A combinatorial proof of the Gordon Conjecture: The sum of two Heegaard splittings is stabilized if and only if one of the two summands is stabilized.
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Dehn surgery homeomorphic pairs contradict a conjecture.
problem Understanding homeomorphic outcomes of Dehn surgeries on knots.
method Analyzing rational surgeries on knots to find counterexamples.
result Dehn surgery functions are never injective.
This paper proves the Boyer-Gordon-Watson conjecture for 1-bridge braids.
problem Proving the Boyer-Gordon-Watson conjecture for a specific class of knots.
method Calculated knot groups and peripheral subgroups, verified the conjecture using a developed criterion.
result Proved the Boyer-Gordon-Watson conjecture for three families of 1-bridge braids.
It has been recently conjectured by Boyer-Gordon-Watson that a closed, orientable, irreducible -manifold is a Heegaard Floer -space if and only if is not left-orderable. In this article, we study this conjecture from the point of view of lattice cohomology, an invariant introduced by Némethi which is…
The Kashaev conjecture is proven for classical signatures and Alexander polynomials of links.
problem Proving the Kashaev conjecture for signatures and Alexander polynomials.
method Relating Kashaev's matrix to Gordon-Litherland's work and Kauffman's model.
result Proven Alexander polynomial and classical signature parts of the conjecture for arbitrary links, and full conjecture for definite knots.
Every knot has infinitely many characterising slopes.
problem Characterizing slopes for knots in 3-sphere.
method Analyzing surgeries on knots and their homeomorphisms.
result Every knot has infinitely many characterising slopes.
C. Gordon conjectured that a connected sum of two Heegaard splittings is stabilized if and only if one of the two factors is stabilized (Problem 3.91 in Kirby's problem list). In this paper, we shall prove this conjecture.
The paper disproves a conjecture about satellite maps not inducing homomorphisms.
problem Satellite maps and their impact on knot concordance groups.
method Casson-Gordon signatures and -solvable filtration analysis.
result Examples of satellite maps that act like homomorphisms but do not induce them.
Cable knots can't have homeomorphic surgeries.
problem Cosmetic surgeries on non-trivial knots.
method Analyzing cable knots specifically.
result Cable knots do not admit purely cosmetic surgeries.
We determine the smooth concordance order of the 3-stranded pretzel knots P(p,q,r) with p,q,r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained by Fintushel-Stern and Casson-Go…
In this paper, we prove a conjecture of Friedl and Powell that their Casson-Gordon type invariant of 2-component link with linking number one is actually an obstruction to being height 3.5 Whitney tower/grope concordant to the Hopf Link. The proof employs the notion of solvable cobordism of 3-manifolds with boundary, w…
We show that any exceptional non-trivial Dehn surgery on a hyperbolic two-bridge knot yields a 3-manifold whose fundamental group is left-orderable. This gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.
Kähler Solvmanifoldsmath.DG
Compact Kähler solvmanifolds are classified up to biholomorphism. A proof of a conjecture Benson and Gordon, that completely solvable compact Kähler solvmanifolds are tori is deduced from this. The main ingredient in the proof is a restriction theorem for polycyclic Kähler groups proved by Nori and the author.
The study connects Dehn surgeries on links to trisections of 4-manifolds.
problem Understanding Dehn surgeries on links via trisections.
method Expanding known connections and introducing new trisection conditions.
result New trisection conditions can obstruct reducibility of trisections.
Minimal surfaces in AdS(4) correspond to elliptic solutions of the cosh-Gordon equation.
problem Minimal surfaces in AdS(4) and their relation to entanglement entropy.
method Inverting Pohlmeyer reduction to construct static minimal surfaces in AdS(4) that correspond to elliptic solutions of the reduced system.
result A two-parameter family of static minimal surfaces in AdS(4) that include helicoids and catenoids as special limits.
