论文信息 - SPINAL KNOTS IN LENS SPACES

SPINAL KNOTS IN LENS SPACES

A knot in a lens is said to be spinal if it can be isotoped on a standard spine (e.g. in ℝP3, spinal knots bound a Mobius band). We prove that a Dehn surgery on a non-spinal knot in a lens space cannot produce 𝕊3. With a view to study the Dehn surgeries that produce lens spaces, the main part is devoted to finding an obstruction for a standard spine to be minimal. We consider the intersection graphs coming from a standard spine and an arbitrary surface. This obstruction is given by the existence of a generalized Scharlemann cycle.

D. Matignon | A. Deruelle

[1] D. Matignon,et al. Thin presentation of knots and lens spaces , 2003, math/0402457.

[2] K. Ichihara. Exceptional surgeries and genera of knots , 2001 .

[3] C. Gordon. Small surfaces and Dehn filling , 1999, math/9911251.

[4] H. Goda,et al. Dehn surgeries on knots which yield lens spaces and genera of knots , 1999, Mathematical Proceedings of the Cambridge Philosophical Society.

[5] James A. Hoffman. THERE ARE NO STRICT GREAT x-CYCLES AFTER A REDUCING OR P2 SURGERY ON A KNOT , 1998 .

[6] D. Matignon,et al. Dehn surgeries and P2-reducible 3-manifolds , 1996 .

[7] Ying‐Qing Wu. Cyclic surgery and satellite knots , 1990 .

[8] John Luecke,et al. Knots are determined by their complements , 1989 .

[9] Shicheng Wang,et al. Cyclic surgery on knots , 1989 .

[10] M. Culler,et al. Dehn surgery on knots , 1985 .

[11] C. Gordon. Dehn surgery and satellite knots , 1983 .

[12] R. Stern,et al. Constructing lens spaces by surgery on knots , 1980 .

[13] L. Moser. Elementary surgery along a torus knot. , 1971 .

[14] C. Gordon. Dehn filling: A survey , 1998 .

[15] David Gabai. Surgery on knots in solid tori , 1989 .