New Witten rigidity theorems for elliptic genus in various dimensions.
problem Proving rigidity theorems for elliptic genus in different dimensions.
method Combining Liu's and Han-Yu's methods to prove Witten rigidity theorems for elliptic genus in even and odd dimensions.
result Several new Witten rigidity theorems for elliptic genus in even and odd dimensions have been established.
Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister t…
Prove integrality of genus-g indices with adjoint Reidemeister torsions for twist knots and meridians.
problem Prove integrality of genus-g indices with adjoint Reidemeister torsions for twist knots and meridians. method Consider the sum of the adjoint Reidemeister torsions and prove integrality for twist knots and meridians.
result Prove integrality of genus-g indices with adjoint Reidemeister torsions for twist knots and meridians. The twisting number of a ribbon knot is at least as large as its doubly slice genus.
problem Proving a lower bound for the twisting number of ribbon knots in terms of their doubly slice genus.
method Analyzing symmetric unions and tangle replacements to establish the bound.
result Ribbon knots have arbitrarily high twisting numbers, matching their doubly slice genus.
Study on nonorientable 4-genus of double twist knots.
problem Determining the nonorientable 4-genus of double twist knots.
method Explicit constructions and obstructions from Donaldson's diagonalization theorem.
result Proved bounds on nonorientable 4-genus for infinite subfamilies of double twist knots.
Study eigenvalues of curvature operators to annihilate cobordism invariants.
problem Annihilating rational cobordism invariants on spin manifolds.
method Linear inequalities on curvature operator eigenvalues.
result Curvature conditions stabilize to annihilate invariants.
Lower bound on stable 4-genus of knots using Casson-Gordon signatures.
problem Finding a lower bound on the stable 4-genus of knots.
method Using Casson-Gordon τ-signatures to compute the lower bound.
result A twist knot is torsion in the knot concordance group if and only if it has vanishing stable 4-genus.
New method to untangle knots using null-homologous twists.
problem Finding the minimum number of twists to convert a knot to the unknot.
method Using null-homologous twists as a generalization of crossing changes.
result The untwisting number is at most twice the surgery description number plus 1.
Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
problem Finding sharp lower bounds for modular invariants and Dehn twist coefficients.
method Analyzing the relation between fractional Dehn twists and modular invariants, classifying pseudo-periodic maps, and proving rigidity properties.
result Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.
problem Estimating spectral distribution of twisted Laplacian on hyperbolic surfaces.
method Estimate spectral distribution by supremum norm of harmonic form; show small supremum norm for high genus surfaces; prove uniform Weyl law.
result Prove uniform Weyl law for real parts of spectrum on high genus hyperbolic surfaces.
Johnson kernel generated by specific Dehn twists on surfaces.
problem Identifying the generating set of Johnson kernel for surfaces.
method Analyzing Dehn twists on specific types of surfaces.
result Johnson kernel generated by Dehn twists on specified surfaces.
We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.
The paper sets genus bounds for twisted quantum invariants.
problem Bounding the degree of twisted quantum invariants for knots.
method Using Reshetikhin-Turaev construction and Drinfeld doubles.
result Degree of polynomials is bounded by 2g(K)⋅d(H). Every knot can be unknotted with two generalized twists; this was first proved by Ohyama. Here we prove that any knot of genus g can be unknotted with 2g null-homologous twists and that there exist genus g knots that cannot be unknotted with fewer than 2g null-homologous twists.
Presentations for involutions on non-orientable surfaces up to genus 5.
problem Representing involutions on non-orientable surfaces.
method Dehn twist--crosscap slide presentations.
result Presentations for involutions on non-orientable surfaces of genera up to 5.
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.
The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops null-homologous twisting operations as a tool for studying the algebraic genus and, co…
Study satellite knots using bordered Floer theory, proving non-thinness and calculating genus.
problem Properties of twisted Mazur pattern satellite knots.
method Use bordered Floer theory to analyze knots and calculate genus.
result Prove non-thinness of Qn(K) and calculate 3-genus in terms of n and K. We show that on a nonorientable surface of genus at least 7 any power of a Dehn twist is equal to a single commutator in the mapping class group and the same is true, under additional assumptions, for the twist subgroup, and also for the extended mapping class group of an orientable surface of genus at least 3.
New rigidity theorems for spin^c manifolds using modular invariance.
problem Establishing rigidity theorems for twisted Dirac and Toeplitz operators.
method Liu's modular invariance method and its odd-dimensional extension.
result New Witten rigidity theorems for even and odd-dimensional spin^c manifolds.
We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative …
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.
Given a genus-g Heegaard splitting of a 3-sphere, the genus-g Goeritz group is defined to be the group of the isotopy classes of orientation preserving homeomorphism of the 3-sphere that preserve the splitting. In this paper, we determine the twisted first (co)homology group of the genus-2 Goeritz group of 3-sphere.
Generates spin structure stabilizers using Dehn twists.
problem Determining generating sets for spin structure stabilizers.
method Using Dehn twists on specific curves.
result Explicit generating sets for stabilizers of spin structures.
The first aim of this paper is to give four types of examples of surface bundles over surfaces with non-zero signature. The first example is with base genus 2, a prescribed signature, a 0-section and the fiber genus greater than a certain number which depends on the signature. This provides a new upper bound on the min…
Study the first homology group for a specific mapping class group.
problem Calculating the first homology group for a specific mapping class group.
method Determined using coefficients in H1(N;Z) for a non-orientable surface. result Determined the first homology group for the mapping class group of a specific surface.
