Fibred closed braids with disc-band fibre surfaces

Marta Rampichini

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 2, page 433-451
  • ISSN: 0392-4041

Abstract

top
A classical result by Stallings provides a necessary and sufficient condition to decide whether a given embedded surface S is a fibre in S 3 - ∂ ⁡ S . In this paper it is described how to find a candidate fibre surface for a a link presented as a closed braid. Also it is described an implemented algorithm to find the main ingredients of the necessary and sufficient condition of Stallings, namely presentations of the fundamental groups of the surface and of its complement in S 3 , and an explicit expression of the homomorphism induced in homotopy by the push-off map. The paper ends with a discussion of the particular properties of the presentation of π 1 ⁢ S 3 ∖ S W .

How to cite

top

Rampichini, Marta. "Fibred closed braids with disc-band fibre surfaces." Bollettino dell'Unione Matematica Italiana 7-B.2 (2004): 433-451. <http://eudml.org/doc/195824>.

@article{Rampichini2004,
abstract = {A classical result by Stallings provides a necessary and sufficient condition to decide whether a given embedded surface $S$ is a fibre in $S^\{3\}-\partial S$. In this paper it is described how to find a candidate fibre surface for a a link presented as a closed braid. Also it is described an implemented algorithm to find the main ingredients of the necessary and sufficient condition of Stallings, namely presentations of the fundamental groups of the surface and of its complement in $S^\{3\}$, and an explicit expression of the homomorphism induced in homotopy by the push-off map. The paper ends with a discussion of the particular properties of the presentation of $\pi_1 (S^\{3\} \setminus S_\{W\})$.},
author = {Rampichini, Marta},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {433-451},
publisher = {Unione Matematica Italiana},
title = {Fibred closed braids with disc-band fibre surfaces},
url = {http://eudml.org/doc/195824},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Rampichini, Marta
TI - Fibred closed braids with disc-band fibre surfaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/6//
PB - Unione Matematica Italiana
VL - 7-B
IS - 2
SP - 433
EP - 451
AB - A classical result by Stallings provides a necessary and sufficient condition to decide whether a given embedded surface $S$ is a fibre in $S^{3}-\partial S$. In this paper it is described how to find a candidate fibre surface for a a link presented as a closed braid. Also it is described an implemented algorithm to find the main ingredients of the necessary and sufficient condition of Stallings, namely presentations of the fundamental groups of the surface and of its complement in $S^{3}$, and an explicit expression of the homomorphism induced in homotopy by the push-off map. The paper ends with a discussion of the particular properties of the presentation of $\pi_1 (S^{3} \setminus S_{W})$.
LA - eng
UR - http://eudml.org/doc/195824
ER -

References

top
  1. BIRMAN, J. S., Braids, links and mapping class groups, Annals of Math. Studies84 (1974), Princeton Univ. Press. Zbl0305.57013MR375281
  2. BIRMAN, J. S.- KO, K. H.- LEE, S. J., A new approach to the word and conjugacy problems in the braid groups, Adv. in Math., 139 (1998), 322-353. Zbl0937.20016MR1654165
  3. BIRMAN, J. S.- MENASCO, W. W., Studying links via closed braids II: On a theorem of Bennequin, Top. Appl., 40 (1991), 71-82. Zbl0722.57001MR1114092
  4. BIRMAN, J. S.- MENASCO, W. W., Studying links via closed braids VI: A non-finiteness theorem, Pacific J. of Math., 156 (1992), 265-285. Zbl0739.57002MR1186805
  5. BIRMAN, J. S.- WILLIAMS, R. F., Knotted periodic orbits in dynamical systems, I: Lorenz's equations, Top., 22 (1983), 47-82. Zbl0507.58038MR682059
  6. EISENBUD, D.- NEUMANN, W., Three-dimensional link theory and invariants of plane curve singularities, Annals of Math. Stud.110, Princeton Univ. Press (1985). Zbl0628.57002MR817982
  7. GABAI, D., Detecting fibred links in S 3 , Comment. Math. Helvetici, 61 (1986), 519-555. Zbl0621.57003MR870705
  8. GABAI, D., Genera of the arborescent links, Memoirs Amer. Math. Soc., 59 (1986), 1-98. Zbl0585.57003MR823442
  9. GABAI, D., Foliations and genera of links, Top., 23 (1982), 381-394. Zbl0567.57021MR780731
  10. HARER, J., How to construct all fibered knots and links, Top., 21 (1982), 263-280. Zbl0504.57002MR649758
  11. KO, K. H.- LEE, S. J., Genera of some clesed 4 -braids, Top. Appl., 78 (1997), 61-77. Zbl0879.57006MR1465025
  12. LYNDON, R. C.- SCHUPP, P. E., Combinatorial Group Theory, Springer, 1977. Zbl0368.20023MR577064
  13. MILNOR, J., Singular points of complex hypersurfaces, Annals of Math. Studies61. Zbl0184.48405
  14. MONTESINOS-AMILIBIA, J. M.- MORTON, H. R., Fibred links from closed braids, Proc. London Math. Soc. (3), 62 (1991), 167-201. Zbl0673.57010MR1078219
  15. MORTON, H. R., Infinitely many fibred knots having the same Alexander polynomial, Top.,7 (1978), 101-104. Zbl0383.57005MR486796
  16. OERTEL, U., HOMOLOGY BRANCHED SURFACES: THURSTON'S NORM ON H 2 ⁢ M 3 , in Low dimensional topology and Kleinian groups, London Math. Soc. Lecture Note Ser.112 (1986) 253-272. Zbl0628.57011MR903869
  17. RAMPICHINI, M., Exchangeable fibred links, doctoral thesis, Univ. of Milan, Italy (1998). 
  18. ROLFSEN, D., Knots and links, Math. Lecture Series7 (1976), Publish or Perish Inc., Berkeley. Zbl0339.55004MR515288
  19. RUDOLPH, L., Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helvetici, 58 (1983), 1-37. Zbl0522.57017MR699004
  20. RUDOLPH, L., Construction of quasipositive knots and links, I, in Noeuds, tresses et singularités, L'einsegnement Mathematique, 31 (1983), 233-245. Zbl0557.57002MR728589
  21. RUDOLPH, L., Special positions for surface bounded by closed braids, Rev. Mat. Iberoamericana, 1 (1985), 93-133. Zbl0603.57004MR836285
  22. RUDOLPH, L., Quasipositive plumbing (Construction of quasipositive knots and links, V), Proc. Amer. Math. Soc., 126 (1998), 257-267. Zbl0888.57010MR1452826
  23. STALLINGS, J. R., On fibering certain 3 -manifolds, in Topology of 3-manifolds and related topics, ed. M. K. Fort, jr., Prentice-Hall inc, 1962, 95-100. MR158375
  24. STALLINGS, J. R., Constructions of fibred knots and links, Proc. Symp. in Pure Math., AMS32 part 2 (1978), 55-60. Zbl0394.57007MR520522
  25. TOLLEFSON, J. L.- WANG, N., Taut normal surfaces, Topology, 35 (1996), 55-75. Zbl0868.57022MR1367275
  26. XU, P., The genus of closed 3 -braids, J. Knot Theory Ram., 1 (1992), 303-326. Zbl0773.57007MR1180404

NotesEmbed ?

top

You must be logged in to post comments.