# Boundary Slopes

For knot K, a rational number p/q is a boundary slope if there exists a boundary incompressible surface in the knot complement having as boundary a set of parallel curves on the peripheral torus, each one representing p(meridian) + q(longitude).

The existence of a Seifert surface implies that 0 = 0/1.

As an example, the (2,3)-torus knot K has boundary slope 0. It lies on a torus T, and T - K is a boundary incompressible annulus having boundary two copies of a (6,1) curve. The full set of boundary slopes is {0,6). At the moment, the only knots for which KnotInfo provides boundary slopes are Montesinos knots, as described in the references.

## Specific Knots

The following list, provided by Nathen Dunfield, describes whether the given list is for the knot or its mirror image. "bs" = "boundary slope", "mirrored" means the data is for the reverse of the knot, and "symmetric" means that the data is symmetric. The data needs to be orientation corrected in the KnotInfo database.

3_1 K(1/3) bs ok

4_1 K(2/5) bs is symmetric

5_1 K(1/5) bs ok

5_2 K(3/7) bs ok

6_1 K(4/9) bs ok

6_2 K(4/11) bs ok

6_3 K(5/13) bs is symmetric

7_1 K(1/7) bs ok

7_2 K(5/11) bs ok

7_3 K(4/13) bs mirrored

7_4 K(4/15) bs mirrored

7_5 K(7/17) bs ok

7_6 K(7/19) bs ok

7_7 K(8/21) bs ok

8_1 K(6/13) bs ok

8_2 K(6/17) bs ok

8_3 K(4/17) bs is symmetric

8_4 K(5/19) bs mirrored

8_5 K(1/3;1/3;1/2) bs ok

8_6 K(10/23) bs ok

8_7 K(9/23) bs ok

8_8 K(9/25) bs ok

8_9 K(7/25) bs is symmetric

8_10 K(1/3;2/3;1/2) bs mirrored

8_11 K(10/27) bs ok

8_12 K(12/29) bs is symmetric

8_13 K(11/29) bs ok

8_14 K(12/31) bs ok

8_15 K(2/3;2/3;1/2) bs mirrored

8_19 K(1/3;1/3;-1/2) bs ok

8_20 K(1/3;2/3;-1/2) bs ok

8_21 K(2/3;2/3;-1/2) bs mirrored

9_1 K(1/9) bs ok

9_2 K(7/15) bs ok

9_3 K(6/19) bs mirrored

9_4 K(5/21) bs ok

9_5 K(6/23) bs mirrored

9_6 K(11/27) bs ok

9_7 K(13/29) bs ok

9_8 K(11/31) bs ok

9_9 K(9/31) bs ok

9_10 K(10/33) bs mirrored

9_11 K(14/33) bs ok

9_12 K(13/35) bs ok

9_13 K(10/37) bs mirrored

9_14 K(14/37) bs ok

9_15 K(16/39) bs ok

9_16 K(1/3;1/3;3/2) bs ok

9_17 K(14/39) bs ok

9_18 K(17/41) bs ok

9_19 K(16/41) bs ok

9_20 K(15/41) bs ok

9_21 K(18/43) bs ok

9_22 K(3/5;1/3;1/2) bs ok

9_23 K(19/45) bs ok

9_24 K(1/3;2/3;3/2) bs mirrored

9_25 K(2/5;2/3;1/2) bs ok, but so is mirror

9_26 K(18/47) bs ok

9_27 K(19/49) bs ok

9_28 K(2/3;2/3;3/2) bs mirrored

9_30 K(3/5;2/3;1/2) bs ok

9_31 K(21/55) bs ok

9_35 K(1/3;1/3;1/3) bs mirrored

9_36 K(2/5;1/3;1/2) bs mirrored

9_37 K(1/3;2/3;2/3) bs mirrored

9_42 K(2/5;1/3;-1/2) bs mirrored

9_43 K(3/5;1/3;-1/2) bs ok

9_44 K(2/5;2/3;-1/2) bs mirrored

9_45 K(3/5;2/3;-1/2) bs ok

9_46 K(1/3;1/3;-1/3) bs ok

9_48 K(2/3;2/3;-1/3) bs mirrored

10_1 K(8/17) bs ok

10_2 K(8/23) bs ok

10_3 K(6/25) bs ok

10_4 K(7/27) bs mirrored

10_5 K(13/33) bs ok

10_6 K(16/37) bs ok

10_7 K(16/43) bs ok

10_8 K(6/29) bs ok

10_9 K(11/39) bs mirrored

10_10 K(17/45) bs ok

10_11 K(13/43) bs ok

