East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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(1,1)-DECOMPOSITIONS OF RATIONAL PRETZEL KNOTS

  • Song, Hyun-Jong (Department of Applied Mathematics, Pukyong National University)
  • Received : 2015.12.18
  • Accepted : 2016.01.28
  • Published : 2016.01.30
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Abstract

We explicitly derive diagrams representing (1,1)-decompositions of rational pretzel knots $K_{\beta}=M((-2,\;1),\;(3,\;1),\;({\mid}6{\beta}+1{\mid},{\beta}))$ from four unknotting tunnels for ${\beta}=1,\;-2$ and 2.

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References

  1. S.-B, Cho and D. MacCullough, The tree of knot tunnels, Geom. Topol. 13 (2009), no. 2, 769815. https://doi.org/10.2140/gt.2009.13.769
  2. M. J. Dunwoody, Cyclic presentations and 3-manifolds, Proceeding of Groups Korea 94' edited by A.C.Kim and D.L.Johnson, 47-55(1994).
  3. K. Ishihara, An algorithm for finding parameters of tunnels, Algebr. Geom. Topol. 11 (2011), no. 4, 21672190. https://doi.org/10.2140/agt.2011.11.2167
  4. H. Goda, H. Mastuda and T. Morifuji, Knot Floer homology of (1,1)-knots, Geometriae Dedicata 2005;112(1):197-214. https://doi.org/10.1007/s10711-004-5403-2
  5. L. Grasselli and M. Mulazzani, Genus one 1-bridge knots and Dunwoody manifolds, Forum Math. 13 (2001), no. 3, 379397. https://doi.org/10.1515/form.2001.013
  6. D. Heath and H.-J. Song, Unknotting tunnels of p(-2, 3, 7) . J. Knot Theory Ramifications 14 (2005), no. 8, 10771085. https://doi.org/10.1142/S0218216505004238
  7. C. Hayashi, 1-genus 1-bridge splittings for knots, Osaka J. Math., Vol.41, (2004), 371-426.
  8. S.-H. Kim and Y.-K. Kim, The Dual and Mirror images of the Dunwoody 3-Manifolds, Hidawi Pub. Corp. International Journal of Mathematics and Mathematical Sciences Volume 2013, Article ID 103209, 7 pages.
  9. S,Y. Lee and H.-J. Song, (1,1)-diagrams and Alexander polynomials of torus knots, in preparation.
  10. J. Rasmussen, Knot Polynomials and Knot Homologies, Geometry and topology of manifolds, 261280, Fields Inst. Commun., 47, Amer. Math. Soc., Providence, RI, 2005.
  11. H.-J. Song, Dunwoody (1,1)-knots, unpublised manuscripts 2000.
  12. H.-J. Song, Morimoto Sakuma Yokota's geometric approach to tunnel number one knots, Topology Appl. 127 (2003), no. 3, 375392. https://doi.org/10.1016/S0166-8641(02)00100-1