Reidemeister Moves

Definition

Reidemeister moves are specific types of manipulations that can be performed on knot diagrams without changing the fundamental topology of the knot. These moves demonstrate how two different knot diagrams can represent the same knot, emphasizing the concept of ambient isotopy and the equivalence of knots through simple transformations.

5 Must Know Facts For Your Next Test

1. There are three types of Reidemeister moves: type I (adding or removing a twist), type II (adding or removing a crossing), and type III (changing the way strands cross).
2. These moves are fundamental in proving that two knot diagrams represent the same knot by showing a sequence of transformations.
3. Every knot can be represented in many different diagrams, and Reidemeister moves provide a systematic way to navigate these representations.
4. The concept of Reidemeister moves is critical for understanding other invariants and properties of knots, such as crossing number and bridge number.
5. Understanding Reidemeister moves lays the foundation for more advanced topics in knot theory, including polynomial invariants like the Jones and HOMFLY polynomials.

Review Questions

• How do Reidemeister moves relate to the concept of ambient isotopy in demonstrating knot equivalence?
  • Reidemeister moves illustrate how knots can be transformed into one another through simple manipulations on their diagrams. This relates to ambient isotopy, which involves continuously deforming a knot in space without cutting it. By applying a series of Reidemeister moves, we can show that two diagrams represent the same knot, emphasizing that they are equivalent under ambient isotopy.
• In what ways do Reidemeister moves assist in determining invariants like crossing number and bridge number?
  • Reidemeister moves play a significant role in calculating knot invariants such as crossing number and bridge number by allowing us to manipulate diagrams into forms that reveal these properties. For instance, applying type II Reidemeister moves can help simplify a diagram to count its crossings more easily. The ability to transform a diagram while preserving its knot type enables us to find minimal representations for these invariants.
• Evaluate the importance of Reidemeister moves in the development of polynomial invariants like the Jones polynomial and Kauffman bracket.
  • Reidemeister moves are crucial for establishing the foundations of polynomial invariants such as the Jones polynomial and Kauffman bracket. These invariants are defined through specific rules that rely on the manipulation of knot diagrams via Reidemeister moves. By ensuring that these polynomials remain unchanged under such transformations, we confirm their effectiveness as knot invariants, which help distinguish between different knots in a more algebraic manner.

Related terms

Knot Diagrams: Visual representations of knots on a plane, showing crossings and over/under relations that help analyze their properties and equivalence.

Ambient Isotopy: A continuous deformation of a knot or link in three-dimensional space, allowing for transformations between different configurations without cutting the strand.

Knot Equivalence: The property of two knots being considered the same if one can be transformed into the other through a series of Reidemeister moves and ambient isotopies.