Seifert Matrix
Given a Seifert form f(x,y), choose a basis e_1, ..., e_(2g) for H_1(M^^) as a Z-module so every element is uniquely expressible as
| n_1e_1+...+n_(2g)e_(2g) |
(1)
|
with n_i integer. Then define the Seifert matrix V as the 2g×2g integer matrix with entries
| v_(ij)=lk(e_i,e_j^+). |
(2)
|
For example, the right-hand trefoil knot has Seifert matrix
| [画像: V=[-1 1; 0 -1]. ] |
(3)
|
![画像: V=[-1 1; 0 -1].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/SeifertMatrix/NumberedEquation3.svg){kind=link}
A Seifert matrix is not a knot invariant, but it can be used to distinguish between different Seifert surfaces for a given knot.
See also
Alexander MatrixExplore with Wolfram|Alpha
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References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 200-203, 1976.Referenced on Wolfram|Alpha
Seifert MatrixCite this as:
Weisstein, Eric W. "Seifert Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SeifertMatrix.html