equal to a direct sum of the g copies of the matrix The 2g × 2g integer Seifert matrix has the linking number in Euclidean 3-space (or in the 3-sphere) of ai and the "pushoff" of aj in the positive direction of . More precisely, recalling that Seifert surfaces are bicollared, meaning that we can extend the embedding of to an embedding of , given some representative loop which is homology generator in the interior of , the positive pushout is and the negative pushout is . With this, we have where V∗ = (v(j, i)) the transpose matrix. Every integer 2g × 2g matrix with arises as the Seifert matrix of a knot with genus g Seifert surface. The Alexander polynomial is computed from the Seifert matrix by which is a polynomial of degree at most 2g in the indeterminate The Alexander polynomial is independent of the choice of Seifert surface and is an invariant of the knot or link. The signature of a knot is the signature of the symmetric Seifert matrix It is again an invariant of the knot or link. Genus of a knot Seifert surfaces are not at all unique: a Seifert surface S of genus g and Seifert matrix V can be modified by a topological surgery, resulting in a Seifert surface S′ of genus g + 1 and Seifert matrix The genus of a knot K is the knot invariant defined by the minimal genus g of a Seifert surface for K. For instance: An unknot—which is, by definition, the boundary of a disc—has genus zero. Moreover, the unknot is the knot with genus zero. The trefoil knot has genus 1, as does the figure-eight knot. The genus of a (p,q)-torus knot is (p − 1)(q − 1)/2 The degree of a knot's Alexander polynomial is a lower bound on twice its genus. A fundamental property of the genus is that it is additive with respect to the knot sum: In general, the genus of a knot is difficult to compute, and the Seifert algorithm usually does not produce a Seifert surface of least genus. For this reason other related invariants are sometimes useful. The canonical genus of a knot is the least genus of all Seifert surfaces that can be constructed by the Seifert algorithm, and the free genus is the least genus of all Seifert surfaces whose complement in is a