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. 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.
New families of twisted torus knots found with essential surfaces.
problem Whether all twisted torus knots have essential tori.
method Analyzing sequences of twists on torus knots.
result Found two new families of toroidal twisted torus knots.
The paper classifies twisted torus links that are unlinks.
problem Identifying conditions for twisted torus links to be unlinks.
method Characterization of parameter families (p,q,r,s) for unlinking twisted torus links. result Complete characterization of parameter families for unlinking twisted torus links.
New examples of positive twisted torus knots with same surface slope found.
problem Finding new examples of positive twisted torus knots with same surface slope.
method Extending Dean's conditions and using new four-parameter family.
result Four-parameter family of positive twisted torus knots with same surface slope.
Lee's work on twisted torus knots with Fibonacci parameters is extended to Horadam parameters.
problem Classifying twisted torus knots with Horadam parameters.
method Using recursive Horadam parameters to generalize Lee's work on Fibonacci parameters.
result Families of twisted torus knots with Horadam parameters are provided.
Study finds infinite non-fibered twisted torus knots.
problem Identifying non-fibered twisted torus knots.
method Explicit formula for Alexander polynomial, leading coefficients analysis.
result Infinite families of non-fibered twisted torus knots found.
New method calculates number of components in twisted torus links.
problem Determining the number of components of twisted torus links.
method Euclidean algorithm-like procedure based on parameters.
result Number of components is a multiple of gcd(p, q, r, s).
Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their …
Jones polynomial for twisted torus knots is trivial if and only if the knot is trivial.
problem Computing Jones polynomial for a specific family of links.
method Computed Jones polynomial for T((p,q),(2,s)) links, focusing on s=2n. result Jones polynomial is trivial if and only if the knot is trivial.
We give a necessary condition for a torus knot to be untied by a single twisting. By using this result, we give infinitely many torus knots that cannot be untied by a single twisting.
Formula for twisted Alexander polynomials of torus links.
problem Computing twisted Alexander polynomials for torus links.
method Explicit formula derivation and character variety analysis.
result Locally constant function on character variety.
The twisted torus knots lie on the standard genus 2 Heegaard surface for S3, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted tor…
Abstract: Generalized full twist formula for superpolynomials of torus knots.
problem Finding a formula for superpolynomials of torus knots.
method Using Mellit's results, generalized full twist formula for superpolynomials of positive toric braids.
result Generalized full twist formula for superpolynomials of torus knots.
Legendrian products can be twist spuns under certain conditions.
problem Understanding when Legendrian products are twist spuns.
method Using Legendrian contact homology and augmentation variety analysis.
result Not all Legendrian products are twist spuns.
Paper calculates the ribbonlength of twisted torus knots.
problem Determining the ribbonlength of twisted torus knots.
method Analyzes the ribbonlength of twisted torus knots Tp,q;r,s and provides upper bounds. result Ribbonlength of Tp,q;r,s is bounded by 2(max{p,q,r}+∣s∣r) and 2(p+(∣s∣−1)r) under specific conditions. New findings on T-links derived from torus links.
problem Difficulty in determining geometric type of T-links.
method Examined T-links obtained by full twists along torus links.
result T-links obtained by full twists are not torus links.
New definition of twisted 1-loop invariant using Ptolemy coordinates.
problem Defining and proving properties of twisted 1-loop invariants.
method Alternative definition via Jacobian of Ptolemy coordinates.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial for hyperbolic once-punctured torus bundles.
In the present note, we will show that there are infinitely many composite twisted torus knots.
We show that twisted torus knots T(p,q,3,s) are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.
Study calculates torsion for iterated torus knots using complex representations.
problem Calculating torsion for complex representations of iterated torus knots.
method Using complex special linear group representations of fundamental groups.
result Twisted Reidemeister torsions appear in colored Jones polynomial asymptotics.
Simplified expressions for torus relations found.
problem Finding simplified expressions for torus relations.
method Explicit factorization of boundary multi-twists as products of Dehn twists.
result Simplified expressions for the k-holed torus relations. Formula found for braid index of n-bridge braids.
problem Finding a formula for the braid index of n-bridge braids. method Elementary, effective, self-contained proof.
result Closed form formula for braid index of n-bridge braids. Study Alexander polynomials of links in 3-torus.
problem Investigate Alexander polynomials of links in 3-torus.
method Diagrammatic approach, Reidemeister moves, fundamental group, homology group, Alexander polynomials, twisted Alexander polynomials.
result Computed Alexander and twisted Alexander polynomials of links in 3-torus.
