10 42

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10 41.gif

10_41

10 43.gif

10_43

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10 42 Quick Notes


10 42 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X11,1,12,20 X5,13,6,12 X15,18,16,19 X13,9,14,8 X17,7,18,6 X7,17,8,16 X19,14,20,15 X9,2,10,3
Gauss code -1, 10, -2, 1, -4, 7, -8, 6, -10, 2, -3, 4, -6, 9, -5, 8, -7, 5, -9, 3
Dowker-Thistlethwaite code 4 10 12 16 2 20 8 18 6 14
Conway Notation [2211112]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 13.2398
A-Polynomial See Data:10 42/A-polynomial

[edit Notes for 10 42's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 0 }[/math]
Topological 4 genus [math]\displaystyle{ 0 }[/math]
Concordance genus [math]\displaystyle{ 0 }[/math]
Rasmussen s-Invariant 0

[edit Notes for 10 42's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+7 t^2-19 t+27-19 t^{-1} +7 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6+z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 81, 0 }
Jones polynomial [math]\displaystyle{ -q^5+3 q^4-6 q^3+10 q^2-12 q+14-13 q^{-1} +10 q^{-2} -7 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -3 z^4-a^4 z^2+3 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} -5 z^2+a^2+3 a^{-2} - a^{-4} -2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^9+z^9 a^{-1} +4 a^2 z^8+3 z^8 a^{-2} +7 z^8+6 a^3 z^7+11 a z^7+9 z^7 a^{-1} +4 z^7 a^{-3} +4 a^4 z^6+2 z^6 a^{-2} +3 z^6 a^{-4} -5 z^6+a^5 z^5-11 a^3 z^5-24 a z^5-18 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4-10 a^2 z^4-11 z^4 a^{-2} -6 z^4 a^{-4} -8 z^4-a^5 z^3+5 a^3 z^3+14 a z^3+10 z^3 a^{-1} -2 z^3 a^{-5} +2 a^4 z^2+6 a^2 z^2+9 z^2 a^{-2} +4 z^2 a^{-4} +9 z^2-a z-z a^{-1} +z a^{-3} +z a^{-5} -a^2-3 a^{-2} - a^{-4} -2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-2 q^{10}+2 q^8-q^6-2 q^4+2 q^2-2+3 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -2 q^{-10} + q^{-12} - q^{-16} }[/math]
The G2 invariant [math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+14 q^{72}-11 q^{70}-2 q^{68}+27 q^{66}-50 q^{64}+72 q^{62}-77 q^{60}+46 q^{58}+9 q^{56}-85 q^{54}+151 q^{52}-176 q^{50}+153 q^{48}-70 q^{46}-43 q^{44}+156 q^{42}-219 q^{40}+209 q^{38}-128 q^{36}+4 q^{34}+105 q^{32}-160 q^{30}+140 q^{28}-53 q^{26}-50 q^{24}+135 q^{22}-152 q^{20}+76 q^{18}+46 q^{16}-181 q^{14}+262 q^{12}-253 q^{10}+150 q^8+16 q^6-182 q^4+299 q^2-322+238 q^{-2} -86 q^{-4} -82 q^{-6} +199 q^{-8} -227 q^{-10} +171 q^{-12} -51 q^{-14} -60 q^{-16} +126 q^{-18} -121 q^{-20} +43 q^{-22} +66 q^{-24} -153 q^{-26} +181 q^{-28} -130 q^{-30} +27 q^{-32} +93 q^{-34} -177 q^{-36} +209 q^{-38} -172 q^{-40} +90 q^{-42} +7 q^{-44} -92 q^{-46} +132 q^{-48} -130 q^{-50} +97 q^{-52} -45 q^{-54} -3 q^{-56} +35 q^{-58} -51 q^{-60} +46 q^{-62} -32 q^{-64} +16 q^{-66} -2 q^{-68} -7 q^{-70} +8 q^{-72} -8 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_42.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{11}+3 q^9-3 q^7+3 q^5-3 q^3+q+2 q^{-1} -2 q^{-3} +4 q^{-5} -3 q^{-7} +2 q^{-9} - q^{-11} }[/math]
