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Rasmussen invariants of Whitehead doubles and other satellites
by L. Lewark and C. Zibrowius
For any c equal to a prime number or 0, the paper defines certain knot invariants \(\vartheta_c\) that govern the behaviour of the Rasmussen invariants \(s_c\) over fields of characteristic \(c\) under satellite operations with patterns of wrapping number 2.

Table of \(\boldsymbol{\vartheta_c}\)-invariants

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Name pretzel torus \(\boldsymbol{\vartheta_{0}}\) \(\boldsymbol{\vartheta_{2}}\) \(\boldsymbol{\vartheta_{3}}\) \(\boldsymbol{\vartheta_{5}}\) \(\boldsymbol{\vartheta_{7}}\) \(\boldsymbol{\vartheta_{2}}\)-rat \(\boldsymbol{\vartheta_{3}}\)-rat \(\boldsymbol{\vartheta_{5}}\)-rat \(\boldsymbol{s_{2}}\)(2,1-cable) \(\boldsymbol{s_{3}}\)(2,1-cable) \(\boldsymbol{s_{5}}\)(2,1-cable) \(\boldsymbol{s_{2}}\) \(\boldsymbol{\sigma}\) \(\boldsymbol{\tau}\) \(\boldsymbol{\varepsilon}\) det amphicheiral Genus-4D
03_1ᶜ 2,3 3 4 3 3 3 0 0 0 4 4 4 2 -2 1 1 3 0 1
04_1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 5 1 1
05_1 2,5 6 8 6 6 6 0 0 0 8 8 8 4 -4 2 1 5 0 2
05_2 3 4 3 3 3 0 0 0 4 4 4 2 -2 1 1 7 0 1
06_1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 9 0 0
06_2 3 4 3 3 3 0 0 0 4 4 4 2 -2 1 1 11 0 1
06_3 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 13 1 1
07_1 9 12 9 9 9 0 0 0 12 12 12 6 -6 3 1 7 0 3
07_2 3 4 3 3 3 0 0 0 4 4 4 2 -2 1 1 11 0 1
07_3 6 8 6 6 6 0 0 0 8 8 8 4 -4 2 1 13 0 2
07_4 3 4 3 3 3 0 0 0 4 4 4 2 -2 1 1 15 0 1
07_5 6 8 6 6 6 0 0 0 8 8 8 4 -4 2 1 17 0 2
07_6 3 4 3 3 3 0 0 0 4 4 4 2 -2 1 1 19 0 1
07_7 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 21 0 1
08_01 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 13 0 1
08_02 6 8 6 6 6 0 0 0 8 8 8 4 -4 2 1 17 0 2
08_03 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 17 1 1
08_04 -3 -4 -3 -3 -3 0 0 0 -2 -2 -2 -2 2 -1 -1 19 0 1
08_05 6 8 6 6 6 0 0 0 8 8 8 4 -4 2 1 21 0 2
08_06 3 4 3 3 3 0 0 0 4 4 4 2 -2 1 1 23 0 1
08_07 -3 -4 -3 -3 -3 0 0 0 -2 -2 -2 -2 2 -1 -1 23 0 1
08_08 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 25 0 0
08_09 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 25 1 0
08_10 -3 -4 -3 -3 -3 0 0 0 -2 -2 -2 -2 2 -1 -1 27 0 1
08_11 3 4 3 3 3 0 0 0 4 4 4 2 -2 1 1 27 0 1
08_12 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 29 1 1
08_13 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 29 0 1
08_14 3 4 3 3 3 0 0 0 4 4 4 2 -2 1 1 31 0 1
08_15 6 8 6 6 6 0 0 0 4 -4 2 1 33 0 2
08_16 -3 -4 -3 -3 -3 0 0 0 -2 2 -1 -1 35 0 1
08_17 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 37 1 1
08_18 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 45 1 1
08_19ᶜ 3,3,-2 3,4 8 8 9 8 8 0 0 0 12 12 6 -6 3 1 3 0 3
08_20 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 9 0 0
08_21 3 4 3 3 3 0 0 0 2 -2 1 1 15 0 1
09_01 12 16 12 12 12 0 0 0 8 -8 4 1 9 0 4
09_02 3 4 3 3 3 0 0 0 2 -2 1 1 15 0 1
09_03 9 12 9 9 9 0 0 0 6 -6 3 1 19 0 3
09_04 6 8 6 6 6 0 0 0 4 -4 2 1 21 0 2
09_05 3 4 3 3 3 0 0 0 2 -2 1 1 23 0 1
09_06 9 12 9 9 9 0 0 0 6 -6 3 1 27 0 3
09_07 6 8 6 6 6 0 0 0 4 -4 2 1 29 0 2
09_08 3 4 3 3 3 0 0 0 2 -2 1 1 31 0 1
