We develop a framework that generalizes Budney-Gabai's $W_3$ invariant on $\pi_0\textrm{Diff}(S^1\times D^3,\partial)$ to 4-manifolds with 1-handles. As applications, we show if $M=(S^1\times D^3)\natural \hat M$ where $\hat either has the form $I\times Y$ or is punctured aspherical manifold, then center of mapping class group $M$ infinite rank.

Let $M=\Sigma\times S^2$ where $\Sigma$ is a closed oriented surface of positive genus. $\operatorname{MCG}(M)$ be the smooth mapping class group $M$, and let $\operatorname{MCG}_0(M)$ denote subgroup consisting elements homotopic to identity. We show that there exists surjection from $\mathbb{Z}^\infty$ such its restriction also infinite rank. As result, every contains infinitely generated. The proof based on generalization Dax invariant embedded \emph{closed} surfaces. Using generalized...

If L is an oriented link with n components, then the rank of its Khovanov homology at least 2^{n} . We classify all links that achieve this lower bound and show such can be obtained by iterated connected sums disjoint unions Hopf unknots. This gives a positive answer to question asked Batson Seed (2015).

We prove an excision theorem for the singular instanton Floer homology that allows surfaces to intersect locus. This is extension of non-singular by Kronheimer and Mrowka genus-zero Street. use define theory sutured manifolds with tangles. As applications, we annular Khovanov (1) detects unlink, (2) closure trivial braid, (3) distinguishes braid closures from other links; also Thurston norm meridional surfaces.

We prove that any link in S 3 whose Khovanov homology is the same as of a Hopf must be isotopic to link.This holds for both reduced and unreduced homology, with coefficients either Z or Z/2Z.Khovanov [Kho00] associates each L ⊂ bigraded group Kh * , (L), graded Euler characteristic recovers Jones polynomial V (q), well variant red (L) [Kho03].It known detect unknot [KM11], n-component unlink all n [BS15], trefoils [BS18].In this note we links H ± which are oriented so two components have...

In this paper, we introduce the annular instanton Floer homology which is defined for links in a thickened annulus. It an analogue of Khovanov homology. A spectral sequence whose second page and converges to constructed. As application sequence, prove that detects unlink annulus (assuming all components are null-homologous). Another new proof Grigsby Ni's result tangle distinguishes braids from other tangles.

Abstract Ordered mesoporous carbons (OMC) loading on sulfonated graphene (OMC/SG) have been fabricated by multi‐components co‐assembly followed thermal polymerization and carbonization. OMC/SG composites possess the hierarchically ordered hexagonal mesostructure with lattice unit parameter porous diameter about 10 nm 4 nm, respectively. Sulfonated is integrated into interpenetrated network structures via covalent bonding hydrogen bonding, as well highly dispersed in OMC matrix. composite...

We classify all links whose Khovanov homology have ranks no greater than 8, and three-component 12, where the coefficient ring is Z/2. The classification based on previous results of Kronheimer-Mrowka, Batson-Seed, Baldwin-Sivek, authors.

Porous cobalt nanowall arrays have been prepared by electrochemical deposition of mono-precursor [Co(NH(3))(5)Cl]Cl(2) on copper substrates. Brunauer-Emmett-Teller (BET) and Barret-Joyner-Halenda (BJH) investigations the surface properties indicate that resulting porous nanomaterials possess high area uniform pore size distribution, which implies potential applications in some fields, such as catalysis, energy, magnetic data storage devices. The magnetism measurements take a good...

If L is an oriented link with $n$ components, then the rank of its Khovanov homology at least $2^n$. We classify all links whose Z/2-coefficients achieves this lower bound, and show that such can be obtained by iterated connected sums disjoint unions Hopf unknots. This gives a positive answer to question asked Batson Seed.

Surgery exact triangles in various 3-manifold Floer homology theories provide an important tool studying and computing the relevant groups. These relate invariants of 3-manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, behavior SU ( N ) -instanton with respect to surgery is studied. particular, it shown that there are tetragons pentagons, respectively, for 3 - 4 homologies. It also conjectured general admits + 1 -gon. An essential step proof construction...

Abstract We prove a rank inequality on the instanton knot homology and Khovanov of link in $S^3$. The key step proof is to construct spectral sequence relating Baldwin–Levine–Sarkar’s pointed singular invariant for links.

We prove a rank inequality on the instanton knot homology and Khovanov of link in $S^3$. The key step proof is to construct spectral sequence relating Baldwin-Levine-Sarkar's pointed singular invariant for links.

We prove that Khovanov homology with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z slash 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">\mathbb {Z}/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–coefficients detects the link L7n1...

Asaeda, Przytycki, and Sikora (2004) defined a generalization of Khovanov homology for links in I -bundles over compact surfaces. We prove that, link L\subset (-1,1)\times T^{2} , the Asaeda–Przytycki–Sikora L has rank 2 with \mathbb{Z}/2 -coefficients if only is isotopic to an embedded knot \{0\}\times . also that detects unlink torus

For each integer $N\geq 2$, Mari\~no and Moore defined generalized Donaldson invariants by the methods of quantum field theory, made predictions about values these invariants. Subsequently, Kronheimer gave a rigorous definition using moduli spaces anti-self-dual connections on hermitian vector bundles rank $N$. In this paper, Moore's are confirmed for simply connected elliptic surfaces without multiple fibers certain general type in case that $N=3$. The primary motivation is to study...

Surgery exact triangles in various 3-manifold Floer homology theories provide an important tool studying and computing the relevant groups. These relate invariants of 3-manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, behavior $SU(N)$-instanton with respect to surgery is studied. particular, it shown that there are tetragons pentagons, respectively, for $SU(3)$- $SU(4)$-instanton homologies. It also conjectured general admits $(N+1)$-gon. An essential...

Suppose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M comma gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>γ<!-- γ --></mml:mi> stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(M, \gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a balanced sutured...

We calculate the ring structure of singular instanton Floer homology $(S^1\times \Sigma, S^1\times \{p_1,\dots,p_n\})$ with C-coefficients, where $\Sigma$ is a closed oriented surface. As an application, we prove excision formula for when n=1. This settles last unknown case homology.