Reidemeister Move III

Definition

Reidemeister Move III is one of the three fundamental moves in knot theory that allows for the manipulation of knot diagrams without changing the knot type. This particular move involves adding or removing a pair of crossings with the same orientation, which helps simplify or modify a knot diagram while preserving its topological properties. Understanding this move is essential for working with polynomials like the HOMFLY and Jones polynomial, as it can affect their calculation and interpretation.

5 Must Know Facts For Your Next Test

1. Reidemeister Move III allows for the addition or removal of crossings in knot diagrams while preserving their equivalence.
2. This move is particularly useful in simplifying calculations when determining invariants such as the HOMFLY and Jones polynomials.
3. Reidemeister Move III is one of three Reidemeister moves, alongside Move I and Move II, which together form the basis for transforming knot diagrams.
4. The move helps identify if two different diagrams represent the same knot, which is essential in knot classification.
5. Using Reidemeister Move III correctly can often lead to an easier computation of polynomial invariants, making it a powerful tool in knot theory.

Review Questions

• How does Reidemeister Move III relate to the process of simplifying knot diagrams?
  • Reidemeister Move III simplifies knot diagrams by allowing the addition or removal of pairs of crossings with the same orientation. This manipulation does not change the fundamental nature of the knot, making it easier to identify equivalent knots. By applying this move, one can potentially reduce a complex knot diagram into a simpler form that can facilitate the computation of invariants such as the HOMFLY or Jones polynomials.
• Discuss how Reidemeister Move III impacts the calculation of polynomial invariants like the HOMFLY and Jones polynomials.
  • Reidemeister Move III can significantly streamline the calculation of polynomial invariants by allowing one to manipulate knot diagrams more freely. When two knots are shown to be equivalent through this move, their associated polynomials must also be identical. Consequently, understanding how to apply this move effectively can lead to more efficient calculations and deeper insights into the relationships between different knots.
• Evaluate the role of Reidemeister Move III within the broader context of knot theory and its applications in mathematics.
  • Reidemeister Move III plays a crucial role in knot theory by providing a method for proving equivalence between knots and simplifying complex diagrams. Its applications extend beyond just calculating polynomials; it aids in understanding topological properties and classifications of knots. By facilitating these transformations, this move contributes to various fields within mathematics, including topology and algebraic geometry, highlighting its importance in studying spatial structures.

Related terms

Knot Diagram: A visual representation of a knot, consisting of a series of curves in a plane that intersect at points called crossings.

HOMFLY Polynomial: A two-variable polynomial invariant of knots and links that generalizes both the Alexander and Jones polynomials, reflecting more complex knot structures.

Jones Polynomial:

A polynomial invariant of knots that assigns a unique polynomial to each knot, providing insight into its structure and properties.