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oai:docta.ucm.es:20.500.14352/647142023-08-27T17:01:11Zcom_20.500.14352_14col_20.500.14352_15

Montesinos Amilibia, José María 2023-06-21T02:03:00Z 2023-06-21T02:03:00Z 1976-02-01 0008-414X 10.4153/CJM-1976-020-5 https://hdl.handle.net/20.500.14352/64714 http://cms.math.ca/cjm/v28/cjm1976v28.0161-0167.pdf http://cms.math.ca/ J. S. Birman [same J. 28 (1976), no. 2, 264–290] has shown that any two plat representations of a link in S3 are stably equivalent and that stabilization is a necessary feature of the equivalence for certain composite knots. She has asked whether all 2n-plat representations of a prime link are equivalent. The author provides a negative answer, by exhibiting an infinite collection of prime knots and links in S3 in which each element L has at least two minimal and inequivalent 6-plat representations. In addition, as an application of another result of Birman [Knots, groups and 3-manifolds (Papers dedicated to the memory of R. H. Fox), pp. 137–164, Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975], the 2-fold cyclic covering spaces of S3 branched over such links L form further examples of closed, orientable, prime 3-manifolds having inequivalent minimal Heegaard splittings, which were first constructed by Birman, F. González-Acuña and the author [Michigan Math. J. 23 (1976), no. 2, 97–103]. eng restricted access Minimal plat representations of prime knots and links are not unique journal article