Knot Theory and Its Applications

  title={Knot Theory and Its Applications},
  author={Krishnendu Gongopadhyay and Rama Mishra},
Knot theory is the embedding of one topological space into another topological space. Knot theory was first proposed as the flawed vortex model of atoms by Lord Kelvin, but was later proved to be incorrect. Some fundamental theorems of involved in knot theory include one involving Reidemeister moves (transforming knots into unknots) and knot invariants such as the Alexander Polynomial, Conway Polynomial and Jones Polynomial that originated in math and physics. It has applications in cell… 
Obstructions to slicing knots and splitting links
In this thesis, we use invariants inspired by quantum field theory to study the smooth topology of links in space and surfaces in space-time. In the first half, we use Khovanov homology to the study
Finite Type Invariants and the Kauffman Bracket Polynomials of Virtual Knots
This paper gives a necessary condition for a virtual knot invariant to be of finite type by using t(a1, · · · , am)–sequences of virtual knots and shows that the higher derivatives f (n) K (a) of the f–polynomial fK(A) of a virtual knots K at any point a are not of finiteType unless n ≤ 1 and a = 1.
The knowledge of knots: an interdisciplinary literature review
This paper presents a review of the literature related to the investigation of knots from the topological, physical, cognitive and computational standpoints, aiming at bridging the gap between pure mathematical work on knot theory and macroscopic physical knots, with an eye to applications and modeling.
On Alexander-Conway polynomials of two-bridge links
This paper will investigate the additivity of the crossing number with respect to the composition of knots. The additivity of the crossing number is a long standing conjecture. The paper presents
Residual Torsion-Free Nilpotence, Bi-Orderability and Pretzel Knots
The residual torsion-free nilpotence of the commutator subgroup of a knot group has played a key role in studying the bi-orderability of knot groups. A technique developed by Mayland provides a
Twisted Alexander polynomials of tunnel number one Montesinos knots
We calculate the twisted Alexander polynomials of all tunnel number one Montesinos knots associated to their $SL_2(\mathbb{C})$-representations and obtain their leading coefficients and degrees. As
Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots
The knotting probability of a knot K is defined by the probability that a given random polygon or self-avoiding polygon of N vertices has the knot K, and a formula for expressing it as a function of the number of segments N is shown, which gives good fitted curves to the data of the knotting probabilities.
A General Method for Computing the Homfly Polynomial of DNA Double Crossover 3-Regular Links
In this paper, a general method is given for computing the chain polynomial of the truncated cubic graph with two different labels from the chain diagonals of the original labeled cubic graph by substitutions and it is obtained that the Homfly polynomials of the double crossover 3-regular link which has relatively large number of crossings are obtained.


The Combinatorial Revolution in Knot Theory
K not theory is usually understood to be the study of embeddings of topological spaces in other topological spaces. Classical knot theory, in particular, is concerned with the ways in which a circle
Knot theory in understanding proteins
This paper aims to enthuse mathematicians, especially topologists, knot theorists and geometers to examine problems in the study of proteins. We have highlighted those advances and breakthroughs in
Applications of Knot Theory
CHAPTER 11 – Geometric Aspects in the Development of Knot Theory
3. On Vortex Motion
Solutio problematis ad geometriam situs pertinentis
  • Commentarii Academiae Scientiarum Imperialis Petropolitanae, 8:128–140,
  • 1736