Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

Definition. A (topological) n-manifold M is a Hausdorff topological space with a countable basis of open sets, such that each point of M lies in an open set homeomorphic to R or R+ = {(x1, . . . , xn) ∈ R n : xn ≥ 0}. The boundary ∂M of M is the set of points not having neighbourhoods homeomorphic to R. The set M − ∂M is the interior of M , denoted int(M). If M is compact and ∂M = ∅, then M is closed.

Three-dimensional manifolds

For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities.

Knot homology groups from instantons

1.1 An observation of some coincidences For a knot or link K in S, the Khovanov homology Kh.K/ is a bigraded abelian group whose construction can be described in entirely combinatorial terms [15]. If we forget the bigrading, then as abelian groups we have, for example, Kh.unknot/ D Z and Kh.trefoil/ D Z Z=2: The second equality holds for both the rightand left-handed trefoils, though the bigrading would distinguish these two cases. The present paper was motivated in large part by the observation that the group Z Z=2 arises in a different context. Pick a basepoint y0 in the complement of the knot or link, and consider the space of all homomorphisms

This paper provides both a detailed study of color-dependence of link homologies, as realized in physics as certain spaces of BPS states, and a broad study of the behavior of BPS states in general. We consider how the spectrum of BPS states varies as continuous parameters of a theory are perturbed. This question can be posed in a wide variety of physical contexts, and we answer it by proposing that the relationship between unperturbed and perturbed BPS spectra is described by a spectral sequence. These general considerations unify previous applications of spectral sequence techniques to physics, and explain from a physical standpoint the appearance of many spectral sequences relating various link homology theories to one another. We also study structural properties of colored HOMFLY homology for links and evaluate Poincare polynomials in numerous examples. Among these structural properties is a novel “sliding” property, which can be explained by using (refined) modular S-matrix. This leads to the identification of modular transformations in Chern-Simons theory and 3d N=2 theory via the 3d/3d correspondence. Lastly, we introduce the notion of associated varieties as classical limits of recursion relations of colored superpolynomials of links, and study their properties.

/pdf/sequencing-bps-spectra-24alyjdxr9.pdf

Sequencing BPS Spectra

https://maths.dur.ac.uk/users/andrew.lobb/slnbound.pdf

A slice genus lower bound from sl(n) Khovanov–Rozansky homology

Given a diagram D of a knot K, we give easily computable bounds for Rasmussen’s concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.

/pdf/computable-bounds-for-rasmussen-s-concordance-invariant-288a8nxp69.pdf

Computable bounds for Rasmussen’s concordance invariant

We show that the information contained in the associated graded vector space to Gornik's version of Khovanov-Rozansky knot homology is equivalent to a single even integer s_n(K). Furthermore we show that s_n is a homomorphism from the smooth knot concordance group to the integers. This is in analogy with Rasmussen's invariant coming from a perturbation of Khovanov homology.

/pdf/a-note-on-gornik-s-perturbation-of-khovanov-rozansky-5e3yq2k3x5.pdf

A note on Gornik's perturbation of Khovanov-Rozansky homology

We show that the information contained in the associated graded vector space to Gornik’s version of Khovanov‐Rozansky knot homology is equivalent to a single even integer sn.K/. Furthermore we show that sn is a homomorphism from the smooth knot concordance group to the integers. This is in analogy with Rasmussen’s invariant coming from a perturbation of Khovanov homology. 57M25

/pdf/a-note-on-gornik-s-perturbation-of-khovanov-rozansky-2ilkffffai.pdf

A note on Gornik's perturbation of Khovanov-Rozansky homology.

Given a diagram D of a knot K, we give easily computable bounds for Rasmussen's concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.

/pdf/computable-bounds-for-rasmussen-s-concordance-invariant-2ioi88muqm.pdf

Andrew Lobb

Papers

A slice genus lower bound from sl(n) Khovanov–Rozansky homology

Computable bounds for Rasmussen’s concordance invariant

A note on Gornik's perturbation of Khovanov-Rozansky homology

A note on Gornik's perturbation of Khovanov-Rozansky homology.

Computable bounds for Rasmussen's concordance invariant