
result Characterizes knots with a zero-twisted Alexander polynomial.
The paper explores methods to decompose periodic maps into Dehn twists.
problem Factoring periodic maps into Dehn twists.
method Various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy.
result Existence of conjugates of periodic maps whose product is pseudo-Anosov.
Twisted Alexander invariants have been defined for any knot and linear representation of its group. The invariants are generalized for any periodic representation of the commutator subgroup of the knot group. Properties of the new twisted invariants are given. Under suitable hypotheses, reciprocality and bounds on the …
Solves generalized twisted rabbit problems for higher degree polynomials.
problem When a quadratic polynomial is twisted by a cyclic subgroup, what polynomial is equivalent?
method Uses d2-adic expansion instead of 4-adic for higher degree polynomials. result Provides a solution that depends on the d2-adic expansion of the power of the mapping class element.