Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Geometric indices and the Alexander polynomial of a knot

Author(s): Hirozumi Fujii
Journal: Proc. Amer. Math. Soc. 124 (1996), 2923-2933.
MSC (1991): Primary 57M25
MathSciNet review: 1343693
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Abstract: It is well-known that any Laurent polynomial $\Delta (t)$ satisfying $\Delta (t)\allowbreak \doteq \Delta (t^{-1})$ and $\Delta (1) = \pm 1$ is the Alexander polynomial of a knot in $S^3$ . We show that $\Delta (t)$ can be realized by a knot which has the following properties simultaneously: (i) tunnel number 1; (ii) bridge index 3; and (iii) unknotting number 1.

References:

1.: A. Bleiler and Y. Moriah, Heegaard splittings and branched coverings of , Math. Ann. 281 (1988), 531--543. MR 89k:57006
2.: J. Conway, An enumeration of knots and links and some of their related properties. In : Computational problems in Abstract Algebra (Oxford 1967). Pergamon Press (1970), pp. 329--358. MR 41:2661
3.: M. Culler, C. Gordon, J. Luecke, and P. Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987), 237--300. MR 88a:57026
4.: P. Freyd, D. Yetter, J. Hoste, W. B. R. Lichorish, K. C. Millett, and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985), 239--246. MR 86e:57007
5.: Hartley, R.: On two-bridged knots polynomials, J. Austral. Math. Soc. 28 (1979), 241--249. MR 81a:57006
6.: T. Kanenobu, Genus and Kauffman polynomial of a -bridge knot, Osaka J. Math. 29 (1992), 635--651. MR 93h:57010
7.: A. Kawauchi, On coefficient polynomials of the skein polynomial of an oriented link, Kobe J. Math. 11 (1994), 49--68. CMP 95:06
8.: H. Kondo, Knots of unknotting number one and their Alexander polynomials, Osaka J. Math. 16 (1979), 551--559. MR 80g:57008
9.: F. B. Kronheimer and T. S.Mrowka, Gauge theory for embedded surfaces, I, Topology. 32 (1993), 773--826. MR 94k:57048
10.: W. B. R. Lickorish, The unknotting number of a classical knot, Comtemp. Math. 44 (1985), pp. 117--121. MR 87a:57012
11.: W. B. R. Lickorish and K. C. Millett, A polynomial invariant of oriented links, Topology. 26 (1987), pp. 107--141. MR 88b:57012
12.: J. M. Montesinos, Surgery on links and double branched covers of , Ann. Math. Studies. 84 (1975), pp. 227--259. MR 52:1699
13.: K. Morimoto, On the additivity of -genus of knots, Osaka J. Math. 31 (1994), 137--145. MR 95d:57006
14.: K. Morimoto and M. Sakuma, On unknotting tunnel number for knot, Math. Ann. 289 (1991), 143--167. MR 92e:57015
15.: J. H. Przytycki and P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1987), 115--189. MR 89h:57006
16.: D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, 1970. MR 58:24236
17.: T. Sakai, A remark on the Alexander polynomials of knots, Math. Sem. Notes Kobe Univ. 5 (1977), 451--456. MR 58:12998
18.: T. Sakai, Polynomials of invertible knots, Math. Ann. 266 (1983), 229--232. MR 85f:57004
19.: M. Scharlemann and A. Thompson, Unknotting number, genus, and companion tori, Math. Ann. 280 (1988), 191--205. MR 89d:57008
20.: H. Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954), 245--288. MR 17:292a

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