The paper calculates knot invariants using Blanchfield forms and obstructs sliceness.
problem Computing and obstructing the sliceness of knots.
method Algorithmic computation of twisted signature invariants using twisted Blanchfield forms and satellite formulas.
result Illustrated algorithm for (2,q)-torus knots and obstruction of sliceness for certain knots. The study examines subgroups of torus mapping class group generated by Dehn twists powers.
problem Characterizing subgroups generated by powers of Dehn twists.
method Using the ping pong lemma and geometric intersection numbers.
result Subgroups can be free groups, direct products, or have specific ranks.
In the present paper, we will show that for any integer n>0 there are infinitely many twisted torus knots with n-string essential tangle decompositions.
The paper conjectures Khovanov homology can distinguish torus and twist knots.
problem Detecting and distinguishing knots using Khovanov homology.
method Examining all prime knots with up to 20 crossings, conjecturing Legendrian simplicity.
result Numerical evidence supports Khovanov homology distinguishing torus and twist knots.
We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in \cite{Du} and a formula for the nonabelian Reidemeister torsion of twist knots that is better than…
Generalizes Hecke algebra for double torus, linking to skein algebra.
problem Understanding algebraic structures on double torus.
method Introducing Heegaard dual operators and Dehn twists.
result Established relationship between Hecke algebra and skein algebra.
Study shows certain knots can be surgically altered to create non-left-orderable 3-manifolds.
problem Understanding non-left-orderable fundamental groups in 3-manifolds.
method Analyzing surgeries on negatively twisted torus knots.
result Certain knots can be surgically altered to create 3-manifolds with non-left-orderable fundamental groups.
The paper certifies projective rigidity for once-punctured torus bundles using twisted Alexander polynomials.
problem Certifying infinitesimal projective rigidity for hyperbolic once-punctured torus bundles.
method Using twisted Alexander polynomials of representations associated with the holonomy.
result The induced action on the tangent space of the character variety matches the group theoretic action.
Explicit matrix presentations of Blanchfield pairings and twisted pairings for torus knots.
problem Computing explicit matrix presentations of Blanchfield and twisted Blanchfield pairings for torus knots.
method Using a taut identity to construct a chain complex with few generators, and describing the twisted Alexander module.
result Explicit matrix presentations of the Blanchfield pairing and twisted pairings for (m,n)-torus knots. The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.
problem Proving the non-triviality of welded knots and ribbon torus-knots.
method By generating examples and determining the fundamental group of the concerned welded knot.
result Non-triviality of welded knots and ribbon torus-knots is demonstrated.
Sharp signature bound for 3-strand torus knots proved.
problem Determining the topological 4-genus of 3-strand torus knots.
method Using McCoy's twisting method and improving upper bounds.
result Signature bound is sharp for 3-strand knots and off by at most 1 for 4- and 6-strand knots.
We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficien…
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.
Geometrically, twist numbers on punctured tori are dense and non-continuous.
problem Understanding twist numbers on hyperbolic punctured tori.
method Hyperbolic geometry and Farey graph analysis.
result The graph of twist numbers is dense in [0,1]x[0,1].
Explains connections between monopoles and modules on elliptic curves.
problem Understanding relationships between different mathematical objects.
method Explains equivalences between monopoles and polystable bundles and modules.
result Monopoles and polystable difference modules on elliptic curves are equivalent.
Given a torus bundle Y over the circle and a cohomology class [ω]∈H2(Y;Z) which evaluates nontrivially on the fiber, we compute the Heegaard Floer homology of Y with twisted coefficients in the universal Novikov ring.
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…
New algebra for twice-punctured torus curves.
problem Constructing a new algebra for skein theory.
method Using Heegaard dual of Iwahori--Hecke operator, Dehn twists are represented.
result Automorphisms correspond to Dehn twists on the twice-punctured torus.
The paper classifies hyperbolic and satellite T-links formed by twisting.
problem Classifying hyperbolic and satellite T-links formed by twisting.
method Classification through Dehn filling and analysis of parent links.
result Classification of hyperbolic and satellite T-links formed by twisting.
We explain the effect of applying a full Lutz twist along a pre-Lagrangian torus in a contact 3-manifold, on the contact invariant in Heegaard Floer homology with twisted coefficients.
Study of knot polynomials for twist satellites, generalizing cabling.
problem Lifting decomposition rule to superpolynomials for positive and negative twistings.
method General decomposition of satellite's colored HOMFLY polynomial in terms of original knot's contributions.
result Knot polynomials for twist satellites are related to Vogel's universality.
Classifies small links in an unmarked solid torus.
problem Classifying knots and links in an unmarked solid torus.
method Invariants and Dehn twists to detect and transform links.
result Classification of all non-split links up to 6 crossings.
Fintushel-Stern surgery on fibered knots in torus bundles yields symplectic manifolds.
problem Understanding when knot surgery in torus bundles results in symplectic manifolds.
method Seiberg-Witten theory and twisted Alexander polynomials.
result Symplectic structures in XK if and only if K is a fibered knot.