Crossing

Definition

In knot theory, a crossing refers to the point where two strands of a knot or braid intersect. Crossings are essential for understanding the structure of knots and braids, as they determine the knot's complexity and properties. The arrangement and type of crossings can influence calculations such as the knot group, polynomials like the Alexander polynomial, and results from braid theory.

5 Must Know Facts For Your Next Test

1. Each crossing in a knot can be classified as either an overcrossing or an undercrossing, which affects the way the knot is represented.
2. In braid theory, the number of crossings provides important information about the braid's complexity and can help determine equivalence between different braids.
3. The calculation of the Alexander polynomial involves considering how crossings are arranged in a knot diagram, affecting the resulting polynomial's coefficients.
4. The Wirtinger presentation uses crossings to define the relations in the fundamental group of a knot, linking algebraic structures to geometric representations.
5. Markov's theorem states that two braids represent the same knot if one can be transformed into another through a series of allowed operations that include changing crossings.

Review Questions

• How do overcrossings and undercrossings contribute to distinguishing different knots?
  • Overcrossings and undercrossings play a crucial role in distinguishing knots by determining how strands interact at crossing points. These classifications help establish the orientation of the strands and influence whether two knots can be transformed into one another. Understanding the arrangement of these crossings allows mathematicians to differentiate between knots and analyze their properties more effectively.
• Discuss how crossings affect the computation of the Alexander polynomial and its significance in knot theory.
  • Crossings are integral to computing the Alexander polynomial, as they directly influence the knot diagram's representation. Each crossing contributes to forming the polynomial's matrix representation, which leads to its coefficients. The Alexander polynomial serves as an invariant that helps distinguish different knots, making crossings not only a structural component but also critical in understanding knots' algebraic properties.
• Evaluate the impact of crossing changes on braids and how this relates to Markov's theorem.
  • Crossing changes have a significant impact on braids, as they can modify the braid's structure while maintaining its fundamental characteristics. According to Markov's theorem, two braids represent the same knot if they can be related through crossing changes and other specific operations. This highlights how manipulating crossings allows for exploration of knot equivalences, revealing deeper connections between braid theory and knot theory.

Related terms

Overcrossing: An overcrossing occurs when one strand of a knot goes over another strand at a crossing point.

Undercrossing: An undercrossing happens when one strand goes under another strand at a crossing point.

Braid: A braid is a collection of strands that are intertwined in a specific manner, defined by their crossings and arrangements.