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3.1 Concept of knot invariants

2 min readLast Updated on July 22, 2024

Knot invariants are mathematical properties that remain constant when a knot is deformed. They help distinguish different knots and include crossing numbers, tricolorability, and knot polynomials. These tools are crucial for understanding knot equivalence.

While knot invariants are powerful for differentiating knots, they have limitations. No single invariant can classify all knots, and knots with the same invariants aren't necessarily equivalent. This ongoing challenge keeps mathematicians searching for new methods to understand knots.

Knot Invariants

Definition of knot invariants

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  • Mathematical properties or quantities associated with a knot that remain constant under continuous deformations (ambient isotopy)
  • Help distinguish different knots from one another even if they appear similar
  • Two knots with different invariants cannot be equivalent
  • Examples include:
    • Crossing number represents the minimum number of crossings in any diagram of the knot
    • Tricolorability determines whether a knot diagram can be colored with three colors while following certain rules
    • Knot polynomial is a polynomial associated with a knot (Jones polynomial, Alexander polynomial)

Invariance under ambient isotopy

  • Ambient isotopy is a continuous deformation of the 3-dimensional space containing a knot without allowing the knot to pass through itself
  • Can be visualized as manipulating a knotted rope without cutting or gluing it
  • Knot invariants are designed to be unaffected by ambient isotopy
  • If two knots have different invariants, they cannot be transformed into each other through ambient isotopy
  • If two knots have the same invariants, they may be equivalent but additional invariants might be needed to confirm this

Types of knot invariants

  • Crossing number:
    • Determines the minimum complexity of a knot
    • Can quickly rule out the equivalence of knots with different crossing numbers
  • Tricolorability:
    • A knot is tricolorable if its diagram can be colored with three colors following the rule that at each crossing, either all three colors are present or only one color is present
    • Non-tricolorable knots cannot be equivalent to tricolorable knots
  • Knot polynomials:
    • Algebraic expressions associated with knots obtained through specific rules and calculations
    • Different knot polynomials (Jones polynomial, Alexander polynomial) can distinguish between knots that other invariants might not
    • Reveal connections between knot theory and other areas of mathematics (statistical mechanics, quantum field theory)

Limitations in knot classification

  • Knot invariants are powerful tools for distinguishing knots but have limitations in providing a complete classification
  • Two knots with the same invariants are not necessarily equivalent; they may be distinguished by other, more sensitive invariants
  • No known single invariant or finite set of invariants can distinguish all knots from one another
  • Mutant knots have the same polynomial invariants but are not equivalent highlighting the need for additional invariants or techniques to fully classify knots
  • Classification of knots remains an ongoing area of research in knot theory
  • Mathematicians continue to develop new invariants and methods to better understand and distinguish knots
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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