09_09 9 12 9 9 9 0 0 0 6 -6 3 1 31 0 3
09_10 6 8 6 6 6 0 0 0 4 -4 2 1 33 0 2
09_11 -6 -8 -6 -6 -6 0 0 0 -4 4 -2 -1 33 0 2
09_12 3 4 3 3 3 0 0 0 2 -2 1 1 35 0 1
09_13 6 8 6 6 6 0 0 0 4 -4 2 1 37 0 2
09_14 0 0 0 0 0 1 1 1 0 0 0 0 37 0 1
09_15 -3 -4 -3 -3 -3 0 0 0 -2 2 -1 -1 39 0 1
09_16 9 12 9 9 9 6 -6 3 1 39 0 3
09_17 3 4 3 3 3 2 -2 1 1 39 0 1
09_18 6 8 6 6 6 4 -4 2 1 41 0 2
09_19 0 0 0 0 0 0 0 0 0 41 0 1
09_20 6 8 6 6 6 4 -4 2 1 41 0 2
09_21 -3 -4 -3 -3 -2 2 -1 -1 43 0 1
09_22 3 4 3 3 2 -2 1 1 43 0 1
09_23 8 6 6 4 -4 2 1 45 0 2
09_24 0 0 0 0 0 0 0 45 0 1
09_25 4 2 -2 1 1 47 0 1
09_26 -3 -4 -3 -3 -3 -2 2 -1 -1 47 0 1
09_27 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 49 0 0
09_28 4 2 -2 1 1 51 0 1
09_29 -2 2 -1 -1 51 0 1
09_30 0 0 0 0 53 0 1
09_31 4 2 -2 1 1 55 0 1
09_32 -2 2 -1 -1 59 0 1
09_33 0 0 0 0 61 0 1
09_34 0 0 0 0 69 0 1
09_35 3 4 3 3 3 2 -2 1 1 27 0 1
09_36 -6 -8 -6 -6 -6 -4 4 -2 -1 37 0 2
09_37 0 0 0 0 45 0 1
09_38 4 -4 2 1 57 0 2
09_39 -2 2 -1 -1 55 0 1
09_40 2 -2 1 1 75 0 1
09_41 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 49 0 0
09_42 0 0 0 0 0 0 2 0 0 7 0 1
09_43 6 8 6 6 6 4 -4 2 1 13 0 2
09_44 0 0 0 0 0 0 0 0 0 17 0 1
09_45 -3 -4 -3 -3 -3 -2 2 -1 -1 23 0 1
09_46 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 9 0 0
09_47 3 4 3 3 3 2 -2 1 1 27 0 1
09_48 -3 -2 2 -1 -1 27 0 1
09_49 8 6 6 6 4 -4 2 1 25 0 2
10_001 0 0 0 0 17 0 1
10_002 6 -6 3 1 23 0 3
10_003 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 25 0 0
10_004 -2 2 -1 -1 27 0 1
10_005 -6 -8 -6 -6 -6 -4 4 -2 -1 33 0 2
10_006 6 8 6 6 6 4 -4 2 1 37 0 2
10_007 2 -2 1 1 43 0 1
10_008 6 8 6 6 6 4 -4 2 1 29 0 2
10_009 3 4 3 3 3 2 -2 1 1 39 0 1
10_010 0 0 0 0 0 0 0 0 0 45 0 1
10_011 3 4 3 3 3 2 -2 1 1 43 0 1
10_012 -3 -4 -3 -3 -3 -2 2 -1 -1 47 0 1
10_013 0 0 0 0 53 0 1
10_014 4 -4 2 1 57 0 2
10_015 -2 2 -1 -1 43 0 1
10_016 2 -2 1 1 47 0 1
10_017 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 41 1 1
10_018 2 -2 1 1 55 0 1
10_019 2 -2 1 1 51 0 1
10_020 2 -2 1 1 35 0 1
10_021 4 -4 2 1 45 0 2
10_022 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 49 0 0
10_023 -2 2 -1 -1 59 0 1
10_024 2 -2 1 1 55 0 1
10_025 4 -4 2 1 65 0 2
10_026 0 0 0 0 61 0 1
10_027 -2 2 -1 -1 71 0 1
10_028 0 0 0 0 53 0 1
10_029 2 -2 1 1 63 0 1
10_030 2 -2 1 1 67 0 1
10_031 0 0 0 0 57 0 1
10_032 0 0 0 0 69 0 1
10_033 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 65 1 1
10_034 0 0 0 0 37 0 1
10_035 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 49 0 0
10_036 2 -2 1 1 51 0 1
10_037 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 53 1 1
10_038 2 -2 1 1 59 0 1
10_039 4 -4 2 1 61 0 2
10_040 -2 2 -1 -1 75 0 1
10_041 2 -2 1 1 71 0 1
10_042 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 81 0 0
10_043 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 73 1 1
10_044 2 -2 1 1 79 0 1
10_045 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 89 1 1
10_046 9 12 9 9 9 6 -6 3 1 31 0 3
10_047 -4 4 -2 -1 41 0 2
10_048 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 49 0 0
10_049 6 -6 3 1 59 0 3
10_050 4 -4 2 1 53 0 2
10_051 -2 2 -1 -1 67 0 1
10_052 2 -2 1 1 59 0 1
10_053 4 -4 2 1 73 0 2
10_054 -2 2 -1 -1 47 0 1
10_055