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THE MODULE OF TWO-CHORD DIAGRAMS FOR ORIENTED KNOTS OF ZERO WINDING NUMBER IN THE SOLID TORUS

KHALED BATAINEH

Jordan University of Science and Technology, Irbid, Jordon

New Mexico State University, Las Cruses, NM, USA

https://doi.org/10.1142/S0218216512500642Cited by:1 (Source: Crossref)

Abstract

In this paper we study the ℤ-module A₂ of two-chord diagrams for knots with zero winding number in the solid torus KST₀, which is needed in studying the type-two invariants for knots in KST₀. We show that this module (or abelian group), which is given as a presentation with infinite number of generators and an infinite number of relations, is a free infinitely generated module. Moreover, we show that this module is isomorphic to the direct sum of three free modules that are easier to understand.

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References

F. Aicardi, C. R. Math. Acad. Sci. Paris Ser. I 321, 81 (1995). Web of Science, Google Scholar
D. Bar-Natan, Topology 34, 423 (1995), DOI: 10.1016/0040-9383(95)93237-2. Crossref, Google Scholar
K. Bataineh, Finite type invariants for knots in the solid torus, Ph.D. thesis, New Mexico State University (2003) . Google Scholar
J. Birman and X.-S. Lin, Invent. Math. 111, 225 (1993), DOI: 10.1007/BF01231287. Crossref, Web of Science, Google Scholar
V. Goryunov, Amer. Math. Soc. Transl. 190, 37 (1999). Google Scholar
E. Kalfagianni, Topology 37, 673 (1998), DOI: 10.1016/S0040-9383(97)00034-7. Crossref, Web of Science, Google Scholar
P. Kirk and C. Livingston, Topology 36, 1333 (1997), DOI: 10.1016/S0040-9383(96)00047-X. Crossref, Web of Science, Google Scholar
P. Kirk and C. Livingston, Pacific J. Math. 183, 305 (1998), DOI: 10.2140/pjm.1998.183.305. Crossref, Web of Science, Google Scholar
M. Kontsevich, Adv. Sov. Math. 16, 137 (1993). Google Scholar