6.1 Definition and properties of the Alexander polynomial

The Alexander polynomial is a powerful tool in knot theory, assigning a unique polynomial to each knot or link. It's calculated from a knot diagram and remains constant under deformation, making it useful for distinguishing between different knots.

This polynomial has interesting properties, like symmetry and reversibility. Its degree and coefficients provide insights into the knot's structure, including a lower bound for the knot's genus. However, it's not a perfect classifier, as different knots can share the same polynomial.

Alexander Polynomial

Definition of Alexander polynomial

• Associates a polynomial invariant $\Delta_K(t)$ with a knot or link $K$ computed from a diagram of the knot or link
• Constructed by creating the Alexander matrix and calculating its determinant
• Laurent polynomial in the variable $t$ with integer coefficients
  • Lowest degree term has a positive power of $t$
  • Highest degree term has a negative power of $t$ (trefoil knot, figure-eight knot)

Invariance under ambient isotopy

• Remains constant under ambient isotopy, continuous deformation without self-intersection (Reidemeister moves)
• Knots or links with identical polynomials may not be equivalent as different knots can share the same Alexander polynomial (knot 5_1 and knot 10_132)
• Knots or links with distinct polynomials are definitely not equivalent (trefoil knot and figure-eight knot)

Symmetry and reversibility properties

• Symmetry property: $\Delta_K(t) = \Delta_K(t^{-1})$ meaning coefficients are the same read from left to right or right to left due to the determinant of the Alexander matrix
• Reversibility property: $\Delta_{K^r}(t) = \Delta_K(t^{-1})$, where $K^r$ is the reverse of knot $K$
  • Obtained by substituting $t^{-1}$ for $t$ in the original polynomial
  • Results from the symmetry property and reversing a knot preserves its type (trefoil knot and its reverse)

Interpretation of degree and coefficients

• Degree of $\Delta_K(t)$ gives a lower bound for the genus $g(K)$ of knot $K$, the minimum number of handles needed to construct a surface bounded by the knot
  • $\text{deg}(\Delta_K(t)) \leq 2g(K)$ (trefoil knot has genus 1, degree 2)
• Coefficients alternate in sign with the constant term (coefficient of $t^0$) and leading/trailing coefficients (coefficients of highest/lowest degree terms) always $\pm 1$
• Absolute values of coefficients do not necessarily decrease from the center (figure-eight knot polynomial $t^{-1} - 3 + t$)