An excision theorem connects Heegaard Floer homology of 3-manifolds.
problem Establishing a connection between Heegaard Floer homology of related 3-manifolds.
method Using Heegaard Floer homology and excision construction.
result Isomorphic twisted Heegaard Floer homology groups of related 3-manifolds.
New bounds on curve distances on surfaces of arbitrary genus.
problem Understanding distances between curves on surfaces of arbitrary genus.
method Analyzing the action of mapping class groups on curve complexes.
result Dehn twists increase distances between certain curves by at least 4.
There exists a (relatively minimal) genus g Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus h iff g>2 and h>1. The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of tw…
Researchers compute cohomology of mapping class groups with Prym representations, showing instability for large genus.
problem Computing the cohomology of mapping class groups with level structures and Prym representations.
method Using twisted cohomology and Prym representations for any positive integer r.
result Cohomology exhibits instability for large genus, but remains stable for r=0 or r=1.
The study examines Heegaard splittings defined by Dehn twists and finds hyperbolic metrics with specific geodesic lengths.
problem Characterizing Heegaard splittings defined by Dehn twists and their geometric properties.
method Examining Heegaard splittings of genus g≥3 defined by the n-th power of a Dehn twist along a pared acylindrical curve. result For n≥14, the Heegaard splitting has a hyperbolic metric with a closed geodesic of length between 0.7/(n2g2) and 34.3/n2. Study on kernels of SO(3) WRT representations for surfaces of genus g≥3.
problem Determine if the kernel of SO(3) WRT representations is generated by p-th powers of Dehn twists.
method Investigate kernels for different genus and prime p values, showing containment in specific subgroups.
result Kernels are contained in subgroups generated by p-th powers of Dehn twists and other specific elements for certain conditions.
Study of knots with generalized Mazur patterns and their invariants.
problem Understanding the invariants and properties of knots with generalized Mazur patterns.
method Computational analysis of τ and ε invariants for n-twisted satellites. result None of the n-twisted patterns from the family act surjectively on the smooth or rational concordance group. The paper shows knots can have zero support genus in certain spaces.
problem Understanding the support genus of Legendrian knots in specific spaces.
method Using open book decompositions and positive Dehn twists.
result Knots and links can have Legendrian representatives with support genus zero in certain spaces.
D. Margalit and S. Schleimer found examples of roots of the Dehn twist about a nonseparating curve in a closed orientable surface, that is, homeomorphisms whose nth power is isotopic to the Dehn twist. Our main theorem gives elementary number-theoretic conditions that describe the values of n for which an nth root exis…
Two Seifert surfaces of links in S3 are said to be twist equivalent if one can be obtained from the other, up to isotopy, by repeatedly performing operations consisting of cutting along an embedded arc, applying a full twist near one copy of the arc, and re-gluing. By using bridge spheres for their boundary links, w…
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…
Study shows Dehn twist coefficients are consistent across different actions on surfaces.
problem Consistency of fractional Dehn twist coefficients under various actions.
method Analyzes left orderings of mapping class groups and uses cofinality properties.
result Fractional Dehn twist coefficients are independent of the underlying action for surfaces with genus > 1.
Classifies 3-braid knots with maximal 4-genus using McCoy's method.
problem Classifying 3-braid knots with maximal 4-genus.
method McCoy's twisting method and Xu normal form.
result Upper bounds for the topological 4-genus of 3-braid knots.
Let Ng be the non-orientable surface with genus g, MCG(Ng) be the mapping class group of Ng, T(Ng) be the index 2 subgroup generated by all Dehn twists of MCG(Ng). We prove that for odd genus, T(Ng) can be generated by three elements of finite orders.
In this short note, we show that the twisted Alexander polynomial associated to a parabolic SL(2,C)-representation detects genus and fibering of the twist knots. As a corollary, a conjecture of Dunfield, Friedl and Jackson is proved for the hyperbolic twist knots.
We compute the genus zero bridge numbers and give lower bounds on the genus one bridge numbers for a large class of sufficiently generic hyperbolic twisted torus knots. As a result, the bridge spectra of these knots have two gaps which can be chosen to be arbitrarily large, providing the first known examples of hyperbo…
Study singular fibers in genus 2 algebraic fibrations and their monodromy factorizations.
problem Classify singular fibers in genus 2 algebraic fibrations and relate them to Lefschetz fibrations.
method Analyze four families of hypersurface singularities in C^3, determine resolutions, and find flat deformations into simpler pieces.
result Establish a dictionary between configurations of curves and monodromy factorizations for some genus 2 fibrations.
Researchers found that the twist subgroup can be generated by two elements for certain surface genera.
problem Generating the twist subgroup of nonorientable surfaces using minimal elements.
method Using generators and commutators, the researchers determined the minimum number of elements needed to generate the twist subgroup for various surface genera.
result The twist subgroup can be generated by two elements for odd genera g≥27 and even genera g≥42. Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are know…
New infinite class of hyperbolic knots with high genus and generalized torsion found.
problem Finding knots with high genus and generalized torsion.
method Demonstrated through (2, 2q+1)-torus knots and their twist families.
result Infinitely many hyperbolic knots with arbitrarily high genus and generalized torsion.
Paper analyzes double twist knots using adjoint hyperbolic torsion polynomial.
problem Determining the genus and fibering of double twist knots.
method Uses adjoint hyperbolic torsion polynomial to analyze double twist knots.
result The adjoint hyperbolic torsion polynomial determines the genus and fibering of double twist knots.
Quantum Lefschetz theorem by Coates and Givental gives a relationship between the genus 0 Gromov-Witten theory of X and the twisted theory by a line bundle L on X. We prove the convergence of the twisted theory under the assumption that the genus 0 theory for original X converges. As a byproduct, we prove the semi-simp…