2 [math]\displaystyle{ q^{32}-3 q^{30}-q^{28}+11 q^{26}-8 q^{24}-12 q^{22}+24 q^{20}-6 q^{18}-27 q^{16}+29 q^{14}+5 q^{12}-31 q^{10}+19 q^8+14 q^6-19 q^4-3 q^2+15+5 q^{-2} -24 q^{-4} +7 q^{-6} +26 q^{-8} -29 q^{-10} -4 q^{-12} +32 q^{-14} -19 q^{-16} -10 q^{-18} +20 q^{-20} -7 q^{-22} -7 q^{-24} +7 q^{-26} - q^{-28} -2 q^{-30} + q^{-32} }[/math]
3 [math]\displaystyle{ -q^{63}+3 q^{61}+q^{59}-7 q^{57}-6 q^{55}+12 q^{53}+22 q^{51}-20 q^{49}-41 q^{47}+17 q^{45}+71 q^{43}-4 q^{41}-106 q^{39}-24 q^{37}+135 q^{35}+66 q^{33}-149 q^{31}-114 q^{29}+143 q^{27}+161 q^{25}-122 q^{23}-189 q^{21}+83 q^{19}+201 q^{17}-39 q^{15}-191 q^{13}-8 q^{11}+161 q^9+56 q^7-123 q^5-91 q^3+67 q+132 q^{-1} -9 q^{-3} -154 q^{-5} -51 q^{-7} +164 q^{-9} +108 q^{-11} -156 q^{-13} -155 q^{-15} +129 q^{-17} +184 q^{-19} -91 q^{-21} -188 q^{-23} +45 q^{-25} +175 q^{-27} -10 q^{-29} -139 q^{-31} -18 q^{-33} +103 q^{-35} +27 q^{-37} -66 q^{-39} -26 q^{-41} +38 q^{-43} +20 q^{-45} -20 q^{-47} -14 q^{-49} +10 q^{-51} +7 q^{-53} -4 q^{-55} -3 q^{-57} + q^{-59} +2 q^{-61} - q^{-63} }[/math]
4 [math]\displaystyle{ q^{104}-3 q^{102}-q^{100}+7 q^{98}+2 q^{96}+2 q^{94}-22 q^{92}-14 q^{90}+29 q^{88}+29 q^{86}+35 q^{84}-75 q^{82}-97 q^{80}+32 q^{78}+112 q^{76}+182 q^{74}-108 q^{72}-297 q^{70}-122 q^{68}+173 q^{66}+532 q^{64}+78 q^{62}-512 q^{60}-556 q^{58}-30 q^{56}+924 q^{54}+611 q^{52}-412 q^{50}-1067 q^{48}-630 q^{46}+970 q^{44}+1230 q^{42}+129 q^{40}-1222 q^{38}-1298 q^{36}+525 q^{34}+1462 q^{32}+778 q^{30}-870 q^{28}-1560 q^{26}-101 q^{24}+1170 q^{22}+1118 q^{20}-274 q^{18}-1313 q^{16}-593 q^{14}+588 q^{12}+1098 q^{10}+294 q^8-781 q^6-899 q^4-63 q^2+876+805 q^{-2} -120 q^{-4} -1083 q^{-6} -735 q^{-8} +503 q^{-10} +1210 q^{-12} +614 q^{-14} -1022 q^{-16} -1298 q^{-18} -82 q^{-20} +1285 q^{-22} +1273 q^{-24} -582 q^{-26} -1464 q^{-28} -715 q^{-30} +869 q^{-32} +1508 q^{-34} +50 q^{-36} -1075 q^{-38} -1009 q^{-40} +213 q^{-42} +1175 q^{-44} +433 q^{-46} -435 q^{-48} -804 q^{-50} -209 q^{-52} +595 q^{-54} +396 q^{-56} -12 q^{-58} -399 q^{-60} -245 q^{-62} +192 q^{-64} +181 q^{-66} +85 q^{-68} -125 q^{-70} -125 q^{-72} +43 q^{-74} +45 q^{-76} +49 q^{-78} -28 q^{-80} -41 q^{-82} +12 q^{-84} +5 q^{-86} +15 q^{-88} -5 q^{-90} -10 q^{-92} +4 q^{-94} +3 q^{-98} - q^{-100} -2 q^{-102} + q^{-104} }[/math]
