Study connects knot polynomials with number theory sums.
problem Alexander polynomials and Dedekind sums of torus knots.
method No specific method mentioned; connects known concepts.
result Established relationship between knot theory and number theory.
Study fibrations over S2 with same singularities, showing monodromies are equivalent up to direct sums.
problem Classifying torus fibrations over S2 up to fibre sum stabilisation. method Analyzing monodromies and using direct sums with certain torus Lefschetz fibrations.
result Global monodromies of fibrations with same singularities are Hurwitz equivalent after direct sums.
Study concordance of alternating torus knots to L-space knots.
problem When are linear combinations of alternating torus knots concordant to L-space knots?
method Proved Allen's conjecture for alternating torus knots and established a necessary condition.
result Linear combinations of alternating torus knots are concordant to L-space knots if and only if they are a single torus knot.
We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.
If a knot is a nontrivial connected sum of positive torus knots, then it is not concordant to an L-space knot.
Formula connects knot complements' invariants.
problem Understanding invariants of knot complements.
method Proposed a connect sum formula for two-variable series invariants.
result Numerical evidence supports the formula for various torus knots.
Proves conjecture about integer sums of torus knot torsions.
problem Integrality of sums of (g-1)st powers of adjoint Reidemeister torsions for torus knots.
method Introduced Verlinde numbers from modular S-matrix, proved integrality through recursion formulas.
result Proven integrality of sums of (g-1)st powers of adjoint Reidemeister torsions for all torus knots and non-negative g.
The AJ conjecture is verified for certain connected sums of torus knots.
problem Verifying the AJ conjecture for specific connected sums of torus knots.
method Analyzing recurrence polynomials and their factorization properties.
result The AJ conjecture requires a modification for certain connected sums of torus knots.
Constructs harmonic spinors and 1-forms on 3-manifold connected sums and torus sums.
problem Constructing harmonic spinors and 1-forms on 3-manifold connected sums and torus sums.
method Parameterized Nash-Moser implicit function theorem and gluing argument.
result Proves existence of infinitely many Z2-harmonic spinors and 1-forms on 3-manifolds. The paper classifies manifolds with free torus actions and positive Ricci curvature.
problem Classifying manifolds with specific geometric properties.
method Using circle bundles and free torus actions on connected sums of products of spheres.
result Infinitely-many manifolds with positive Ricci curvature and torus actions are classified.
The study characterizes 3D manifolds using specific Morse-Bott functions.
problem Characterizing 3D manifolds represented as connected sums of Lens spaces, S2imesS1, and torus bundles. method Using Morse-Bott functions to classify the manifolds.
result Explicit characterization of the manifolds via certain Morse-Bott functions.
We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus.
Sum of Lagrange numbers equals a specific formula.
problem Proving the Markov Uniqueness Conjecture (MUC).
method Combining McShane's identity and Schmutz's work.
result MUC is equivalent to the given sum formula.
We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying…
It is known that connected sums of positive torus knots are not concordant to L-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial L-space knots other than the torus knots themselves…
The main results of the paper is that we give a characteristics for an annulus sum and a once-punctured torus sum of two handlebodies to be a handlebody as follows: 1. The annulus sum H=H1∪AH2 of two handlebodies H1 and H2 is a handlebody if and only if the core curve of A is a longitude for either $H_…
Study cobordism distances between 3-braid links and trefoil knots.
problem Understanding the geometric relationship between 3-braid links and trefoil knots.
method Determined cobordism distances between 3-braid links and trefoil knots, and explored limits of Coxeter's braid group result.
result Found cobordism distances between 3-braid links and trefoil knots, up to a constant error.
We prove a Torres-like formula for the L2-Alexander torsions of links, as well as formulas for connected sums and cablings of links. Along the way we compute explicitly the L2-Alexander torsions of torus links inside the three-sphere, the solid torus and the thickened torus.
