Geodesic
Inequivalent handlebody-knots with homeomorphic complements
Lee, Jung Hoon1 ; Lee, Sangyop2
1Department of Mathematics and Inst. of Pure and Applied Math., Chonbuk National University, Jeonju 561-756, Korea
2Department of Mathematics, Chung-Ang University, 221 Heukseok-dong, Dongjak-gu, Seoul 156-756, South Korea
Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 1059-1079
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We distinguish the handlebody-knots 51,64 and 52,613 in the table, due to Ishii et al, of irreducible handlebody-knots up to six crossings. Furthermore, we construct two infinite families of handlebody-knots, each containing one of the pairs 51,64 and 52,613, and show that any two handlebody-knots in each family have homeomorphic complements but they are not equivalent.

DOI : 10.2140/agt.2012.12.1059
Classification : 57M50
Keywords: handlebody-knot, essential annuli

Lee, Jung Hoon  1   ; Lee, Sangyop  2

1 Department of Mathematics and Inst. of Pure and Applied Math., Chonbuk National University, Jeonju 561-756, Korea
2 Department of Mathematics, Chung-Ang University, 221 Heukseok-dong, Dongjak-gu, Seoul 156-756, South Korea 
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Lee, Jung Hoon; Lee, Sangyop. Inequivalent handlebody-knots with homeomorphic complements. Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 1059-1079. doi: 10.2140/agt.2012.12.1059

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[2] A Ishii, Moves and invariants for knotted handlebodies, Algebr. Geom. Topol. 8 (2008) 1403

[3] A Ishii, K Kishimoto, H Moriuchi, M Suzuki, A table of genus two handlebody-knots up to six crossings, J. Knot Theory Ramifications 21 (2012)

[4] T Kobayashi, Structures of the Haken manifolds with Heegaard splittings of genus two, Osaka J. Math. 21 (1984) 437

[5] M Motto, Inequivalent genus 2 handlebodies in S3 with homeomorphic complement, Topology Appl. 36 (1990) 283

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