6.1 Definition and properties of the Alexander polynomial
2 min read•july 22, 2024
The is a powerful tool in knot theory, assigning a unique polynomial to each knot or link. It's calculated from a and remains constant under deformation, making it useful for distinguishing between different knots.
This polynomial has interesting properties, like symmetry and reversibility. Its and provide insights into the knot's structure, including a for the knot's . However, it's not a perfect classifier, as different knots can share the same polynomial.
Alexander Polynomial
Definition of Alexander polynomial
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Associates a polynomial invariant ΔK(t) with a knot or link K computed from a diagram of the knot or link
Constructed by creating the and calculating its
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 (, )
Invariance under ambient isotopy
Remains constant under , continuous deformation without self-intersection ()
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 are definitely not equivalent (trefoil knot and figure-eight knot)
Symmetry and reversibility properties
: ΔK(t)=ΔK(t−1) meaning coefficients are the same read from left to right or right to left due to the determinant of the Alexander matrix
: ΔKr(t)=ΔK(t−1), where Kr 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 ΔK(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
deg(ΔK(t))≤2g(K) (trefoil knot has genus 1, degree 2)
Coefficients alternate in sign with the constant term (coefficient of t0) and leading/trailing coefficients (coefficients of highest/lowest degree terms) always ±1
do not necessarily decrease from the center (figure-eight knot polynomial t−1−3+t)