5 [math]\displaystyle{ -q^{155}+3 q^{153}+q^{151}-7 q^{149}-2 q^{147}+2 q^{145}+8 q^{143}+14 q^{141}+5 q^{139}-29 q^{137}-43 q^{135}-7 q^{133}+47 q^{131}+89 q^{129}+57 q^{127}-58 q^{125}-196 q^{123}-170 q^{121}+68 q^{119}+320 q^{117}+372 q^{115}+63 q^{113}-474 q^{111}-756 q^{109}-344 q^{107}+565 q^{105}+1229 q^{103}+948 q^{101}-391 q^{99}-1810 q^{97}-1903 q^{95}-170 q^{93}+2229 q^{91}+3151 q^{89}+1336 q^{87}-2225 q^{85}-4533 q^{83}-3102 q^{81}+1548 q^{79}+5677 q^{77}+5301 q^{75}-19 q^{73}-6209 q^{71}-7619 q^{69}-2245 q^{67}+5844 q^{65}+9550 q^{63}+4969 q^{61}-4528 q^{59}-10682 q^{57}-7654 q^{55}+2405 q^{53}+10794 q^{51}+9848 q^{49}+88 q^{47}-9869 q^{45}-11136 q^{43}-2576 q^{41}+8143 q^{39}+11462 q^{37}+4612 q^{35}-5987 q^{33}-10853 q^{31}-6023 q^{29}+3703 q^{27}+9594 q^{25}+6835 q^{23}-1598 q^{21}-8003 q^{19}-7094 q^{17}-287 q^{15}+6249 q^{13}+7153 q^{11}+1985 q^9-4606 q^7-7051 q^5-3577 q^3+2890 q+7018 q^{-1} +5234 q^{-3} -1168 q^{-5} -6918 q^{-7} -6933 q^{-9} -774 q^{-11} +6617 q^{-13} +8636 q^{-15} +2955 q^{-17} -5908 q^{-19} -10096 q^{-21} -5332 q^{-23} +4625 q^{-25} +10986 q^{-27} +7707 q^{-29} -2705 q^{-31} -11057 q^{-33} -9720 q^{-35} +310 q^{-37} +10131 q^{-39} +10999 q^{-41} +2233 q^{-43} -8250 q^{-45} -11262 q^{-47} -4511 q^{-49} +5741 q^{-51} +10462 q^{-53} +6029 q^{-55} -3016 q^{-57} -8707 q^{-59} -6674 q^{-61} +602 q^{-63} +6489 q^{-65} +6318 q^{-67} +1130 q^{-69} -4149 q^{-71} -5300 q^{-73} -2075 q^{-75} +2207 q^{-77} +3927 q^{-79} +2256 q^{-81} -802 q^{-83} -2572 q^{-85} -1957 q^{-87} +1474 q^{-91} +1447 q^{-93} +325 q^{-95} -728 q^{-97} -922 q^{-99} -364 q^{-101} +285 q^{-103} +524 q^{-105} +288 q^{-107} -92 q^{-109} -263 q^{-111} -173 q^{-113} +12 q^{-115} +113 q^{-117} +96 q^{-119} +9 q^{-121} -53 q^{-123} -43 q^{-125} - q^{-127} +19 q^{-129} +13 q^{-131} +6 q^{-133} -8 q^{-135} -11 q^{-137} +4 q^{-139} +5 q^{-141} - q^{-143} -3 q^{-149} + q^{-151} +2 q^{-153} - q^{-155} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{16}+q^{14}+2 q^{12}-2 q^{10}+2 q^8-q^6-2 q^4+2 q^2-2+3 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -2 q^{-10} + q^{-12} - q^{-16} }[/math]
1,1 [math]\displaystyle{ q^{44}-6 q^{42}+20 q^{40}-50 q^{38}+103 q^{36}-188 q^{34}+312 q^{32}-474 q^{30}+659 q^{28}-840 q^{26}+1000 q^{24}-1096 q^{22}+1083 q^{20}-946 q^{18}+670 q^{16}-278 q^{14}-211 q^{12}+752 q^{10}-1268 q^8+1714 q^6-2021 q^4+2168 q^2-2124+1902 q^{-2} -1526 q^{-4} +1038 q^{-6} -516 q^{-8} +8 q^{-10} +430 q^{-12} -762 q^{-14} +968 q^{-16} -1034 q^{-18} +1006 q^{-20} -898 q^{-22} +746 q^{-24} -578 q^{-26} +419 q^{-28} -288 q^{-30} +182 q^{-32} -108 q^{-34} +58 q^{-36} -28 q^{-38} +12 q^{-40} -4 q^{-42} + q^{-44} }[/math]
