Key Knot Polynomials

Why This Matters

Knot polynomials are the workhorses of knot theory—they transform the visual complexity of tangled loops into algebraic expressions you can actually compute and compare. When you're asked whether two knots are equivalent or distinct, these polynomials give you concrete tools to answer. You're being tested on understanding how each polynomial is constructed, what information it captures, and crucially, where each one succeeds or fails as an invariant.

The deeper concept here is the relationship between topology, algebra, and computation. Each polynomial encodes geometric information differently—some detect chirality, others relate to the knot complement's fundamental group, and still others connect to quantum mechanics and statistical physics. Don't just memorize the variable names and discoverers—know what makes each polynomial powerful, what it misses, and how the polynomials relate to one another through generalization and specialization.

---

Single-Variable Classical Invariants

These polynomials use one variable and emerged from classical algebraic topology, connecting knot diagrams to group theory and recursive computation.

Alexander Polynomial

• First knot polynomial ever discovered (1928)—James W. Alexander introduced it using the knot group presentation, making it the foundation for all later polynomial invariants
• Laurent polynomial in one variable $\Delta(K, t)$—computed from the knot diagram using the fundamental group of the knot complement
• Limited distinguishing power—different knots can share the same Alexander polynomial, so it's not a complete invariant, but it reveals information about the knot's genus and fibering

Conway Polynomial

• Reformulation of Alexander via skein relations—John Horton Conway showed the Alexander polynomial could be computed recursively from crossing changes, simplifying calculations dramatically
• Laurent polynomial $\nabla(K, z)$—related to Alexander by a variable substitution, making conversion between them straightforward
• Connects to knot genus—provides bounds on the minimum genus of a spanning surface, linking algebraic invariants to geometric properties

---

Quantum-Inspired Polynomials

These invariants emerged from connections to statistical mechanics and quantum groups, offering stronger distinguishing power through state-sum models.

Jones Polynomial

• Revolutionary 1984 discovery—Vaughan Jones found this invariant through operator algebras, unexpectedly connecting knot theory to quantum physics and statistical mechanics
• Laurent polynomial $V(K, t)$ computed via bracket polynomial—uses a state-sum model that counts ways to "smooth" crossings, with each state weighted by powers of $t$
• Detects chirality—can distinguish many knots that Alexander cannot, including telling a knot from its mirror image in cases where Alexander fails

Kauffman Polynomial

• Two-variable generalization $F(K, a, z)$—Louis Kauffman (1987) developed this using an unoriented state-sum model, capturing different information than Jones
• Includes both regular and framed isotopy versions—the bracket polynomial is a building block, but Kauffman's full polynomial handles writhe normalization differently
• Distinguishes some knots Jones misses—particularly useful for unoriented links and provides independent information from the Jones polynomial

---

Unifying Multi-Variable Polynomials

These two-variable polynomials generalize the single-variable invariants, revealing deeper structural relationships.

HOMFLY-PT Polynomial

• Grand unification of Alexander and Jones—discovered independently by multiple groups in 1985, this two-variable polynomial $P(K, l, m)$ contains both classical invariants as special cases
• Computed via skein relation—the relation $lP(L_+) + l^{-1}P(L_-) + mP(L_0) = 0$ connects any three links differing at one crossing
• Strongest polynomial for oriented links—setting specific variable values recovers Alexander ($l = i, m = i(t^{1/2} - t^{-1/2})$) or Jones ($l = it^{-1}, m = i(t^{1/2} - t^{-1/2})$), making it the go-to invariant for systematic study

---

Beyond Polynomials: Finite-Type Invariants

This framework organizes all polynomial invariants into a unified hierarchy based on singularity theory.

Vassiliev Invariants

• Infinite sequence of invariants indexed by degree—arising from studying singular knots (with self-intersections), these invariants form a filtration that systematically captures knot complexity
• Polynomial invariants decompose into Vassiliev invariants—the coefficients of Jones, HOMFLY-PT, and other polynomials can be expressed as combinations of finite-type invariants
• Conjectured to be complete—while unproven, mathematicians believe Vassiliev invariants together can distinguish all knots, making them the theoretical "ultimate" invariant system

---

Quick Reference Table

---

Self-Check Questions

1. Which two polynomials can both be recovered as special cases of the HOMFLY-PT polynomial, and what variable substitutions accomplish this?

2. Compare and contrast the Jones and Kauffman polynomials: what computational approach do they share, and what key difference makes them complementary invariants?

3. A knot and its mirror image have the same Alexander polynomial. Which polynomial should you try next to distinguish them, and why?

4. How do Vassiliev invariants relate to polynomial invariants—are they an alternative, a generalization, or something else entirely?

5. If you needed to compute a knot polynomial by hand from a diagram, which approach (group-theoretic or skein relation) would be more practical, and which polynomials support that method?