Related papers: Linking number and Milnor invariants
The ropelength of a knot or link is the minimal number of inches of 1-inch-thick rope that it takes to tie it. The relationship of this measurement to knot and link invariants has been studied by various authors. We give the first results…
In this study, depending on the upper and the lower indices of the hyperharmonic number $h_{n}^{(r)}$, nonlinear recurrence relations are obtained. It is shown that generalized harmonic number and hyperharmonic number can be obtained from…
The linking number of an oriented two-component link is an invariant indicating how intertwined the two components are. Tuler proved that the linking number of a two-component rational $\frac{p}{q}$-link is $$\sum^{\frac{|p|}{2}}_{k=1}…
The conventional topological description given by the fundamental group of nematic order parameter does not adequately explain the entangled defect line structures that have been observed in nematic colloids. We introduce a new topological…
In the previous article we introduced the new concept of mixed representations of quivers and described the generators of their algebras of invariants. In this article we describe the defining relations of these algebras. Some applications…
We extend several classical invariants of links in the 3-sphere to links in so-called quasi-cylinders. These invariants include the linking number, the Seifert form, the Alexander module, the Alexander-Conway polynomial and the…
An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain…
In the paper we present some new inversion formulas and two new formulas for Stirling numbers.
The knots-quivers correspondence is a relation between knot invariants and enumerative invariants of quivers, which in particular translates the knot operations of linking and unlinking to a certain mutation operation on quivers. In this…
In this paper we introduce and study two generalizations of Lah numbers, analogous to the Stirling numbers with higher level - a combinatorial one (Lah numbers with higher level) and an algebraic one (Lah numbers of order $s$). We define…
Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a…
In this paper we describe the algebra of differential invariants for GL(n,C)-structures. This leads to classification of almost complex structures of general positions. The invariants are applied to the existence problem of…
We define a notion of r-generalized column distances for the j-truncation of a convolutional code. Taking the limit as j tends to infinity allows us to define r-generalized column distances of a convolutional code. We establish some…
Using the concept of constant evasion to different sorts of suitable binary relations, we establish many cardinal invariants derived from the established cardinal invariants $\mathfrak{e}^\mathrm{const}_{n}$ and…
A handlebody-link is a disjoint union of embeddings of handlebodies in $S^3$ and an HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. The second author and Ryo Nikkuni classified the set of…
A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of extensor fields is present using algebraic and analytical tools developed in previous papers. Several important formulas are derived.
We explore the notion of sectional number of a group homomorphism, leading to a generalization of the covering number of a group, and present several characterizations when the sectional number is finite, providing estimates for computing…
A geometric argument is given to prove that the Seifert genus of a positive knot equals its slice genus. A combinatorial invariant, giving a lower bound for the slice genus, is formulated for arbitrary knots. Properties and applications of…
This paper extends the construction of invariants for virtual knots to virtual long knots and introduces two new invariant modules of virtual long knots. Several interesting features are described that distinguish virtual long knots from…
Some relations between cohomological dimensions and depths of linked ideals are investigated and discussed by various examples.