Definition:Knot Polynomial
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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