×

Obstructions to trivializing a knot. (English) Zbl 1074.57002

J. Birman and W. Menasco showed that a closed braid representing the unknot can be reduced to a 1-braid by a finite sequence of “exchange moves” and “reducing moves”. Using the work of Krammer-Lawrence and Bigelow on faithful matrix representations of the braid group \(B_n\) the authors prove algebraic criteria for the detection of such moves. The tool is an intersection pairing defined with the help of the above mentioned representations of \(B_n\). The criteria imply so-called “simple” homology classes represented by simple loops. It remains an unsolved question how to give an algebraic description of these simple homology classes. This would suffice to make this approach into an algorithm to recognize the unknot.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bigelow, S., The Burau representation of B_5 is not faithful, Geometry and Topology, 3, 397-404 (1999) · Zbl 0942.20017
[2] Bigelow, S., Braid groups are linear, Journal of the American Mathematical Society, 14, 471-486 (2001) · Zbl 0988.20021
[3] Birman, J. S.; Menasco, W., Studying links via closed braids IV: Closed braid representatives of split and composite links, Inventiones Mathematicae, 102, 115-139 (1990) · Zbl 0711.57006
[4] Birman, J. S.; Menasco, W., Studying links via closed braids V: Closed braid representatives of the unlink, Transaction of the American Mathematical Society, 329, 585-606 (1992) · Zbl 0758.57005
[5] Birman, J. S.; Menasco, W., Studying links via closed braids VI: A nonfiniteness theorem, Pacific Journal of Mathematics, 156, 265-285 (1992) · Zbl 0739.57002
[6] J. Birman,Braids, Links and Mapping Class Groups, Annals of Mathematics Studies 82, Princeton University Press, 1974.
[7] Birman, J.; Hirsch, M., A new algorithm for recognizing the unknot, Geometry and Topology, 2, 175-220 (1998) · Zbl 0955.57005
[8] Birman, J.; Lubotzky, A.; McCarthy, J., Abelian and solvable subgroups of the mapping class group, Duke Mathematical Journal, 50, 1107-1120 (1983) · Zbl 0551.57004
[9] J. Birman, D. Long and J. Moody,Finite-dimensional representations of Artin’s braid group, inThe Mathematical Legacy of Wilhelm Magnus, Contemporary Mathematics169 (1994), 123-132. · Zbl 0847.20035
[10] Burau, W., Uber Zopfgruppen und gleichsinning verdrillte Verkettunger, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 11, 171-178 (1936)
[11] A. Fathi, F. Laudenbach and V. Peonaru,Traveaux de Thurston sur les surfaces, Asterisque66-67 (1979).
[12] J. Fehrenbach,Quelques aspects géométriques et dynamiques du mapping class group, PhD thesis, Université de Nice, January 1998.
[13] Fiedler, T., A small state sum for knots, Topology, 32, 281-294 (1993) · Zbl 0787.57007
[14] Gassner, B. J., On braid groups, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25, 19-22 (1961)
[15] Ivanov, N., Subgroups of the Teichmuller Modular Group (1992), Providence, RI: American Mathematical Society, Providence, RI
[16] Jones, V. F. R., Hecke algebra representations of braid groups and link polynomials, Annals of Mathematics, 126, 335-388 (1987) · Zbl 0631.57005
[17] Krammer, D., The braid group B_4 is linear, Inventiones Mathematicae, 142, 451-486 (2000) · Zbl 0988.20023
[18] Krammer, D., Braid groups are linear, Annals of Mathematics, 155, 2, 131-156 (2002) · Zbl 1020.20025
[19] Lawrence, R. J., Homological representations of the Hecke algebra, Communications in Mathematical Physics, 135, 141-191 (1990) · Zbl 0716.20022
[20] Long, D., Constructing representations of braid groups, Communications in Mathematical Analysis and Geometry, 2, 217-238 (1994) · Zbl 0845.20028
[21] Long, D.; Patton, M., The Burau representation is not faithful for n ≥ 6, Topology, 32, 439-447 (1993) · Zbl 0810.57004
[22] McCarthy, J., A Tits alternative for subgroups of mapping class groups, Transactions of the American Mathematical Society, 291, 583-612 (1985) · Zbl 0579.57006
[23] McCool, J., On reducible braids, Word Problems II, 261-295 (1980), Amsterdam: North-Holland, Amsterdam · Zbl 0434.20021
[24] Menasco, W. W., On iterated torus knots and transversal knots, Geometry and Topology, 5, 651-682 (2001) · Zbl 1002.57025
[25] Moody, J., The faithfulness question for the Burau representation, Proceedings of the American Mathematical Society, 119, 671-679 (1993) · Zbl 0796.57004
[26] Morton, H., An irreducible 4-braid with unknotted closure, Mathematical Proceedings of the Cambridge Philosophical Society, 93, 259-261 (1983) · Zbl 0522.57006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.