How do you find the Jones polynomial?
The Jones Polynomial It can be calculated by anyone with high-school algebra and a cool head. This invariant is a generalized polynomial: an expression like t-2 + 2t-1 + 3 – 2t2, where both positive and negative powers can appear.
Is the Jones polynomial an isotopy invariant?
Instead, a number of quantities have been discovered that are isotopy invariant. While these invariants are not perfect, they are powerful tools for distinguishing knots. This paper will describe a number of such invariants, including the knot group, some elementary invariants, and the Jones polynomial.
What would the bracket polynomial of the usual projection of the trivial link of N components be?
The bracket polynomial of the usual projection of the trivial link of n components will be < ○∪ ○∪ …∪ ○> = (−1)n-1(A2 + A-2)n-1. and the type III Reidemeister move. Therefore, the bracket polynomial is not a knot invariant. The writhe of an oriented link, denoted by w(L), is the sum of all the signs of its crossings.
What are the applications of knot theory?
In biology, we can use knots to examine the ability of topoiso- merase enzymes to add or remove tangles from DNA; in chemistry, knots allow us to describe the structure of topological stereoisomers, or molecules with the same atoms but different configurations; and in physics, we use graphs used in knot theory to …
Why is the knot theory important?
Knot theory provides insight into how hard it is to unknot and reknot various types of DNA, shedding light on how much time it takes the enzymes to do their jobs.
What are DNA knots?
Just like any long polymer chain, DNA tends to form knots. Using technology that allows them to stretch DNA molecules and image the behavior of these knots, MIT researchers have discovered, for the first time, the factors that determine whether a knot moves along the strand or “jams” in place.
Why can’t you have knots in more than 4 dimensions?
A knot is a closed curve in space. A knot is called trivial, if one can deform it to a simple unknotted circle without having any selfintersections at any time. It is quite easy to see that in four dimensions, there are no nontrivial knots. You would not be able to tie a shoe in four dimensional space.
How do you know if knots are equivalent?
In math, we say two knots are equivalent or are the same knot if you can move the strands around to make one look like the other one.
What is the enzyme Unknots DNA?
The procaryotic enzyme, DNA gyrase, is also capable of unknotting DNA. We propose that these unknotting enzymes constitute a new general class of DNA topoisomerases (type II DNA topoisomerases).
Do knots exist in higher dimensions?
Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.
How many dimensions are there?
The world as we know it has three dimensions of space—length, width and depth—and one dimension of time. But there’s the mind-bending possibility that many more dimensions exist out there. According to string theory, one of the leading physics model of the last half century, the universe operates with 10 dimensions.
How do you prove a knot is not Tricolorable?
Example of a non-tricolorable knot If three of the strands had the same color, then all strands would be forced to be the same color. Otherwise each of these four strands must have a distinct color. Since tricolorability is a knot invariant, none of its other diagrams can be tricolored either.
How is the Hopf link not a hyperbolic link?
This space has a locally Euclidean geometry, so the Hopf link is not a hyperbolic link. The knot group of the Hopf link (the fundamental group of its complement) is Z2 (the free abelian group on two generators), distinguishing it from an unlinked pair of loops which has the free group on two generators as its group.
Which is the fundamental group of the Hopf link?
The knot group of the Hopf link (the fundamental group of its complement) is Z2 (the free abelian group on two generators), distinguishing it from an unlinked pair of loops which has the free group on two generators as its group. The Hopf-link is not tricolorable.
What is the property of the Hopf fibration?
The Hopf fibration is a continuous function from the 3-sphere (a three-dimensional surface in four-dimensional Euclidean space) into the more familiar 2-sphere, with the property that the inverse image of each point on the 2-sphere is a circle.
Who was the Hopf link topology named after?
The Hopf link topology is highly conserved in proteins and adds to their stability. The Hopf link is named after topologist Heinz Hopf, who considered it in 1931 as part of his research on the Hopf fibration. However, in mathematics, it was known to Carl Friedrich Gauss before the work of Hopf.