Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Languages: English
Types: Doctoral thesis
Subjects: QA

Classified by OpenAIRE into

arxiv: Mathematics::Geometric Topology
This work provides the topological background and a preliminary study for the analogue of the 2-variable Jones polynomial as an invariant of oriented links in arbitrary 3- manifolds via normalized traces of appropriate algebras, and it is organized as follows:\ud \ud Chapter 1: Motivated by the study of the Jones polynomial, we produce and present a new algorithm for turning oriented link diagrams in S3 into braids. Using this algorithm we then provide a new, short proof of Markov's theorem and its relative version.\ud \ud Chapter 2: The objective of the first part of Chapter 2 is to state and prove an analogue of Markov's theorem for oriented links in arbitrary 3-manifolds. We do this by modifying first our algorithm, so as to produce an analogue of Alexander's theorem for oriented links in arbitrary 3-manifolds. In the second part we show that the study of links (up to isotopy) in a 3-manifold can be restricted to the study of cosets of the braid groups Bn,m, which are subgroups of the usual braid groups Bn+m .\ud \ud Chapter 3: In this chapter we try to use the above topological set-up in a procedure analogous to the way V.F.R. Jones derived his famous link invariant. The analogy amounts to the following: We observe that Bn,1 - the braid group related to the solid torus and to the lens spaces L(p, 1) - is the Artin group of the Coxeter group of Bn-type. This implies the existence of an epimorphism of eEn,1 onto the Hecke algebra of Bn-type. Then we give an analogue of Ocneanu's trace function for the above algebras. This trace, after being properly normalized, yields a HOMFLY-PTtype isotopy invariant for oriented links inside a solid torus. Finally, by forcing a strong condition, we normalize this trace, so as to obtain a link invariant in SI x S2.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • • If now c is of type (i) ,i.e. c E 'Hn-.(q, Q) and b is of type (ii) , i.e. b = X9n-lY , x, Y E 'Hn-1(q, Q), then: t r (9nX9n-lY9nC) B=.r t r (X9n9n-19nYC ) B=.r t r (X9n-19n9n-lYC ) (=3') [1] [2] [4] [5] [6] [7] [8] [9] [13] [14] [15] J.W. Alexander, A lemma on systems of knotted curves, Proe. Nat. Aead. SeL U.S.A. 9 , pp. 93-95, 1923.
    • Soc. 30 , pp. 275-306, 1928.
    • [3] J.W. Alexander, G.B. Briggs On types of knotted curves, Annals of Mathematics 2 , vol. 28, pp. 563-586, 1927.
    • E. Artin, Theorie der Zopfe, Abh. Math. Sem. Univ. Hamburg 4 , pp. 47-72, 1926.
    • E. Artin, Theory of braids, Ann. of Math. 48 , pp. 101-126,643-649, 1947.
    • D. Bennequin, Entrlacements et equations de Pfaffe, Asterisque 107-108 , pp.
    • 82 , Princeton University Press, Princeton 1974.
    • N. Bourbaki, Groupes et algebres de Lie, Chap. 4-6 Hermann, Paris, 1968.
    • P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millet, A. Ocneanu, A new polynomial invariant of knots and links, Bull. AMS 12 , pp. 239-246, 1985.
    • 4 , pp. 115-139, 1987.
    • Univ. Hamburg 5, pp. 24-32, 1927.
    • K. Reidemeister, Knotentheorie (Reprint), Chelsea, New York, 1948.
    • C.P. Rourke, B.J. Sanderson, Springer-Verlag, 1972, 1982.
    • R. Skora, Closed braids in 9-manifolds, Math. Zeitschrift 211 , pp. 173-187, 1992.
    • P.A. Sundheim, Reidemeister's theorem for 9-manifolds, Math. Proc. Camb.
    • Phil. Soc. 110 , pp. 281-292, 1991.
    • Nauchn. Sem. Lomi 167 , pp. 79-89, 1988. English translation: J. Soviet Math.
    • Helvetici 65 , pp. 104-113, 1990.
    • A.D. Wallace, Modifications and cobounding manifolds, Can. J. Math. 12 , pp.
  • No related research data.
  • Discovered through pilot similarity algorithms. Send us your feedback.

Share - Bookmark

Cite this article