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We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q:=H_q(N;Z)$. Our main result is a complete readily calculable classification of embeddings $N\to R^7$, up to the…

A knot semigroup is defined by A. Vernitski. A. Vernitski conjectured that the knot semigroup of the 2-bridge knot is isomorphic to an alternating sum semigroup. To support this conjecture, and as a first main result, we prove that the knot…

We prove the following results (1) (2) (3) on relations between $n$-links and their components. (1) Let L=(L_1, L_2) be a (4k+1)-link (4k+1\geq 5). Then we have Arf L=Arf L_1+Arf L_2. (2) Let L=(L_1, L_2) be a (4k+3)-link (4k+3\geq3). Then…

We present a new short proof of the explicit formula for the group of links (and also link maps) in the 'quadruple point free' dimension. Denote by $L^m_{p,q}$ (respectively, $C^{m-p}_p$) the group of smooth embeddings $S^p\sqcup S^q\to…

Let $B_n$ denote the classical braid group on $n$ strands and let the {\em mixed braid group} $B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for which the first $m$ strands form the identity braid. Let…

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points…

We obtain estimations for isotopy classes of embeddings of closed k-connected n-manifolds into R^{2n-k-1} for n>2k+5 and k\ge0. This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action…

We study the connected sum of Hopf links in $S^3$. Particularly, we compute the entanglement entropy (EE) as a function of the number of link components. We find evidence of lower and upper bounds for the entanglement entropy. We show that…

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; \mathbb Z)$. Our main result is a readily calculable classification of embeddings $N\to\mathbb R^7$ up to…

Let $K\subset S^3$ be a knot and $\eta, \gamma \subset S^3\backslash K$ be simple closed curves. Denote by $\Sigma_q(K)$ the $q$-fold cyclic branched cover of $K$. We give an explicit formula for computing the linking numbers between lifts…

We define an isotopy invariant of embeddings N -> R^m of manifolds into Euclidean space. This invariant together with the \alpha-invariant of Haefliger-Wu is complete in the dimension range where the \alpha-invariant could be incomplete. We…

The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the R\'enyi entropy of index $m$, which captures the higher moments of the reduced density matrix. In this work, we…

This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings $N\to S^m$. We study the specific case of knotted tori, i. e. the embeddings $S^p \times S^q \to S^m$.…

A fixed knot $K$ acts via Murasugi sum on the space $\mathcal{S}$ of isotopy classes of knots. This operation endows $\mathcal{S}$ with a directed graph structure denoted by $M\kern-1pt SG(K)$. We show that any given family of knots in…

Suppose $R$ is a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$ such that $q+q^{-1}$ is invertible. For an oriented surface $\Sigma$, let $\mathcal{S}(\Sigma;R)$ denote the Kauffman bracket skein algebra of…

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…

This is the first paper in a series of three devoted to studying twisted linking forms of knots and three-manifolds. Its function is to provide the algebraic foundations for the next two papers by describing how to define and calculate…

This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\times S^q\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high…

Suppose there are two framed links in a compact, connected 3-manifold (possibly with boundary, or non-orientable) such that the associated 3-manifolds obtained by surgery are homeomorphic (relative to their common boundary, if there is…

In this paper we investigate the relationship between isotopy classes of knots and links in S^3 and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the…