Related papers: Integer Knot Invariants: Inequalities, Computation…
We describe a new method for combinatorially computing the transverse invariant in knot Floer homology. Previous work of the authors and Stone used braid diagrams to combinatorially compute knot Floer homology of braid closures. However,…
This paper contains the first knot polynomials which can distinguish the orientations of classical knots and which make no excplicit use of the knot group. But they make extensive use of the meridian and of the longitude in a geometric way.…
A knot invariant is called skein if it is determined by a finite number of skein relations. In the paper we discuss some basic properties of skein invariants and mention some known examples of skein invariants.
New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising…
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…
A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain…
We explore the application of automated reasoning techniques to unknot detection, a classical problem of computational topology. We adopt a two-pronged experimental approach, using a theorem prover to try to establish a positive result…
Ascending numbers are determined for 64 knots with at most n=10 crossings. After proving the theorem about the signature of alternating knot families, we distinguished all families of knots obtained from generating alternating knots with at…
Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly…
The main goal of the present paper is to construct new invariants of knots with additional structure by adding new gradings to the Khovanov complex. The ideas given below work in the case of virtual knots, closed braids and some other cases…
The topological framework of circuit topology has recently been introduced to complement knot theory and to help in understanding the physics of molecular folding. Naturally evolved linear molecular chains, such as proteins and nucleic…
A series invariant of a complement of a knot was introduced recently. The invariant for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure…
Jablan and Radovi\'c originally defined two invariants called the Meander number and OGC number of knots for certain classes of knots. We generalize these definitions to all knots and name the straight number and contained straight number…
Knots are fascinating topological structures that have been observed in various contexts, ranging from micro-worlds to macro-systems, and are conjectured to play a fundamental role in their respective fields. In order to characterize their…
I present a summary of the recent progress made in field and string theory which has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be described in…
To each rail knotoid we associate two unoriented knots along with their oriented counterparts, thus deriving invariants for rail knotoids based on these associations. We then translate them to invariants of rail isotopy for rail arcs.
In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguish knots. In particular, we recast and extend by geometric means a recent result of Silver and…
We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of…
The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of…
We compute the involutive concordance invariants for the 10- and 11-crossing (1,1)-knots.