We study properties of the signature function of the torus knot Tp,q. First we provide a very elementary proof of the formula for the integral of the signatures over the circle. We obtain also a closed formula for the Tristram--Levine signature of a torus knot in terms of Dedekind sums.
Derives adjoint polynomials of torus knots in explicit form.
problem Understanding adjoint invariants of torus knots.
method Closed-form double sum expression derivation.
result Explicit double sum form of adjoint polynomials.
Derives curvature formulas for convex metric sums and conditions for positive average variation.
problem Understanding how the curvature of a convex sum of metrics changes and whether it can increase the average curvature.
method Explicit formulae for curvature of convex sums of Riemannian metrics, studying total geodesic flat torus.
result Necessary and sufficient conditions for positive average variation of curvature of \(g_t\).
The study examines Morse diagrams and their behavior under Murasugi sums, leading to contact structure classifications.
problem Understanding Morse diagrams and their behavior under Murasugi sums.
method Examination of combinatorial Morse structures, open book decompositions, and contact structures.
result Diagrammatic criterion for detecting overtwisted contact structures and classification of Morse diagrams for one-holed torus pages.
Efficient cobordisms show minimal signature values on certain links.
problem Finding minimal signature values for specific link types.
method Constructing topological cobordisms between torus links and connected sums of trefoil knots.
result The signature invariant σω at ω=ζ6 takes minimal values on torus links. Triple-crossing number bound for knots and links, especially torus knots.
problem Finding bounds for triple-crossing numbers of knots and links.
method Using the genus of a knot or link, we derive bounds for the triple-crossing number.
result Triple-crossing number of torus knots and many other knots is at least twice their genus.
Closed formulas for η-corrections in the once-punctured torus identified.
problem Identifying η-corrections in the Kauffman bracket skein algebra of the once-punctured torus.
method Explicit closed formulas for Chebyshev-threaded families and η-corrections.
result Explicit Chebyshev expansions and coefficients for η-corrections.
The paper proves conditions for closed minimally immersed hypersurfaces in a 5-sphere.
problem Conditions for closed minimally immersed hypersurfaces in a 5-sphere.
method Analyzes hypersurfaces with constant scalar curvature and A3. result Closed minimally immersed hypersurfaces in a 5-sphere are isoparametric and can only have specific scalar curvature values.
Study on quantum invariants from surgeries on torus knots.
problem Quantum invariants of three-manifolds from surgeries along torus knots.
method Analysis of Witten-Reshetikhin-Turaev invariant and Chern-Simons invariants.
result Quantum invariants can be described as sums of Chern-Simons invariants and twisted Reidemeister torsions.
In this paper, we shall prove that any Heegaard splitting of a ∂-reducible 3-manifold M, say M=W∪V, can be obtained by doing connected sums, boundary connected sums and self-boundary connected sums from Heegaard splittings of n manifolds M1,...,Mn where Mi is either a solid torus or a $…
Classifies multigraphs for torus actions on 6D manifolds with isolated fixed points.
problem Classifying torus actions on 6D manifolds with isolated fixed points.
method Associate multigraphs to fixed point data, study operations, and prove classification.
result Classifies multigraphs for 6D manifolds by converting them into the empty graph.
Researchers establish a connection between knot homology and Lie algebra actions.
problem Understanding the HOMFLY-PT homology of (n,n+1) torus knots. method Constructing an explicit isomorphism and computing tautological class actions.
result The tautological class action extends to Hamiltonian vector fields and differentials in spectral sequences.
Certain torus knots have infinitely many slopes that do not uniquely identify them.
problem Identifying non-characterizing slopes for certain torus knots.
method Applying a condition from Baker and Motegi to show infinitely many non-characterizing slopes.
result The knots T2,2n+3#T−2,2n+1 have infinitely many non-characterizing slopes. New bases constructed for KBSM of lens spaces.
problem Computing Kauffman bracket skein modules of lens spaces.
method Using a fibered torus basis, new bases constructed for L(p,2) and L(4k,2k+1). result New generating sets found for KBSM of L(0,1). The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. …
Paper disproves a theorem about Kauffman bracket skein module structure.
problem Disproving a 22-year-old theorem about Kauffman bracket skein module structure.
method Analyzing handle slidings on compressing discs in handlebodies.
result More relations found than previously predicted for connected sum of handlebodies.
