Definition:Alexander Polynomial

Definition

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

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

Also see

• Definition:Alexander-Conway Polynomial
• Definition:Jones Polynomial

• Results about Alexander polynomials can be found here.

Source of Name

This entry was named for James Waddell Alexander II.

Historical Note

The Alexander polynomial was discovered in $1928$ by James Waddell Alexander II using homology theory.

In the late $1960$s, John Horton Conway discovered a simpler and explicit method of finding the Alexander polynomial by using transformations applied to the planar representation of the knot.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Alexander polynomial
• 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): Alexander polynomial
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): knot polynomial
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Alexander, James Waddell (1888-1971)