6.3 Applications and limitations of the Alexander polynomial

The Alexander polynomial is a powerful tool in knot theory, helping us tell different knots apart. It's calculated using knot diagrams and has cool properties like being multiplicative for connected sums. This makes it super useful for studying knots and links.

But it's not perfect. Some different knots can have the same Alexander polynomial, and it can't tell apart a knot from its mirror image. Still, it's connected to other important concepts in knot theory and has real-world applications in biology, physics, and chemistry.

Applications and Properties of the Alexander Polynomial

Alexander polynomial for knot distinction

• Powerful knot invariant distinguishes between different knots and links
  • Different Alexander polynomials prove knots are distinct (trefoil knot $\Delta(t) = t^2 - t + 1$, figure-eight knot $\Delta(t) = t^2 - 3t + 1$)
• Calculated using knot diagram and determinant formula
  • Independent of chosen diagram for a given knot, making it well-defined invariant
• Multiplicative under connected sum operation
  • Connected sum $K_1 \# K_2$ has Alexander polynomial $\Delta_{K_1 \# K_2}(t) = \Delta_{K_1}(t) \cdot \Delta_{K_2}(t)$
• Distinguishes between different links
  • Links with different numbers of components have different Alexander polynomials (Hopf link $\Delta(t) = t^{1/2} - t^{-1/2}$, unlink $\Delta(t) = 0$)

Limitations of Alexander polynomial

• Not a complete invariant, distinct knots can have same Alexander polynomial
  • Conway knot and Kinoshita-Terasaka knot both have $\Delta(t) = 1$ but are distinct
• Cannot detect chirality of a knot
  • Knot and its mirror image have same Alexander polynomial (right-handed and left-handed trefoil knots $\Delta(t) = t^2 - t + 1$)
• Does not provide information about genus of a knot
  • Knots with different genera can have same Alexander polynomial
• Cannot detect unknotting number of a knot
  • Minimum number of crossings changed to transform knot into unknot
  • Knots with different unknotting numbers can have same Alexander polynomial

Alexander polynomial vs other invariants

• Related to Jones polynomial, another important knot invariant
  • Jones polynomial obtained from Alexander polynomial by variable substitution and normalization
  • Jones polynomial is stronger invariant, can distinguish knots with same Alexander polynomial
• Connected to knot group, fundamental group of knot complement
  • Alexander polynomial derived from Alexander matrix, obtained from presentation of knot group
  • Knot group contains more information than Alexander polynomial but more difficult to compute and compare
• Related to signature of a knot, another numerical invariant
  • Signature obtained from Alexander polynomial by evaluating at $t = -1$
  • Signature provides information about cobordism class of the knot

Applications of Alexander polynomial

• Biology: studies topology of DNA molecules
  • DNA forms knots and links during replication and recombination
  • Classifies and distinguishes different DNA knot types
  • Analyzes knotting probability and complexity of DNA in viral capsids
• Physics: appears in quantum knot invariants and topological quantum field theories
  • Quantum knot invariants generalize classical knot polynomials (Alexander and Jones) in quantum mechanics context
  • Related to Chern-Simons theory, describes behavior of knots and links in 3D space
• Chemistry: applications in molecular knots and links
  • Molecules (catenanes, rotaxanes) exhibit non-trivial topological structures described using knot theory
  • Characterizes and distinguishes different molecular knot types
  • Studies synthesis and properties of molecular Borromean rings (link of three interlocked rings)