Study on unknotting twisted knots using arc shift and region arc shift moves.
problem Unknotting twisted knots and finding bounds for region arc shift number.
method Introduced arc shift move and region arc shift move for twisted knots.
result Found families of twisted knots with specific arc shift and region arc shift numbers.
We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinat…
We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of q-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of …
We show that for a large class of hyperbolic knots and links, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions…
In this paper, we consider domino tilings of regions of the form D×[0,n], where D is a simply connected planar region and n∈N. It turns out that, in nontrivial examples, the set of such tilings is not connected by flips, i.e., the local move performed by removing two adjace…
New method proves uniqueness of even plats diagrams.
problem Proving uniqueness of even plats diagrams.
method New arguments and restrictions on twist regions.
result Minimal width even plats diagrams are unique.
Improves inference-time alignment for diffusion models without updating weights.
problem Aligning diffusion models without updating weights for high-reward outputs.
method Trust-Region Iterative Twisted Sequential Monte Carlo (TRI-TSMC) for variance reduction and efficiency.
result Improves primary alignment objectives on text generation tasks.
Special knots with many twists have no certain type of surgery.
problem Proving certain knots have no chirally cosmetic surgeries.
method Analyzing the number of twist regions and using invariants to bound surgeries.
result Special alternating knots with more than 63 twist regions have no chirally cosmetic surgeries.
In this thesis, we consider domino tilings of three-dimensional regions, especially those of the form D×[0,N]. In particular, we investigate the connected components of the space of tilings of such regions by flips, the local move performed by removing two adjacent dominoes and placing them back in t…
New families of knots with more straight segments than crossings.
problem Understanding knots with more straight segments than crossings.
method Presented two families of knots, computed straight number for one, and provided a theorem about alternating knots.
result Explicit computation of straight number for one family of knots and a general theorem about alternating knots.
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 …
New proof shows certain knots are hyperbolic.
problem Characterizing knots with highly twisted diagrams.
method New approach involving Euler characteristic argument.
result Complements of certain knots are unannular and atoroidal.
A horospherical torus about a cusp of a hyperbolic manifold inherits a Euclidean similarity structure, called a cusp shape. We bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp. The bounds are in terms of the change in structure in a neighb…
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.
New formulas for colored Jones polynomials of double twist knots and related series.
problem Calculating colored Jones polynomials and related series for double twist knots.
method Utilized Takata's result and Bailey pairs, along with Walsh's formulas.
result Found new families of q-hypergeometric series generalizing the Kontsevich-Zagier series. Proves geodesic connections on 2-torus without invariant tori.
problem Existence of geodesic connections on 2-torus without invariant tori.
method Uses J. Mather's result on connecting orbits for monotone twist maps.
result Proves existence of connecting geodesics on unit tangent bundle of 2-torus.
We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibere…
The paper studies symmetries and hidden symmetries of twisted knot complements.
problem Analyzing symmetries and hidden symmetries of (ε,dL)-twisted knot complements. method Analysis via long Dehn fillings on fully augmented links complements.
result No hidden symmetries in (ε,dL)-twisted knot complements, implying at most two commensurability classes. Knot invariants and quiver stability linked through full twists.
problem Relating knot invariants to quiver stability under twists.
method Using HOMFLY-PT skein relations and linking/unlinking operations on symmetric quivers.
result Full twists on knots correspond to unlinking or linking of augmented symmetric quivers, confirming stable growth in both.
Tanaka shows amphichiral symmetric unions of the unknot are trivial.
problem Understanding amphichiral symmetric unions and their Jones polynomials.
method Analyzing the Jones polynomial of amphichiral symmetric unions of the unknot and generalizing to other knots.
result Amphichiral symmetric unions of any knot with one twist region are trivial.
Paper proves unique canonical form for certain highly twisted knots and links.
problem Classifying knots and links with specific plat projections.
method Analyzes 2m-plat projections with specific constraints on crossings and heights. result Unique canonical form for certain knots and links with plat projections.
An axis of a link projection is a closed curve which lies symmetrically on each region of the link projection. In this paper we define axis systems of link projections and characterize axis systems of the standard projections of twist knots.
Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface in…
Paper proves zero stability for one-row colored sl₃-Jones polynomials.
problem Stability of coefficients in colored Jones polynomials.
method Linear skein theory based on Kuperberg's sl₃-webs.
result Zero stability for B-adequate links with anti-parallel twist regions.
Study on Fox's trapezoidal conjecture for specific alternating links.
problem Investigating Fox's trapezoidal conjecture for alternating links.
method Diagrammatic Murasugi sums, Alexander polynomial, and concordance analysis.
result Established inequalities and conditions for the Alexander polynomial and trapezoidal conjecture.
The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to gr…
Quantum approach to volume computation from colored Jones polynomials.
problem Computing volumes of hyperbolic 3-manifolds from knot polynomials.
method Categorification of Jones polynomials and asymptotic analysis of skein elements.
result Asymptotic growth rate of Kauffman bracket relates to volumes of ideal octahedra.
