# zbMATH — the first resource for mathematics

Knots with unknotting number 1 and essential Conway spheres. (English) Zbl 1129.57009
Summary: For a knot $$K$$ in $$S^3$$, let $$\mathbf T(K)$$ be the characteristic toric sub-orbifold of the orbifold $$(S^3,K)$$ as defined by Bonahon-Siebenmann. If $$K$$ has unknotting number one, we show that an unknotting arc for $$K$$ can always be found which is disjoint from $$\mathbf T(K)$$, unless either $$K$$ is an EM-knot (of Eudave-Muñoz) or $$(S^3,K)$$ contains an EM-tangle after cutting along $$\mathbf T(K)$$. As a consequence, we describe exactly which large algebraic knots (i.e., algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway’s table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsváth-Szabó, this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram.
As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
unknotting number 1; Conway spheres; algebraic knots; mutation
Full Text:
##### References:
 [1] S A Bleiler, A note on unknotting number, Math. Proc. Cambridge Philos. Soc. 96 (1984) 469 · Zbl 0556.57004 · doi:10.1017/S0305004100062381 [2] M Boileau, B Zimmermann, The $$\pi$$-orbifold group of a link, Math. Z. 200 (1989) 187 · Zbl 0663.57006 · doi:10.1007/BF01230281 · eudml:174001 [3] F Bonahon, L C Siebenmann, The characteristic toric splitting of irreducible compact $$3$$-orbifolds, Math. Ann. 278 (1987) 441 · Zbl 0629.57007 · doi:10.1007/BF01458079 · eudml:164300 [4] J H Conway, An enumeration of knots and links, and some of their algebraic properties, Pergamon (1970) 329 · Zbl 0202.54703 [5] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. $$(2)$$ 125 (1987) 237 · Zbl 0633.57006 · doi:10.2307/1971311 [6] M Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35 · Zbl 0889.57023 [7] M Eudave-Muñoz, On hyperbolic knots with Seifert fibered Dehn surgeries (2002) 119 · Zbl 1009.57010 · doi:10.1016/S0166-8641(01)00114-6 [8] C M Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983) 687 · Zbl 0519.57005 · doi:10.2307/1999046 [9] C M Gordon, J Luecke, Only integral Dehn surgeries can yield reducible manifolds, Math. Proc. Cambridge Philos. Soc. 102 (1987) 97 · Zbl 0655.57500 · doi:10.1017/S0305004100067086 [10] C M Gordon, J Luecke, Reducible manifolds and Dehn surgery, Topology 35 (1996) 385 · Zbl 0859.57016 · doi:10.1016/0040-9383(95)00016-X [11] C M Gordon, J Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004) 417 · Zbl 1062.57006 · doi:10.4310/CAG.2004.v12.n1.a18 · euclid:cag/1088454730 [12] W Haken, Theorie der Normalflächen, Acta Math. 105 (1961) 245 · Zbl 0100.19402 · doi:10.1007/BF02559591 [13] R Hartley, Knots and involutions, Math. Z. 171 (1980) 175 · Zbl 0411.57002 · doi:10.1007/BF01176707 · eudml:172931 [14] W H Jaco, P B Shalen, Seifert fibered spaces in $$3$$-manifolds, Mem. Amer. Math. Soc. 21 (1979) · Zbl 0415.57005 [15] T Kanenobu, H Murakami, Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986) 499 · Zbl 0613.57002 · doi:10.2307/2046210 [16] R Kirby, Problems in low-dimensional topology (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35 [17] T Kobayashi, Minimal genus Seifert surfaces for unknotting number $$1$$ knots, Kobe J. Math. 6 (1989) 53 · Zbl 0688.57007 [18] P Kohn, Two-bridge links with unlinking number one, Proc. Amer. Math. Soc. 113 (1991) 1135 · Zbl 0734.57007 · doi:10.2307/2048793 [19] W B R Lickorish, The unknotting number of a classical knot, Contemp. Math. 44, Amer. Math. Soc. (1985) 117 · Zbl 0607.57002 [20] W H Meeks III, P Scott, Finite group actions on $$3$$-manifolds, Invent. Math. 86 (1986) 287 · Zbl 0626.57006 · doi:10.1007/BF01389073 · eudml:143397 [21] W Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984) 37 · Zbl 0525.57003 · doi:10.1016/0040-9383(84)90023-5 [22] W Menasco, M Thistlethwaite, The classification of alternating links, Ann. of Math. $$(2)$$ 138 (1993) 113 · Zbl 0809.57002 · doi:10.2307/2946636 [23] W Menasco, X Zhang, Notes on tangles, 2-handle additions and exceptional Dehn fillings, Pacific J. Math. 198 (2001) 149 · Zbl 1049.57013 · doi:10.2140/pjm.2001.198.149 [24] J M Montesinos, Surgery on links and double branched covers of $$S^3$$ (editor L Neuwirth), Ann. of Math. Studies 84, Princeton Univ. Press (1975) 227 · Zbl 0325.55004 [25] K Motegi, A note on unlinking numbers of Montesinos links, Rev. Mat. Univ. Complut. Madrid 9 (1996) 151 · Zbl 0884.57007 · eudml:44205 [26] Y Nakanishi, Unknotting numbers and knot diagrams with the minimum crossings, Math. Sem. Notes Kobe Univ. 11 (1983) 257 · Zbl 0549.57003 [27] P Ozsváth, Z Szabó, Knots with unknotting number one and Heegaard Floer homology, Topology 44 (2005) 705 · Zbl 1083.57013 · doi:10.1016/j.top.2005.01.002 [28] M G Scharlemann, Unknotting number one knots are prime, Invent. Math. 82 (1985) 37 · Zbl 0576.57004 · doi:10.1007/BF01394778 · eudml:143276 [29] M Scharlemann, Producing reducible $$3$$-manifolds by surgery on a knot, Topology 29 (1990) 481 · Zbl 0727.57015 · doi:10.1016/0040-9383(90)90017-E [30] M Scharlemann, A Thompson, Unknotting number, genus, and companion tori, Math. Ann. 280 (1988) 191 · Zbl 0616.57003 · doi:10.1007/BF01456051 · eudml:164360 [31] M Scharlemann, A Thompson, Link genus and the Conway moves, Comment. Math. Helv. 64 (1989) 527 · Zbl 0693.57004 · doi:10.1007/BF02564693 · eudml:140170 [32] C Sundberg, M Thistlethwaite, The rate of growth of the number of prime alternating links and tangles, Pacific J. Math. 182 (1998) 329 · Zbl 0901.57008 · doi:10.2140/pjm.1998.182.329 · nyjm.albany.edu:8000 [33] M B Thistlethwaite, On the algebraic part of an alternating link, Pacific J. Math. 151 (1991) 317 · Zbl 0745.57003 · doi:10.2140/pjm.1991.151.317 [34] M Thistlethwaite, On the structure and scarcity of alternating links and tangles, J. Knot Theory Ramifications 7 (1998) 981 · Zbl 0971.57010 · doi:10.1142/S021821659800053X [35] J L Tollefson, Involutions of Seifert fiber spaces, Pacific J. Math. 74 (1978) 519 · Zbl 0395.57022 · doi:10.2140/pjm.1978.74.519 [36] F Waldhausen, Über Involutionen der $$3$$-Sphäre, Topology 8 (1969) 81 · Zbl 0185.27603 · doi:10.1016/0040-9383(69)90033-0 [37] Y Q Wu, Sutured manifold hierarchies, essential laminations, and Dehn surgery, J. Differential Geom. 48 (1998) 407 · Zbl 0917.57015 [38] X Zhang, Unknotting number one knots are prime: a new proof, Proc. Amer. Math. Soc. 113 (1991) 611 · Zbl 0728.57008 · doi:10.2307/2048550
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.