2,0 [math]\displaystyle{ q^{42}-q^{40}-3 q^{38}+6 q^{34}+4 q^{32}-9 q^{30}-4 q^{28}+11 q^{26}+4 q^{24}-15 q^{22}-6 q^{20}+16 q^{18}+4 q^{16}-17 q^{14}+q^{12}+17 q^{10}-4 q^8-7 q^6+8 q^4+3 q^2-7+3 q^{-2} +6 q^{-4} -13 q^{-6} -6 q^{-8} +17 q^{-10} +3 q^{-12} -18 q^{-14} +4 q^{-16} +18 q^{-18} -2 q^{-20} -13 q^{-22} +2 q^{-24} +10 q^{-26} -3 q^{-28} -7 q^{-30} +2 q^{-32} +3 q^{-34} - q^{-36} -2 q^{-38} + q^{-42} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{34}-3 q^{32}+q^{30}+6 q^{28}-11 q^{26}+5 q^{24}+13 q^{22}-22 q^{20}+6 q^{18}+20 q^{16}-25 q^{14}+3 q^{12}+23 q^{10}-17 q^8-5 q^6+12 q^4-2 q^2-9-4 q^{-2} +17 q^{-4} -5 q^{-6} -16 q^{-8} +27 q^{-10} + q^{-12} -23 q^{-14} +22 q^{-16} +2 q^{-18} -19 q^{-20} +11 q^{-22} +2 q^{-24} -9 q^{-26} +4 q^{-28} + q^{-30} -2 q^{-32} + q^{-34} }[/math]
1,0,0 [math]\displaystyle{ -q^{21}+q^{19}+2 q^{15}-2 q^{13}+3 q^{11}-2 q^9+q^7-2 q^5+q^3-q- q^{-1} +2 q^{-3} - q^{-5} +3 q^{-7} +4 q^{-11} -2 q^{-13} + q^{-15} - q^{-17} - q^{-21} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{34}+3 q^{32}-7 q^{30}+12 q^{28}-17 q^{26}+23 q^{24}-27 q^{22}+30 q^{20}-28 q^{18}+24 q^{16}-15 q^{14}+3 q^{12}+11 q^{10}-27 q^8+39 q^6-50 q^4+56 q^2-57+52 q^{-2} -41 q^{-4} +29 q^{-6} -14 q^{-8} + q^{-10} +13 q^{-12} -21 q^{-14} +28 q^{-16} -28 q^{-18} +27 q^{-20} -23 q^{-22} +18 q^{-24} -13 q^{-26} +8 q^{-28} -5 q^{-30} +2 q^{-32} - q^{-34} }[/math]
1,0 [math]\displaystyle{ q^{56}-3 q^{52}-3 q^{50}+4 q^{48}+9 q^{46}-q^{44}-14 q^{42}-8 q^{40}+16 q^{38}+20 q^{36}-8 q^{34}-29 q^{32}-8 q^{30}+27 q^{28}+23 q^{26}-16 q^{24}-30 q^{22}+2 q^{20}+29 q^{18}+11 q^{16}-21 q^{14}-15 q^{12}+14 q^{10}+17 q^8-10 q^6-19 q^4+5 q^2+19-2 q^{-2} -21 q^{-4} -2 q^{-6} +22 q^{-8} +10 q^{-10} -21 q^{-12} -17 q^{-14} +18 q^{-16} +29 q^{-18} -5 q^{-20} -31 q^{-22} -10 q^{-24} +26 q^{-26} +22 q^{-28} -11 q^{-30} -25 q^{-32} -4 q^{-34} +17 q^{-36} +11 q^{-38} -7 q^{-40} -11 q^{-42} - q^{-44} +6 q^{-46} +3 q^{-48} -2 q^{-50} -2 q^{-52} + q^{-56} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+14 q^{72}-11 q^{70}-2 q^{68}+27 q^{66}-50 q^{64}+72 q^{62}-77 q^{60}+46 q^{58}+9 q^{56}-85 q^{54}+151 q^{52}-176 q^{50}+153 q^{48}-70 q^{46}-43 q^{44}+156 q^{42}-219 q^{40}+209 q^{38}-128 q^{36}+4 q^{34}+105 q^{32}-160 q^{30}+140 q^{28}-53 q^{26}-50 q^{24}+135 q^{22}-152 q^{20}+76 q^{18}+46 q^{16}-181 q^{14}+262 q^{12}-253 q^{10}+150 q^8+16 q^6-182 q^4+299 q^2-322+238 q^{-2} -86 q^{-4} -82 q^{-6} +199 q^{-8} -227 q^{-10} +171 q^{-12} -51 q^{-14} -60 q^{-16} +126 q^{-18} -121 q^{-20} +43 q^{-22} +66 q^{-24} -153 q^{-26} +181 q^{-28} -130 q^{-30} +27 q^{-32} +93 q^{-34} -177 q^{-36} +209 q^{-38} -172 q^{-40} +90 q^{-42} +7 q^{-44} -92 q^{-46} +132 q^{-48} -130 q^{-50} +97 q^{-52} -45 q^{-54} -3 q^{-56} +35 q^{-58} -51 q^{-60} +46 q^{-62} -32 q^{-64} +16 q^{-66} -2 q^{-68} -7 q^{-70} +8 q^{-72} -8 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} }[/math]

