Lernaean knots and band surgery

The paper is devoted to a line of the knot theory related to the conjecture on the additivity of the crossing number for knots under connected sum. A series of weak versions of this conjecture are proved. Many of these versions are formulated in terms of the band surgery graph also called the H ( 2 ) H(2) -Gordian graph.

Virtual parity Alexander polynomial

In this paper, we define the virtual Alexander polynomial following the works of Boden et al. (2016) [Alexander invariants for virtual knots, J. Knot Theory Ramications 24(3) (2015) 1550009] and Kaestner and Kauffman [Parity biquandles, in Knots in Poland. III. Part 1, Banach Center Publications, Vol. 100 (Polish Academy of Science Mathematical Institute, Warsaw, 2014), pp. 131–151]. The properties of this invariant are explored and some examples are computed. In particular, the invariant demonstrates that many virtual knots cannot be unknotted by crossing changes on only odd crossings.

The Khovanov homology of knots

<p>The Khovanov homology is a knot invariant which first appeared in Khovanov's original paper of 1999, titled ``a categorification of the Jones polynomial.'' This thesis aims to give an exposition of the Khovanov homology, including a complete background to the techniques used. We start with basic knot theory, including a definition of the Jones polynomial via the Kauffman bracket. Next, we cover some definitions and constructions in homological algebra which we use in the description of our title. Next we define the Khovanov homology in an analogous way to the Kauffman bracket, using only the algebraic techniques of the previous chapter, followed closely by a proof that the Khovanov homology is a knot invariant. After this, we prove an isomorphism of categories between TQFTs and Frobenius objects, which finally, in the last chapter, we put in the context of the Khovanov homology. After this application, we discuss some topological techniques in the context of the Khovanov homology.</p>

The Khovanov homology of knots

<p>The Khovanov homology is a knot invariant which first appeared in Khovanov's original paper of 1999, titled ``a categorification of the Jones polynomial.'' This thesis aims to give an exposition of the Khovanov homology, including a complete background to the techniques used. We start with basic knot theory, including a definition of the Jones polynomial via the Kauffman bracket. Next, we cover some definitions and constructions in homological algebra which we use in the description of our title. Next we define the Khovanov homology in an analogous way to the Kauffman bracket, using only the algebraic techniques of the previous chapter, followed closely by a proof that the Khovanov homology is a knot invariant. After this, we prove an isomorphism of categories between TQFTs and Frobenius objects, which finally, in the last chapter, we put in the context of the Khovanov homology. After this application, we discuss some topological techniques in the context of the Khovanov homology.</p>

Knot the Usual Suspects: Finding the Diagrammatic Representations of Physical Knots

In the last few hundred years, mathematicians have been attempting to describe the topological and algebraic properties of mathematical knots. Regarding the study of knots, there exists a disconnect between examining a knot’s mathematical and physical definitions. This is due to the inherent difference in the topology of an open-ended physical knot and a closed mathematical knot. By closing the ends of a physical knot, this paper presents a method to break this discontinuity by establishing a clear relation between physical and mathematical knots. By joining the ends and applying Reidemeister moves, this paper will calculate the equivalent mathematical prime or composite knots for several commonly used physical knots. In the future, it will be possible to study the physical properties of these knots and their potential to expand the field of mathematical knot theory.

Polynomial Invariant of Molecular Circuit Topology

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 acids, often fold into 3D conformations with critical chain entanglements and local or global structural symmetries stabilised by formation contacts between different parts of the chain. Circuit topology captures the arrangements of intra-chain contacts within a given folded linear chain and allows for the classification and comparison of chains. Contacts keep chain segments in physical proximity and can be either mechanically hard attachments or soft entanglements that constrain a physical chain. Contrary to knot theory, which offers many established knot invariants, circuit invariants are just being developed. Here, we present polynomial invariants that are both efficient and sufficiently powerful to deal with any combination of soft and hard contacts. A computer implementation and table of chains with up to three contacts is also provided.