10_12 K(17/47) bs ok

10_13 K(22/53) bs ok

10_14 K(22/57) bs ok

10_15 K(19/43) bs ok

10_16 K(14/47) bs mirrored

10_17 K(9/41) bs is symmetric

10_18 K(23/55) bs ok

10_19 K(14/51) bs is symmetric

10_20 K(16/35) bs ok

10_21 K(16/45) bs ok

10_22 K(13/49) bs mirrored

10_23 K(23/59) bs ok

10_24 K(24/55) bs ok

10_25 K(24/65) bs ok

10_26 K(17/61) bs mirrored

10_27 K(27/71) bs ok

10_28 K(19/53) bs ok

10_29 K(26/63) bs ok

10_30 K(26/67) bs ok

10_31 K(25/57) bs ok

10_32 K(29/69) bs ok

10_33 K(18/65) bs is symmetric

10_34 K(13/37) bs ok

10_35 K(20/49) bs ok

10_36 K(20/51) bs ok

10_37 K(23/53) bs is symmetric

10_38 K(25/59) bs ok

10_39 K(22/61) bs ok

10_40 K(29/75) bs ok

10_41 K(26/71) bs ok, but so is mirror

10_42 K(31/81) bs is symmetric

10_43 K(27/73) bs is symmetric

10_44 K(30/79) bs ok

10_45 K(34/89) bs is symmetric

10_46 K(1/5;1/3;1/2) bs ok

10_47 K(1/5;2/3;1/2) bs mirrored

10_48 K(4/5;1/3;1/2) bs ok

10_49 K(4/5;2/3;1/2) bs mirrored

10_50 K(3/7;1/3;1/2) bs ok

10_51 K(3/7;2/3;1/2) bs mirrored

10_52 K(4/7;1/3;1/2) bs ok

10_53 K(4/7;2/3;1/2) bs mirrored

10_54 K(2/7;1/3;1/2) bs ok, but so is mirror

10_55 K(2/7;2/3;1/2) bs mirrored

10_56 K(5/7;1/3;1/2) bs ok

10_57 K(5/7;2/3;1/2) bs mirrored

10_58 K(2/5;2/5;1/2) bs ok, but so is mirror

10_59 K(2/5;3/5;1/2) bs ok

10_60 K(3/5;3/5;1/2) bs ok

10_61 K(1/4;1/3;1/3) bs ok

10_62 K(1/4;1/3;2/3) bs mirrored

10_63 K(1/4;2/3;2/3) bs mirrored

10_64 K(3/4;1/3;1/3) bs ok

10_65 K(3/4;1/3;2/3) bs mirrored

10_66 K(3/4;2/3;2/3) bs mirrored

10_67 K(2/5;1/3;2/3) bs mirrored

10_68 K(3/5;1/3;1/3) bs ok

10_69 K(3/5;2/3;2/3) bs mirrored

10_70 K(2/5;1/3;3/2) bs mirrored

10_71 K(2/5;2/3;3/2) bs ok, but so is mirror

10_72 K(3/5;1/3;3/2) bs ok

10_73 K(3/5;2/3;3/2) bs ok

10_74 K(1/3;1/3;5/3) bs mirrored

10_75 K(2/3;2/3;5/3) bs ok, but so is mirror

10_76 K(1/3;1/3;5/2) bs ok

10_77 K(1/3;2/3;5/2) bs mirrored

10_78 K(2/3;2/3;5/2) bs mirrored

10_124 K(1/5;1/3;-1/2) bs ok

10_125 K(1/5;2/3;-1/2) bs mirrored

10_126 K(4/5;1/3;-1/2) bs ok

10_127 K(4/5;2/3;-1/2) bs mirrored

10_128 K(3/7;1/3;-1/2) bs ok

10_129 K(3/7;2/3;-1/2) bs mirrored

10_130 K(4/7;1/3;-1/2) bs ok

10_131 K(4/7;2/3;-1/2) bs mirrored

10_132 K(2/7;1/3;-1/2) bs ok

10_133 K(2/7;2/3;-1/2) bs mirrored

10_134 K(5/7;1/3;-1/2) bs ok

10_135 K(5/7;2/3;-1/2) bs mirrored

10_136 K(2/5;2/5;-1/2) bs mirrored

10_137 K(2/5;3/5;-1/2) bs mirrored

10_138 K(3/5;3/5;-1/2) bs ok

10_139 K(1/4;1/3;-2/3) bs ok

10_140 K(1/4;1/3;-1/3) bs mirrored

10_141 K(1/4;2/3;-1/3) bs mirrored

10_142 K(3/4;1/3;-2/3) bs ok

10_143 K(3/4;1/3;-1/3) bs ok

10_144 K(3/4;2/3;-1/3) bs mirrored

10_145 K(2/5;1/3;-2/3) bs ok

10_146 K(2/5;2/3;-1/3) bs ok

10_147 K(3/5;1/3;-1/3) bs ok

## References

[1] Dunfield, N., "A table of boundary slopes of Montesinos knots Topology," 40 (2001), 309-315

[2] Hatcher, A. and Oertel, U., "Boundary slopes for Montesinos knots," Topology 28 (1989), 453-480.

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