.

Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 42"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+7 t^2-19 t+27-19 t^{-1} +7 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6+z^4+1 }[/math]
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 81, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+3 q^4-6 q^3+10 q^2-12 q+14-13 q^{-1} +10 q^{-2} -7 q^{-3} +4 q^{-4} - q^{-5} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -3 z^4-a^4 z^2+3 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} -5 z^2+a^2+3 a^{-2} - a^{-4} -2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^9+z^9 a^{-1} +4 a^2 z^8+3 z^8 a^{-2} +7 z^8+6 a^3 z^7+11 a z^7+9 z^7 a^{-1} +4 z^7 a^{-3} +4 a^4 z^6+2 z^6 a^{-2} +3 z^6 a^{-4} -5 z^6+a^5 z^5-11 a^3 z^5-24 a z^5-18 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4-10 a^2 z^4-11 z^4 a^{-2} -6 z^4 a^{-4} -8 z^4-a^5 z^3+5 a^3 z^3+14 a z^3+10 z^3 a^{-1} -2 z^3 a^{-5} +2 a^4 z^2+6 a^2 z^2+9 z^2 a^{-2} +4 z^2 a^{-4} +9 z^2-a z-z a^{-1} +z a^{-3} +z a^{-5} -a^2-3 a^{-2} - a^{-4} -2 }[/math]