Research confirms non-trivial knots in S^3 do not admit purely cosmetic surgeries.
problem Non-trivial knots in S^3 do not admit purely cosmetic surgeries.
method Analysis of JSJ-structures to confirm Gordon's conjecture.
result Connected sums of knots do not admit purely cosmetic surgeries.
We prove two conjectures of C. Gordon. We show that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10, and that the maximal intersection number between exceptional slopes is 8. The proof uses a combination of new geometric techniques and a rigorous computer-assisted calculation.
New solutions found for elliptic sinh-Gordon and sine-Gordon equations.
problem Elliptic sinh-Gordon and sine-Gordon equations on the real plane.
method Backlund transformation connecting the equations.
result New families of solutions introduced.
New foliations show knot meridians are detectable.
problem Detecting knots in 3-manifolds.
method Constructing co-oriented taut foliations intersecting knot meridians.
result Evidence supports conjecture related to L-space conjecture.
We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil, yields a 3-manifold whose fundamental group is left-orderable. This is a generalization of a result of Clay, Lidman and Watson, and also gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.
The virtual flyping theoremmath.GT
The flyping theorem is extended to virtual links and surfaces.
problem Proving the flyping theorem for virtual links and surfaces.
method Adapting geometric proof for thickened surfaces and using diagrammatic correspondence.
result Established an isomorphism between Gordon-Litherland pairing and 4-manifold intersection form.
We show that the resulting manifold by -surgery on a large class of two-bridge knots has left-orderable fundamental group if the slope satisfies certain conditions. This result gives a supporting evidence to a conjecture of Boyer, Gordon and Watson that relates -spaces and the left-orderability of their funda…
We give a short proof of the (known) result that there are no Kaehler structures on exotic tori. This yields a negative solution to a problem posed by Benson and Gordon. W discuss the symplectic version of the problem and analyze results which yield an evidence for the conjecture that there are no symplectic structures…
Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.
problem Finding solutions for specific types of equations.
method Providing superposition formulae for Bäcklund transformations.
result Algebraically obtain infinitely many solutions after first integration.
Let M_1 and M_2 be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism $φ:\bdy M_1 \to \bdy M_2$. We analyze the relationship between the sets of low genus Heegaard splittings of M_1, M_2, and M, assuming the map φis "sufficiently complicated." This a…
We show that the fundamental group of the double branched cover of an infinite family of homologically thin, non-quasi-alternating knots is not left-orderable, giving further support for a conjecture of Boyer, Gordon, and Watson that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental…
Study improves dividend discount model using VAR process.
problem Improving dividend discount models for better predictions.
method Introduced a Gordon growth model based on Vector Autoregressive Process (VAR).
result Two Propositions related to the new model.
Study proves future stability of perturbed Milne model in Einstein-Klein-Gordon system.
problem Stability of perturbed Milne model in Einstein-Klein-Gordon system.
method Worked within CMC gauge, focused on 1+3 splitting of Bianchi-Klein-Gordon equations, established energy scheme.
result Proved nonlinear future stability of perturbed spacetimes.
New solutions to Einstein-Klein-Gordon equations emerge from Kerr black holes.
problem Stability of Kerr black holes under Klein-Gordon fields.
method Constructing time-periodic scalar field solutions.
result Kerr black holes are not asymptotically stable for certain Klein-Gordon masses.
Rediscovered by a systematic search, a forgotten class of integrable surfaces is shown to disprove the Finkel-Wu conjecture. The associated integrable nonlinear partial differential equation possesses a zero curvature representation, a third-order symmetry, and a nonlocal transformatio…
Enhanced Gordon growth model for valuing financial products.
problem Valuation of financial products with time-varying interest rates and dividends.
method Dynamic Gordon growth model with time-varying spot interest rate and dividends, risk-neutral valuation, locally risk-minimizing strategy.
result Pricing and hedging formulas for dividend-paying European options and equity-linked life insurance products.
New harmonic maps to hyperbolic plane via Bäcklund transformation.
problem Constructing new harmonic maps to the hyperbolic plane.
method Using Bäcklund transformation to connect solutions of sinh-Gordon and sine-Gordon equations.
result Construction of new harmonic maps.
