Murasugi sum

In knot theory, a Murasugi sum is a way of combining the Seifert surfaces of two knots or links, given with embeddings in space of each knot and of a Seifert surface for each knot, to produce another Seifert surface of another knot or link. It was introduced by Kunio Murasugi, who used it to compute the genus^[1] and Alexander polynomials^[2] of certain alternating knots. When the two given Seifert surfaces have the minimum genus for their knot, the same is true for their Murasugi sum.^[3] However, the genus of non-minimal-genus Seifert surfaces does not behave as predictably under Murasugi sums.^[4]