In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring (or "unknot").

A knot diagram is the regular projection of a knot to the plane with broken lines indicating where one part of the knot undercrosses the other part. Informally, an orientation of a knot can be thought of as a direction of travel around the knot.

knot theory, in mathematics, the study of closed curves in three dimensions, and their possible deformations without one part cutting through another. Knots may be regarded as formed by interlacing and looping a piece of string in any fashion and then joining the ends.

An alternating knot is a knot with a projection that has crossings that alternate between over- and undercrossings as you traverse the knot in a fixed direction.

May 1, 2023 · In this presesntation, we address two important questions in Knot Theory: How can we distinguish knots? How can we tell if two knots are the same? We use a knot’s projection in the 2D plane to represent it. We can use arrows to denote a knot’s direction that one is traveling in while they trace the knot.

When drawing knots, the way in which strands cross each other are represented by creating a break in the strand that goes underneath (under strand) at where is crosses with the strand that goes above (over strand). Examples of knots include the unknot and the trefoil knot.

May 29, 2024 · Knot theory is a branch of mathematics that studies mathematical knots, which are closed loops in three-dimensional space. These knots can be formed by intertwining and looping segments of a rope without cutting or overlapping.

A knot is a one-dimensional subset of R3 that is homeomorphic to S1. We can specify a knot K by specifying an embedding (smooth injective) f : S1 ! R3 so that K = f(S1). For f to be smooth, all of its derivatives must exist. Example 2. Examples of …

Oct 31, 2022 · Knot theory has crisscrossed the mathematical landscape over the centuries. It began as an applied area of mathematics, with Thomson attempting to use knots to …

This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in three-dimensional space. Knots can be studied at many levels and from many points of view.