Generalized crossing changes in satellite knots

Balm, Cheryl

doi:10.1090/S0002-9939-2014-12235-0

Generalized crossing changes in satellite knots
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by Cheryl Jaeger Balm PDF

Proc. Amer. Math. Soc. 143 (2015), 447-458 Request permission

Abstract:

We show that if $K$ is a satellite knot in the 3-sphere $S^3$ which admits a generalized cosmetic crossing change of order $q$ with $|q| \geq 6$, then $K$ admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot in $S^3$ which admits a torus knot as a pattern cannot admit a generalized cosmetic crossing change of order $q$ with $|q| \geq 6$. We also show that if there is any knot in $S^3$ admitting a generalized cosmetic crossing change of order $q$ with $|q| \geq 6$, then there must be such a knot which is hyperbolic.

References

Rob Kirby (ed.), Problems in low-dimensional topology, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35–473. MR 1470751, DOI 10.1090/amsip/002.2/02
Cheryl Balm, Stefan Friedl, Efstratia Kalfagianni, and Mark Powell, Cosmetic crossings and Seifert matrices, Comm. Anal. Geom. 20 (2012), no. 2, 235–253. MR 2928712, DOI 10.4310/CAG.2012.v20.n2.a1
Cheryl Balm and Efstratia Kalfagianni, Cosmetic crossings of twisted knots, preprint, arXiv:1301.6369 [math.GT].
David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
C. McA. Gordon, Boundary slopes of punctured tori in $3$-manifolds, Trans. Amer. Math. Soc. 350 (1998), no. 5, 1713–1790. MR 1390037, DOI 10.1090/S0002-9947-98-01763-2
John Hempel, 3-manifolds, AMS Chelsea Publishing, Providence, RI, 2004. Reprint of the 1976 original. MR 2098385, DOI 10.1090/chel/349
Efstratia Kalfagianni, Cosmetic crossing changes of fibered knots, J. Reine Angew. Math. 669 (2012), 151–164. MR 2980586, DOI 10.1515/crelle.2011.148
Efstratia Kalfagianni and Xiao-Song Lin, Knot adjacency, genus and essential tori, Pacific J. Math. 228 (2006), no. 2, 251–275. MR 2274520, DOI 10.2140/pjm.2006.228.251
W. B. Raymond Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997. MR 1472978, DOI 10.1007/978-1-4612-0691-0
Darryl McCullough, Homeomorphisms which are Dehn twists on the boundary, Algebr. Geom. Topol. 6 (2006), 1331–1340. MR 2253449, DOI 10.2140/agt.2006.6.1331
Kimihiko Motegi, Knotting trivial knots and resulting knot types, Pacific J. Math. 161 (1993), no. 2, 371–383. MR 1242205
V. V. Prasolov and A. B. Sossinsky, Knots, links, braids and 3-manifolds, Translations of Mathematical Monographs, vol. 154, American Mathematical Society, Providence, RI, 1997. An introduction to the new invariants in low-dimensional topology; Translated from the Russian manuscript by Sossinsky [Sosinskiĭ]. MR 1414898, DOI 10.1090/mmono/154
Martin Scharlemann and Abigail Thompson, Link genus and the Conway moves, Comment. Math. Helv. 64 (1989), no. 4, 527–535. MR 1022995, DOI 10.1007/BF02564693
Horst Schubert, Knoten und Vollringe, Acta Math. 90 (1953), 131–286 (German). MR 72482, DOI 10.1007/BF02392437
William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
Ichiro Torisu, On nugatory crossings for knots, Topology Appl. 92 (1999), no. 2, 119–129. MR 1669827, DOI 10.1016/S0166-8641(97)00238-1
Chad Wiley, Nugatory crossings in closed 3-braid diagrams, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–University of California, Santa Barbara. MR 2712035