Alexander's and Markov's theorems in dimension four

Author:
Seiichi Kamada

Journal:
Bull. Amer. Math. Soc. **31** (1994), 64-67

MSC:
Primary 57Q45; Secondary 57M25

DOI:
https://doi.org/10.1090/S0273-0979-1994-00505-1

MathSciNet review:
1254074

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Abstract: Alexander's and Markov's theorems state that any link type in is represented by a closed braid and that such representations are related by some elementary operations called Markov moves. We generalize the notion of a braid to that in 4-dimensional space and establish an analogue of these theorems.

**[A]**J. W. Alexander,*A lemma on systems of knotted curves*, Proc. Nat. Acad. Sci. U.S.A.**9**(1923), 93-95.**[B]**J. Birman,*Braids, links and mapping class group*, Ann. of Math. Stud., vol. 82, Princeton Univ. Press, Princeton, NJ, 1974. MR**0375281 (51:11477)****[BS]**E. Brieskorn and K. Saito,*Artin Gruppen und Coxeter Gruppen*, Invent. Math.**17**(1972), 245-271. MR**0323910 (48:2263)****[D]**D. Dahm,*A generalization of braid theory*, Ph.D. thesis, Princeton Univ., Princeton, NJ, 1962.**[G]**F. González-Acuña,*A characterization of*2-*knot groups*, preprint.**[G1]**D. L. Goldsmith,*The theory of motion groups*, Michigan Math. J.**28**(1981), 3-17. MR**600411 (82h:57007)****[G2]**-,*Motion of links in the*3-*sphere*, Math. Scand.**50**(1982), 167-205. MR**672923 (83k:57002)****[J]**V. F. R. Jones,*A polynomial invariant for knots via von Neumann algebras*, Bull. Amer. Math. Soc. (N.S.)**12**(1985), 103-111. MR**766964 (86e:57006)****[K1]**S. Kamada,*A characterization of groups of closed orientable surfaces in*4-*space*, Topology**33**(1944), 113-122. MR**1259518 (95a:57002)****[K2]**-,*Surfaces in**of braid index three are ribbon*, J. Knot Theory Ramifications**1**(1992), 137-160. MR**1164113 (93h:57039)****[K3]**-, 2-*dimensional braids and chart descriptions*, Topics in Knot Theory, Proceedings of the NATO ASI on Topics in Knot Theory, Turkey, 1992 (M. E. Bozhüyük, ed.), pp. 277-287. MR**1257915****[MS]**Yu. I. Manin and V. V. Schechtman,*Arrangements of hyperplanes, higher braid groups and higher Bruhat orders*, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1986, pp. 289-308. MR**1097620 (91m:32042)****[Mo]**H. R. Morton,*Threading knot diagrams*, Math. Proc. Cambridge Philos. Soc.**99**(1986), 247-260. MR**817666 (87c:57007)****[R1]**L. Rudolph,*Braided surfaces and Seifert ribbons for closed braids*, Comment. Math. Helv.**58**(1983), 1-37. MR**699004 (84j:57006)****[R2]**-,*Special positions for surfaces bounded by closed braids*, Rev. Mat. Iberoamericana**1**(1985) 93-133 MR**836285 (88a:57018)**

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DOI:
https://doi.org/10.1090/S0273-0979-1994-00505-1

Article copyright:
© Copyright 1994
American Mathematical Society