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Factorization of homotopies of nanophrases

Published online by Cambridge University Press:  19 October 2011

ANDREW GIBSON
ANDREW GIBSON*
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan. e-mail: [email protected]
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Abstract

Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. For any homotopy given by a composite homotopy data triple we define a complete invariant of nanophrases. This invariant is used to show that equivalence of nanophrases under such a homotopy can be calculated just by using the homotopies given by its prime factors.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 1 , January 2012 , pp. 55 - 90
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Cohn, P. M.Algebra, second edition, volume 3. (John Wiley & Sons, 1997).Google Scholar
[2]Fukunaga, T. Homotopy classification of nanophrases with less than or equal to four letters, arXiv:0904.3478.Google Scholar
[3]Fukunaga, T. On generalization of homotopy of words and its applications. arXiv:0908.2899.Google Scholar
[4]Fukunaga, T.Homotopy classification of nanophrases in Turaev's theory of words. J. Knot Theory Ramifications 18 (2009), no. 7, 901–915.CrossRefGoogle Scholar
[5]Gibson, A.On tabulating virtual strings. Acta Math. Vietnam. 33 (2008), no. 3, 493–518.Google Scholar
[6]Gibson, A., Homotopy invariants of Gauss phrases. Indiana Univ. Math. J. 59 (2010), no. 1, 207–229.CrossRefGoogle Scholar
[7]Gibson, A., Homotopy invariants of Gauss words. Math. Ann. 349 (2011), no. 4, 871–887.CrossRefGoogle Scholar
[8]Kadokami, T.Detecting non-triviality of virtual links, J. Knot Theory Ramifications 12 (2003), no. 6, 781–803.CrossRefGoogle Scholar
[9]Manturov, V. O. On free knots, arXiv:math.GT/0901.2214.Google Scholar
[10]Newman, M. H. A.On theories with a combinatorial definition of “equivalence”. Ann. Math. 43 (1942), no. 2, 223–243.CrossRefGoogle Scholar
[11]Turaev, V.Virtual strings. Ann. Inst. Fourier (Grenoble) 54 (2004), no. 7, 2455–2525.CrossRefGoogle Scholar
[12]Turaev, V. Knots and words. Int. Math. Res. Not. (2006), Art. ID 84098, 23.Google Scholar
[13]Turaev, V.Topology of words. Proc. London Math. Soc. (3) 95 (2007), no. 2, 360–412.CrossRefGoogle Scholar