4-manifolds with nonnegative sectional curvature are area-extremal.
problem Finding extremal properties of 4-manifolds with curvature constraints.
method Analyzing sections in the kernel of a twisted Dirac operator and using the Finsler--Thorpe trick.
result Large classes of compact 4-manifolds are area-extremal.
A short proof for a theorem about composite knots.
problem Proving a theorem about composite knots with symmetric union presentations.
method Presenting a concise proof of Tanaka's theorem.
result Composite knots with symmetric union presentations have non-trivial connected summands.
Explicitly describes pseudospherical helicoids and their topological embeddings.
problem Solving a problem posed by Popov regarding pseudospherical surfaces.
method Explicitly describes pseudospherical helicoids in terms of elliptic functions.
result Countably many continuous families of topologically embedded pseudospherical helicoids constructed.
Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.
problem Spectral gap for convex cocompact hyperbolic surfaces and their covers.
method Using thermodynamic formalism for twisted Selberg zeta functions.
result Uniform resonance-free regions for convex cocompact hyperbolic surfaces and expanders.
Study of naked singularities in Einstein vacuum equations using new self-similarity.
problem Mathematical study of naked singularities in the Einstein vacuum equations.
method Introduction of new self-similarity and geometric twisting for singularity formation.
result Construction of solutions corresponding to the exterior region of a naked singularity.
In this paper, we determine geometric information on slope lengths of a large class of knots in the 3-sphere, based only on diagrammatical properties of the knots. In particular, we show such knots have meridian length strictly less than 4, and we find infinitely many families with meridian length approaching 4 from be…
Study links in 3-manifolds, linking volume to polynomial coefficients.
problem Understanding the volume of hyperbolic links in 3-manifolds.
method Using Kauffman bracket functions and polynomial invariants, linking volume to polynomial coefficients.
result Coefficients of polynomial provide 2-sided linear bounds on the volume of hyperbolic links.
In the present paper we provide new examples of marginally trapped surfaces and tubes in FLRW spacetimes by using a basic relation between these objects and CMC surfaces in 3-manifolds. We also provide a new method to construct marginally trapped surfaces in closed FLRW spacetimes, which is based on the classical Hopf …
Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number c grows exponentially with c, and to date computer-assisted proofs can only classify diagrams up to around twenty crossings. Instead, we consider diagrams enumerated by bridge number, following the lead of S…
The paper discusses knot colorings and their invariants using Goeritz matrices.
problem Distinguishing knots using coloring methods.
method Elementary approach to equivalence between coloring and Goeritz matrices.
result Computing knot determinant and nullity of pretzel knots.
In this paper, we show that the volumes for a family of A-adequate closed braids can be bounded above and below in terms of the twist number, the number of braid strings, and a quantity that can be read from the combinatorics of a given closed braid diagram. We also show that the volumes for many of these closed braids…
We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds wit…
New infinite families of twisted torus knots found.
problem Identifying new types of twisted torus knots.
method Finding new infinite families of twisted torus knots with a single negative twist.
result Eight new infinite families of twisted torus knots are discovered.
The study finds hyperbolic twisted torus links for certain twists.
problem Identifying hyperbolic twisted torus links.
method Analyzing (p,q)-torus links with r strands twisted s times. result All hyperbolic twisted torus links are identified for ∣s∣>3. The paper introduces a method to deform submanifolds in Euclidean space using Drinfel'd twists.
problem Constructing noncommutative deformations of submanifolds in Euclidean space.
method Using Drinfel'd twist deformation of differential geometry, the paper proposes a general procedure to construct noncommutative deformations of an embedded submanifold.
result The method allows for consistent projection of connections and can be applied to various submanifolds like cylinders and hyperboloids.
Paper constructs Chern character for higher twists and shows isomorphism between K-theory and cohomology.
problem Mapping higher twisted K-theory to higher twisted cohomology.
method Constructing Chern character for higher twists and showing isomorphism.
result Chern character gives isomorphism between higher twisted K-theory and higher twisted cohomology.
New algebraic structure for distinguishing twisted virtual handlebody-links.
problem Distinguishing twisted virtual handlebody-links.
method Introducing twisted virtual bikeigebras and using them to define invariants.
result New invariants distinguish some twisted virtual handlebody-links.
The paper studies Gluck twists on 2-knots with periodic monodromy.
problem Determining the resulting 2-knot after a Gluck twist on a 2-knot with periodic monodromy.
method Analyzes the Gluck twist on branched twist spins of 2-knots.
result Provides infinitely many pairs of inequivalent branched twist spins with homeomorphic complements.
Construct noncommutative deformations of algebraic submanifolds in R^n.
problem Deforming algebraic submanifolds in noncommutative geometry.
method Using twisted differential geometry and Drinfel'd twists, constructing noncommutative deformations of algebraic submanifolds.
result Explicitly worked out deformations of quadrics in R^3.
The paper studies the twisted Calabi flow on Kähler manifolds.
problem Analyzing the behavior of the twisted Calabi flow on compact Kähler manifolds.
method Establishing convexity, proving short-time existence, and demonstrating stability of the flow.
result The stability of the twisted Calabi flow near twisted constant scalar curvature Kähler metrics.
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.