Vassiliev invariants

V2 and V3: (0, 1)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{16}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ -\frac{88}{3} }[/math] [math]\displaystyle{ -\frac{16}{3} }[/math] [math]\displaystyle{ -32 }[/math]

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 10 42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -5-4-3-2-1012345χ
11          1-1
9         2 2
7        41 -3
5       62  4
3      64   -2
1     86    2
-1    67     1
-3   47      -3
-5  36       3
-7 14        -3
-9 3         3
-111          -1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[10, 42]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 42]]
Out[3]=  
PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 20], X[5, 13, 6, 12], 

 X[15, 18, 16, 19], X[13, 9, 14, 8], X[17, 7, 18, 6], X[7, 17, 8, 16], 



  X[19, 14, 20, 15], X[9, 2, 10, 3]]
In[4]:=
GaussCode[Knot[10, 42]]
Out[4]=  
GaussCode[-1, 10, -2, 1, -4, 7, -8, 6, -10, 2, -3, 4, -6, 9, -5, 8, -7, 
  5, -9, 3]
In[5]:=
BR[Knot[10, 42]]
Out[5]=  
BR[5, {-1, -1, 2, -1, 2, -3, 2, 4, -3, 4}]
In[6]:=
alex = Alexander[Knot[10, 42]][t]
Out[6]=  
      -3   7    19             2    3

27 - t   + -- - -- - 19 t + 7 t  - t



           2   t


           t
In[7]:=
Conway[Knot[10, 42]][z]
Out[7]=  
     4    6
1 + z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 42], Knot[10, 75]}
In[9]:=
{KnotDet[Knot[10, 42]], KnotSignature[Knot[10, 42]]}
Out[9]=  
{81, 0}
In[10]:=
J=Jones[Knot[10, 42]][q]
Out[10]=  
      -5   4    7    10   13              2      3      4    5

14 - q   + -- - -- + -- - -- - 12 q + 10 q  - 6 q  + 3 q  - q



           4    3    2   q


           q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 42]}
In[12]:=
A2Invariant[Knot[10, 42]][q]
Out[12]=  
      -16    -14    2     2    2     -6   2    2       2    4    6

-2 - q    + q    + --- - --- + -- - q   - -- + -- + 3 q  - q  + q  + 



                   12    10    8          4    2
                  q     q     q          q    q

    8      10    12    16


  3 q  - 2 q   + q   - q
In[13]:=
Kauffman[Knot[10, 42]][a, z]
Out[13]=  
                                                   2      2

     -4   3     2   z    z    z            2   4 z    9 z       2  2



-2 - a   - -- - a  + -- + -- - - - a z + 9 z  + ---- + ---- + 6 a  z  + 



           2         5    3   a                  4      2
          a         a    a                      a      a

              3       3                                         4
    4  2   2 z    10 z          3      3  3    5  3      4   6 z
 2 a  z  - ---- + ----- + 14 a z  + 5 a  z  - a  z  - 8 z  - ---- - 
             5      a                                          4
            a                                                 a

     4                         5      5       5
 11 z        2  4      4  4   z    5 z    18 z          5       3  5
 ----- - 10 a  z  - 7 a  z  + -- - ---- - ----- - 24 a z  - 11 a  z  + 
   2                           5     3      a
  a                           a     a

                   6      6                7      7
  5  5      6   3 z    2 z       4  6   4 z    9 z          7
 a  z  - 5 z  + ---- + ---- + 4 a  z  + ---- + ---- + 11 a z  + 
                  4      2                3     a
                 a      a                a

                     8              9
    3  7      8   3 z       2  8   z       9
 6 a  z  + 7 z  + ---- + 4 a  z  + -- + a z
                    2              a


                    a
In[14]:=
{Vassiliev[2][Knot[10, 42]], Vassiliev[3][Knot[10, 42]]}
Out[14]=  
{0, 1}
In[15]:=
Kh[Knot[10, 42]][q, t]
Out[15]=  
7           1        3       1       4       3       6       4

- + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 
q          11  5    9  4    7  4    7  3    5  3    5  2    3  2



         q   t    q  t    q  t    q  t    q  t    q  t    q  t

  7      6               3        3  2      5  2      5  3      7  3
 ---- + --- + 6 q t + 6 q  t + 4 q  t  + 6 q  t  + 2 q  t  + 4 q  t  + 
  3     q t
 q  t

  7  4      9  4    11  5


  q  t  + 2 q  t  + q   t
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