Related papers: Symmetry Breaking and Link Homologies I

In the first part of this paper, we constructed a filtered U(r)-equivariant stable homotopy type called the spectrum of strict broken symmetries sB(L) of links L given by closing a braid with r strands. We further showed that evaluating…

In the first part of this paper, we constructed a filtered U(r)-equivariant stable homotopy type called the spectrum of strict broken symmetries sB(L) of links L given by closing a braid with r strands. Evaluating this filtered spectrum on…

For a word w in the braid group on n-strands, we denote by T_w the corresponding transverse braid in the rotational symmetric tight contact structure on S^3. We exhibit a map on link Floer homology which sends the transverse invariant…

Let G denote a group and let W be an algebra over a commutative ring R. We will say that W is a G-graded twisted algebra (not necessarily commutative, neither associative) if there exists a G-grading W=\bigoplus_{g \in G}W_{g} where each…

We investigate structural and rigidity properties of \emph{Lie skew braces} (LSBs), objects essentially known in the literature as \emph{post--Lie groups}, obtained by endowing a manifold with two compatible group laws that share the same…

For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to \Aut \,(G, \cdot),~~a \mapsto \lambda_a,$ where $\lambda_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $\lambda$ can also be…

Let $A$ be either a simplicial complex $K$ or a small category $\mathcal C$ with $V(A)$ as its set of vertices or objects. We define a twisted structure on $A$ with coefficients in a simplicial group $G$ as a function $$ \delta\colon…

The WRT invariant of a link L in S2xS1 at sufficiently high values of the level r can be expresses as an evaluation of a special polynomial invariant of L at 2r-th root of unity. We categorify this polynomial invariant by associating to L a…

There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the…

Mixed states can exhibit two distinct kinds of symmetries, either on the level of the individual states (strong symmetry), or only on the level of the ensemble (weak symmetry). Strong symmetries can be spontaneously broken down to weak…

Let G be a group and let W be an algebra over a field K. We will say that W is a G-graded twisted algebra if W can be written as a direct sum over the elements of G of one dimensional K-vector spaces. It is also assumed that W has no…

In their recent preprint, Baldwin, Ozsv\'{a}th and Szab\'{o} defined a twisted version (with coefficients in a Novikov ring) of a spectral sequence, previously defined by Ozsv\'{a}th and Szab\'{o}, from Khovanov homology to Heegaard-Floer…

Symmetry in mixed quantum states can manifest in two distinct forms: strong symmetry, where each individual pure state in the quantum ensemble is symmetric with the same charge, and weak symmetry, which applies only to the entire ensemble.…

Crystallographic symmetries enforcing band touchings (BTs) in the Brillouin zone (BZ) have been utilized to classify and predict the topological semimetals. Though the early proposed topological semimetals contain isolated nodal points in…

For a positive braid $\beta \in \mathrm{Br}^{+}_{k}$, we consider the braid variety $X(\beta)$. We define a family of open sets $\mathcal{U}_{r, w}$ in $X(\beta)$, where $w \in S_k$ is a permutation and $r$ is a positive integer no greater…

We study the Seiberg-Witten invariant $\lambda_{\rm{SW}} (X)$ of smooth spin $4$-manifolds $X$ with integral homology of $S^1\times S^3$ defined by Mrowka, Ruberman, and Saveliev as a signed count of irreducible monopoles amended by an…

By means of Rasmussen's formulation of Khovanov-Rozansky homology originally given over $\mathbb{Q}$ in arXiv:math/0607544, we compare different flavors of $\mathfrak{sl}(n)$ link homology with the link invariants obtained by Kitchloo in…

Real Bruhat cells give an important and well studied stratification of such spaces as $GL_{n+1}$, $Flag_{n+1} = SL_{n+1}/B$, $SO_{n+1}$ and $Spin_{n+1}$. We study the intersections of a top dimensional cell with another cell (for another…

The topological Hochschild homology of a ring (or ring spectrum) $R$ is an $S^1$-spectrum, and the fixed points of THH($R$) for subgroups $C_n\subset S^1$ have been widely studied due to their use in algebraic K-theory computations.…

Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual…