Definition:Knot Polynomial

Definition

Context: knot theory.

A knot polynomial is a polynomial associated with a knot.

Two knots are equivalent if and only if their knot polynomials are equal.

Alexander Polynomial

For a knot $K$ with Seifert matrix $V$, the Alexander polynomial of $K$ is defined as:

$\map {\Delta_K} t = \map \det {V - t V^\intercal}$

Alexander-Conway Polynomial

An Alexander-Conway polynomial is a polynomial associated with a knot.

Jones Polynomial

A Jones Polynomial is a polynomial associated with a knot.

Also see

• Results about knot polynomials can be found here.

Historical Note

As of time of writing, a common framework within which to understand the knot polynomials has not yet been discovered.

A possible strategy has been suggested by Victor Anatolyevich Vassiliev, who introduced the concept of the Vassiliev invariant in $1990$, based on his work in singularity theory.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): knot polynomial
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): knot polynomial