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The tunnel number and the cutting number with constituent handlebody-knots. (English) Zbl 1459.57013

Summary: We give lower bounds for the tunnel number of knots and handlebody-knots. We also give a lower bound for the cutting number, which is a “dual” notion to the tunnel number in handlebody-knot theory. We provide necessary conditions to be constituent handlebody-knots by using \(G\)-family of quandles colorings. The above two evaluations are obtained as the corollaries. Furthermore, we construct handlebody-knots with arbitrary tunnel number and cutting number.

MSC:

57K10 Knot theory
57K12 Generalized knots (virtual knots, welded knots, quandles, etc.)
57K31 Invariants of 3-manifolds (including skein modules, character varieties)
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[1] Cho, S.; McCullough, D., Cabling sequences of tunnels of torus knots, Algebraic Geom. Topol., 9, 1, 1-20 (2009) · Zbl 1170.57005
[2] Cho, S.; McCullough, D., The tree of knot tunnels, Geom. Topol., 13, 2, 769-815 (2009) · Zbl 1191.57005
[3] Cho, S.; McCullough, D., Constructing knot tunnels using giant steps, Proc. Am. Math. Soc., 138, 1, 375-384 (2010) · Zbl 1192.57004
[4] Cho, S.; McCullough, D., Tunnel leveling, depth, and bridge numbers, Trans. Am. Math. Soc., 363, 1, 259-280 (2011) · Zbl 1210.57004
[5] Ishii, A., Moves and invariants for knotted handlebodies, Algebraic Geom. Topol., 8, 3, 1403-1418 (2008) · Zbl 1151.57007
[6] Ishii, A., The Markov theorems for spatial graphs and handlebody-knots with Y-orientations, Int. J. Math., 26, 14, Article 1550116 pp. (2015) · Zbl 1337.57004
[7] Ishii, A.; Iwakiri, M., Quandle cocycle invariants for spatial graphs and knotted handlebodies, Can. J. Math., 64, 1, 102-122 (2012) · Zbl 1245.57014
[8] Ishii, A.; Iwakiri, M.; Jang, Y.; Oshiro, K., A G-family of quandles and handlebody-knots, Ill. J. Math., 57, 3, 817-838 (2013) · Zbl 1306.57011
[9] Ishii, A.; Kishimoto, K.; Moriuchi, H.; Suzuki, M., A table of genus two handlebody-knots up to six crossings, J. Knot Theory Ramif., 21, 4, Article 1250035 pp. (2012) · Zbl 1236.57015
[10] Ishii, A.; Nelson, S., Partially multiplicative biquandles and handlebody-knots, (Knots, Links, Spatial Graphs, and Algebraic Invariants. Knots, Links, Spatial Graphs, and Algebraic Invariants, Contemp. Math., vol. 689 (2017), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 159-176 · Zbl 1387.57026
[11] Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra, 23, 1, 37-65 (1982) · Zbl 0474.57003
[12] Kim, J., On critical Heegaard splittings of tunnel number two composite knot exteriors, J. Knot Theory Ramif., 22, 11, Article 1350065 pp. (2013) · Zbl 1288.57017
[13] Matveev, S. V., Distributive groupoids in knot theory, Mat. Sb. (N.S.), 119(161), 1, 78-88 (1982), 160
[14] Moriah, Y.; Rubinstein, H., Heegaard structures of negatively curved 3-manifolds, Commun. Anal. Geom., 5, 3, 375-412 (1997) · Zbl 0890.57025
[15] Morimoto, K., On the additivity of tunnel number of knots, Topol. Appl., 53, 1, 37-66 (1993) · Zbl 0816.57004
[16] Morimoto, K., On the super additivity of tunnel number of knots, Math. Ann., 317, 3, 489-508 (2000) · Zbl 0981.57005
[17] Morimoto, K., On Heegaard splittings of knot exteriors with tunnel number degenerations, Topol. Appl., 196, part B, 719-728 (2015) · Zbl 1342.57009
[18] Murao, T., A relationship between multiple conjugation quandle/biquandle colorings, Kobe J. Math., 36, 1-2, 57-78 (2019) · Zbl 1448.57027
[19] Rolfsen, D., Knots and Links, Mathematics Lecture Series, vol. 7 (1976), Publish or Perish, Inc.: Publish or Perish, Inc. Berkeley, Calif. · Zbl 0339.55004
[20] Scharlemann, M.; Schultens, J., The tunnel number of the sum of n knots is at least n, Topology, 38, 2, 265-270 (1999) · Zbl 0929.57003
[21] Yang, G.; Lei, F., Some sufficient conditions for tunnel numbers of connected sum of two knots not to go down, Acta Math. Sin. Engl. Ser., 27, 11, 2229-2244 (2011) · Zbl 1238.57014
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