Mathematics.

geometric topology

Knot Theory

Topology100 minDifficulty8 out of 10

You should know: fundamental group, manifolds

Overview

Knot theory is the mathematical study of knots — closed curves embedded in three-dimensional space — and their properties up to continuous deformation (ambient isotopy). The central problem is to determine when two knots are equivalent. This is addressed through knot invariants: algebraic or polynomial quantities that do not change under deformation. Knot theory has applications to DNA topology, quantum field theory, and the study of 3-manifolds.

Intuition

A knot is a closed loop in 3-space that may or may not be untangleable into a simple circle (the unknot). Two knots are equivalent if one can be continuously deformed into the other without cutting. To tell knots apart, we compute invariants. The fundamental group of the knot complement is a powerful invariant; polynomial invariants (Alexander, Jones, HOMFLY) give computable algebraic fingerprints. Reidemeister moves provide the combinatorial calculus for knot diagrams.

Formal Definition

Definition

A knot is a smooth embedding of S^1 into R^3 (or S^3), considered up to ambient isotopy. A link is a disjoint union of knots.

K:S1↪R3K : S^1 \hookrightarrow \mathbb{R}^3
Knot as embedding
K1∼K2  ⟺  ∃ ambient isotopy H:R3×[0,1]→R3 with H0=id, H1(K1)=K2K_1 \sim K_2 \iff \exists \text{ ambient isotopy } H : \mathbb{R}^3 \times [0,1] \to \mathbb{R}^3 \text{ with } H_0 = \mathrm{id},\ H_1(K_1) = K_2
Knot equivalence
π1(R3∖K)\pi_1(\mathbb{R}^3 \setminus K)
Knot group (fundamental group of the complement)
⟨t−1+t−1⟩(Alexander polynomial of trefoil)\langle t - 1 + t^{-1} \rangle \quad (\text{Alexander polynomial of trefoil})
Alexander polynomial example

Notation

NotationMeaning
KKA knot (smooth embedding of S^1 in R^3)
ΔK(t)\Delta_K(t)Alexander polynomial of knot K
VK(t)V_K(t)Jones polynomial of knot K
π1(S3∖K)\pi_1(S^3 \setminus K)Knot group

Theorems

Theorem 1: Reidemeister's Theorem
Two knot diagrams represent ambient isotopic knots if and only if they are related by a finite sequence of Reidemeister moves RI, RII, RIII
Theorem 2: Alexander's Theorem
EverylinkinS3canberepresentedastheclosureofabraidEvery link in S^3 can be represented as the closure of a braid
Theorem 3: Skein Relation for Jones Polynomial
t−1VL+(t)−tVL−(t)=(t1/2−t−1/2)VL0(t)t^{-1} V_{L_+}(t) - t V_{L_-}(t) = (t^{1/2} - t^{-1/2}) V_{L_0}(t)

Worked Examples

  1. 1

    RI: Add or remove a simple twist (a curl) in a strand. This does not change the topology of the knot.

    RI: a loop can be created or cancelled\text{RI: a loop can be created or cancelled}
  2. 2

    RII: Pass two strands over each other, creating or cancelling two crossings of opposite sign.

    RII: two crossings of opposite sign cancel\text{RII: two crossings of opposite sign cancel}
  3. 3

    RIII: Slide a strand over a crossing. This moves a strand past a crossing without changing connectivity.

    RIII: strand slides past crossing\text{RIII: strand slides past crossing}

✓ Answer

Each Reidemeister move corresponds to a local ambient isotopy of R^3, so it does not change the knot type.

Practice Problems

Mediumproof writing

Show that the trefoil knot is not equivalent to the unknot using the Alexander polynomial.

Hardfree response

What is the knot group of the trefoil, and why is it not abelian?

Common Mistakes

Common Mistake

Thinking knot equivalence is just homeomorphism of the knot itself

Two knots are equivalent under ambient isotopy: a deformation of the entire surrounding space R^3. The knot itself (as a circle) is always homeomorphic to S^1; the embedding in 3-space is what matters.

Common Mistake

Assuming the knot group completely classifies knots

The knot group does not distinguish all knots. For example, a knot and its mirror image have isomorphic groups in some cases. However, the knot group determines the knot for prime knots by Gordon-Luecke.

Quiz

Two knot diagrams represent equivalent knots if and only if they are related by:
Which polynomial invariant was discovered by Vaughan Jones in 1984?
The knot group of a knot K is:

Historical Background

Lord Kelvin's 19th-century vortex atom theory motivated Peter Guthrie Tait to begin systematic tabulation of knots. The mathematical theory developed through the work of Alexander (who introduced the Alexander polynomial in 1928), Conway (skein relations, 1969), and Jones (the Jones polynomial, 1984). The Jones polynomial sparked connections to quantum field theory and led to a revolution in knot invariants.

  1. 1867

    Tait begins tabulating knots inspired by Kelvin's vortex atom theory

    Peter Guthrie Tait

  2. 1928

    Alexander introduces the Alexander polynomial, the first polynomial knot invariant

    James W. Alexander

  3. 1969

    Conway introduces skein relations for computing knot polynomials

    John Conway

  4. 1984

    Jones discovers the Jones polynomial, winning the Fields Medal

    Vaughan Jones

  5. 1988

    Witten interprets the Jones polynomial via quantum Chern-Simons theory

    Edward Witten

Summary

  • A knot is a smooth embedding of S^1 into R^3, studied up to ambient isotopy.
  • Reidemeister's theorem reduces knot equivalence to three local moves on diagrams.
  • Knot invariants include the knot group, Alexander polynomial, and Jones polynomial.
  • The trefoil is the simplest non-trivial knot; the Hopf link is the simplest non-trivial link.
  • The Jones polynomial connects knot theory to quantum field theory via Chern-Simons theory.

References

  1. BookAdams, C. — The Knot Book (2004), American Mathematical Society
  2. WebsiteWikipedia — Knot theory
  3. WebsiteMathWorld — Knot Theory
  4. WebsitenLab — knot theory