We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery slope is zero, or the connected sum of a (2,p)-torus knot and a (2,q)-torus kno…
We compute the Heegaard Floer homology of S13(K) (the (+1) surgery on the torus knot Tp,q) in terms of the semigroup generated by p and q, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath--Szabo d-invariant. We relate the result to known knot invariants of Tp,q as …
The Alexander polynomials Δ_{n,3}(t) and Δ_{n,4}(t) are presented as a sum of the Alexander polynomials Δ_{k,2}(t). These polynomials are also expressed in the form of a sum of Chebyshev polynomials of the second kind. These expansions allow one to introduce the "coordinates" in corresponding bases, which are proposed …
The paper explores algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.
problem Investigating algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.
method Analyzing twisted Alexander polynomials and Reidemeister torsions of torus knots associated with irreducible SLn(C)-representations. result Proves that coefficients of twisted Alexander polynomials are locally constant functions on the SLn(C)-character variety. Symmetric function LM,N lifts torus link homology.
problem Computing the triply-graded Khovanov-Rozansky homology of torus links.
method Defined a symmetric function LM,N and showed it satisfies a recursion for torus link homology. result Triply-graded Khovanov-Rozansky homology of torus links is a specialization of LM,N. An upper bound of the superbridge index of the connected sum of two knots is given in terms of the braid index of the summands. Using this upper bound and minimal polygonal presentations, we give an upper bound in terms of the superbridge index and the bridge index of the summands when they are torus knots. In contrast…
Characterizes a specific type of alternating knot.
problem Identifying a special class of genus g alternating knots.
method Uses Ozsváth and Szabó's work on alternating knots.
result Shows that if the coefficients of Alexander polynomial satisfy a certain condition, the knot is a specific torus knot.
The paper calculates a knot invariant for 3-braid knots.
problem Calculating the concordance invariant for 3-braid knots.
method Constructing cobordisms between 3-braid knots and torus knots.
result Explicit formulas and values for the invariant υ(K) for 3-braid knots. The paper characterizes and contrasts knots with high 4D clasp numbers.
problem Characterizing knots with high 4-dimensional clasp numbers.
method Topological category analysis and construction of counterexamples.
result Characterization and contrast of knots with high 4D clasp numbers.
We show that for each even integer m≥2, every reduced shadow with sufficiently many crossings is a shadow of a torus knot T(2,m+1), or of a twist knot Tm, or of a connected sum of m trefoil knots.
Two non-Morse-Bott Chern-Simons functions on homology 3-spheres.
problem Finding homology 3-spheres with non-Morse-Bott Chern-Simons functions.
method Constructing specific surgeries on torus knots.
result Examples of homology 3-spheres with non-Morse-Bott Chern-Simons functions.
Study of curves in rational surfaces using multisections and torus actions.
problem Understanding curves in rational surfaces using multisections and torus actions.
method Analysis of multisections of embedded surfaces in rational 4-manifolds with torus actions.
result Every smooth, complex curve in CP^1 × CP^1 can be put in efficient bridge position with respect to a genus one 4-section.
Classifies T2-bundles over closed surfaces up to isomorphisms.
problem Classifying T2-bundles over closed orientable surfaces. method Using group homomorphisms and mapping class groups.
result Any orientable T2-bundle over Σg with g≥1 is isomorphic to the fiber connected sum of g pieces of T2-bundles over T2. We explicitly construct the twistor spaces of Joyce metrics with torus action that are not treated in Part I (math.DG/0603242). This finishes a construction of all the twistor spaces of Joyce metrics on the connected sum of four complex projective planes.