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]
References
edit- ^ Murasugi, Kunio (1958), "On the genus of the alternating knot. I, II", Journal of the Mathematical Society of Japan, 10: 94–105, 235–248, doi:10.2969/jmsj/01010094, MR 0099664
- ^ Murasugi, Kunio (1963), "On a certain subgroup of the group of an alternating link", American Journal of Mathematics, 85: 544–550, doi:10.2307/2373107, MR 0157375
- ^ Gabai, David (1983), "The Murasugi sum is a natural geometric operation", Low-dimensional topology (San Francisco, Calif., 1981), Contemporary Mathematics, vol. 20, American Mathematical Society, pp. 131–143, doi:10.1090/conm/020/718138, ISBN 0-8218-5016-4, MR 0718138
- ^ Thompson, Abigail (1994), "A note on Murasugi sums", Pacific Journal of Mathematics, 163 (2): 393–395, MR 1262303
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