The sinh-Gordon equation is solved on finite, symmetric graphs.
problem Solving the sinh-Gordon equation with nonzero prescribed functions on finite graphs.
method Uniform a priori estimate to define topological degree, case-by-case calculation of degree, classical sinh-Gordon equation analysis.
result The classical sinh-Gordon equation with nonzero prescribed function is always solvable on finite, symmetric graphs.
Knot Floer homology detects ribbon concordance.
problem Detecting ribbon concordance between knots.
method Using knot Floer homology to study ribbon concordance.
result The map induced by ribbon concordance on knot Floer homology is injective.
New examples of knots with special bridge positions found.
problem Understanding special types of bridge positions for knots.
method Derived examples of knots with unperturbed weakly reducible non-minimal bridge positions.
result Connected sum of unperturbed bridge positions is unperturbed (a new conjecture).
We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical …
Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since large classes of L-spaces can be produced from Dehn surgery on knots in the 3-sphere, it is natural to ask what conditions on the knot group are suffi…
Characterizes alternating links in thickened surfaces using Gordon-Litherland pairing.
problem Identifying alternating links in thickened surfaces.
method Extension of Gordon-Litherland pairing to thickened surfaces.
result A non-split link in a thickened surface is alternating if and only if it bounds two definite surfaces of opposite sign.
Simplified proof for a theorem about graphs.
problem Proving a generalized Conway--Gordon--Sachs theorem for complete graphs.
method Provided a shorter proof over integers.
result A simpler proof of the generalized theorem.
Gordon-Litherland pairing connects combinatorics and topology.
problem Unifying quadratic forms in link theory.
method Picture proof using Kirby diagrams.
result Their theorem has numerous applications in low-dimensional topology.
Study rational homology balls using Casson-Gordon invariants to measure complexity.
problem Measuring complexity of rational homology balls.
method Use Casson-Gordon invariants and Levine-Tristram signatures.
result Obtain lower bounds on the number of 1-handles in handle decompositions.
New right-angled Artin subgroups found in Artin groups.
problem Finding large right-angled Artin subgroups in Artin groups.
method Examining centers of irreducible spherical special subgroups and their powers.
result Conjecture verified for certain classes of Artin groups, leading to hyperbolic surface subgroup conclusions.
Direct proofs of implications between three theorems on maps of simplex.
problem Understanding relations between three theorems on maps of simplex.
method Direct proofs using interesting relations between van Kampen and Conway-Gordon-Sachs numbers.
result Exhibited relations and direct proofs of implications between the theorems.
The paper proves a conjecture about satellite knots and their slice genus.
problem The topological slice genus of satellite knots and its bounds.
method Establishes the conjecture for a variant of the topological slice genus, the Z-slice genus.
result The topological slice genus of a satellite knot is bounded above by the sum of the slice genera of the knot and the pattern.
Guichard's transformations generate Voss surfaces from sine-Gordon solutions.
problem Generating Voss surfaces from sine-Gordon solutions.
method Using Guichard transformations and recursion operators for sine-Gordon symmetries.
result Explicit derivation of Voss nets and length of Guichard sequences.
Positive knots are minimal in a specific knot ordering.
problem Understanding the minimal structure of positive knots in a specific knot ordering.
method Analyzing ribbon concordance and using properties of rational Alexander modules.
result Positive knots are minimal in the ribbon concordance ordering.
A famous example of Casson and Gordon shows that a Haken 3-manifold can have an infinite family of irreducible Heegaard splittings with different genera. In this paper, we prove that a closed non-Haken 3-manifold has only finitely many irreducible Heegaard splittings, up to isotopy. This is much stronger than the Waldh…
Study periodic solutions using spectral data and Darboux coordinates.
problem Elliptic sinh-Gordon equation periodic solutions.
method Spectral data and Darboux coordinates.
result Existence of Darboux coordinates.