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I have been asked to compute the seifert form of a knot, the twist knot.

I know how to compute the seifert surface, and then the seifert matrix seems to be defined accordingly (according to all the sources that I could find):

$V_{ij} = lk(x_i,x_j^+)$ (I believe this is called the linking number)

My problem is that I dont know what the $x_i$ and $x_j^+$ mean, Thank you very much in advance for answering those. May I add that I have been searching the internet for several hours with no clear explanation found, and that is why I'd be most grateful for your answer.

Aleks J
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The Seifert surface $\Sigma\subset S^3$ is oriented, so the normal bundle $\nu$ is trivial; choose a nowhere-vanishing section $s\in \Gamma(\nu)$. For a curve $\gamma$, let $\gamma^+$ denote the curve obtained by pushing $\gamma$ off $\Sigma$ a small distance along $s$. Basically, we're taking a tubular neighborhood around $\Sigma$ and pushing $\gamma$ outward, but codimension $1$ and orientability mean that there's a well-defined direction in which to push at each point along $\gamma$. Descending to homology gives $\operatorname{lk}(x_i, x_j^+)$.

anomaly
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  • Could I ask for a lower level explanation? I am not really familiar with the homology groups, and we have only been introduced to the seifert algorithm. I need to know the practical way of computing the $\gamma$ and the $\gamma^+$ as in your example. I have seen example on the internet ( even on this website) where people 'magically' get some circles instead of $x_i$'s and I am asking how do people get those circles, see: http://math.stackexchange.com/questions/191402/seifert-matrices-and-arf-invariant-cinquefoil-knot How did he "push off" the x's so they started overlapping? – Aleks J Jun 23 '16 at 22:13
  • There might be more concrete versions of the algorithm, but there's no way to get away from homology; the Seifert matrix is actually a bilinear form on $H_1(\Sigma)$. Surfaces do have a canonical choice of basis for $H_1(\Sigma)$ as embedded curves, and the method of pushing off is more or less putting the curves in general position. – anomaly Jun 24 '16 at 02:06