Reidemeister Move II

Definition

Reidemeister Move II is one of the three fundamental types of moves in knot theory that allows for the manipulation of knot diagrams without changing their equivalence. This particular move involves adding or removing a pair of twist crossings in the same direction, thus changing the diagram but preserving the underlying knot type. Understanding this move is crucial as it underlies many properties and calculations associated with knot invariants, such as the HOMFLY and Jones polynomials.

5 Must Know Facts For Your Next Test

1. Reidemeister Move II specifically adds or removes crossings while maintaining the orientation of the strands, which is essential for certain calculations in knot theory.
2. This move does not affect the topological type of the knot, meaning that if two knots can be transformed into one another using Reidemeister Move II, they are considered equivalent.
3. Understanding how to apply Reidemeister Move II is fundamental for proving results related to both the HOMFLY and Jones polynomials.
4. When constructing or simplifying knot diagrams, applying Reidemeister Move II can help reduce complexity and make it easier to compute invariants.
5. Reidemeister Move II can also interact with other Reidemeister moves, providing a framework for demonstrating equivalence between more complex knots.

Review Questions

• How does Reidemeister Move II affect a knot diagram, and what implications does this have for calculating knot invariants?
  • Reidemeister Move II alters a knot diagram by adding or removing twist crossings while preserving the overall structure. This means that when calculating invariants such as the HOMFLY or Jones polynomials, one can use this move to simplify diagrams without changing their knot type. Therefore, applying Reidemeister Move II is key for developing more efficient methods to compute these important invariants.
• Discuss how Reidemeister Move II relates to the concept of knot equivalence and its role in determining knot invariants.
  • Reidemeister Move II is crucial in understanding knot equivalence because it allows two knots that appear different at first glance to be transformed into one another without altering their fundamental properties. This move ensures that when we apply it to diagrams, we can systematically prove which knots are equivalent. The significance of this relationship becomes evident when using moves like Reidemeister Move II in conjunction with other moves to derive and compute knot invariants such as the HOMFLY and Jones polynomials.
• Evaluate the importance of Reidemeister Move II in the broader context of knot theory and its applications in other fields.
  • Reidemeister Move II plays a vital role in knot theory as it provides a foundational tool for manipulating and simplifying knot diagrams, which are essential for defining and calculating knot invariants. Beyond mathematics, these concepts find applications in fields like biology (understanding DNA topology) and physics (analyzing particle trajectories). The ability to systematically utilize moves like Reidemeister Move II facilitates advancements in these areas by improving our understanding of complex systems through the lens of knot theory.

Related terms

Knot Diagram: A visual representation of a knot that is drawn on a plane, showing the crossings and structure of the knot.

Knot Invariant: A property of a knot that remains unchanged under Reidemeister moves and other equivalent transformations, allowing for classification of knots.

HOMFLY Polynomial: A two-variable polynomial that serves as a powerful invariant for distinguishing different knots and links, incorporating additional information compared to the Jones polynomial.