Twisting operations and composite knots

Teragaito, Masakazu

doi:10.1090/S0002-9939-1995-1254855-X

Twisting operations and composite knots
HTML articles powered by AMS MathViewer

by Masakazu Teragaito PDF

Proc. Amer. Math. Soc. 123 (1995), 1623-1629 Request permission

Abstract:

Suppose that a composite knot K in ${S^3}$ can be changed to a trivial knot by 1/n-surgery along a trivial loop C. We show that $|n| \leq 2$. Moreover, if there is a decomposing sphere of K which meets C in two points, then $|n| \leq 1$.

References

Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237–300. MR 881270, DOI 10.2307/1971311

When are tori created by Dehn surgery

C. McA. Gordon, Combinatorial methods in knot theory, Algebra and topology 1990 (Taejon, 1990) Korea Adv. Inst. Sci. Tech., Taejŏn, 1990, pp. 1–23. MR 1098718
Yves Mathieu, Unknotting, knotting by twists on disks and property $(\textrm {P})$ for knots in $S^3$, Knots 90 (Osaka, 1990) de Gruyter, Berlin, 1992, pp. 93–102. MR 1177414
Katura Miyazaki and Akira Yasuhara, Knots that cannot be obtained from a trivial knot by twisting, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 139–150. MR 1282760, DOI 10.1090/conm/164/01590
Kimihiko Motegi, Primeness of twisted knots, Proc. Amer. Math. Soc. 119 (1993), no. 3, 979–983. MR 1181171, DOI 10.1090/S0002-9939-1993-1181171-5
Kimihiko Motegi and Tetsuo Shibuya, Are knots obtained from a plain pattern always prime?, Kobe J. Math. 9 (1992), no. 1, 39–42. MR 1189955
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
Martin G. Scharlemann, Unknotting number one knots are prime, Invent. Math. 82 (1985), no. 1, 37–55. MR 808108, DOI 10.1007/BF01394778
Martin Scharlemann and Abigail Thompson, Unknotting number, genus, and companion tori, Math. Ann. 280 (1988), no. 2, 191–205. MR 929535, DOI 10.1007/BF01456051
Masakazu Teragaito, Composite knots trivialized by twisting, J. Knot Theory Ramifications 1 (1992), no. 4, 467–470. MR 1194998, DOI 10.1142/S0218216592000239