9 31

9_30

9_32

Visit 9 31's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 31's page at Knotilus! Visit 9 31's page at the original Knot Atlas! 9 31 Quick Notes

9 31 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,18 X5,13,6,12 X17,7,18,6 X7,14,8,15 X13,16,14,17 X15,8,16,9 X9,2,10,3
Gauss code	-1, 9, -2, 1, -4, 5, -6, 8, -9, 2, -3, 4, -7, 6, -8, 7, -5, 3
Dowker-Thistlethwaite code	4 10 12 14 2 18 16 8 6
Conway Notation	[2111112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-2] Hyperbolic Volume 11.6863 A-Polynomial See Data:9 31/A-polynomial

[edit Notes for 9 31's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant -2

[edit Notes for 9 31's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 55, -2 }
Jones polynomial	[math]\displaystyle{ -q^2+3 q-5+8 q^{-1} -9 q^{-2} +10 q^{-3} -8 q^{-4} +6 q^{-5} -4 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^6-2 z^4 a^4-4 z^2 a^4-2 a^4+z^6 a^2+4 z^4 a^2+7 z^2 a^2+4 a^2-z^4-2 z^2-1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8+4 z^5 a^7-4 z^3 a^7+6 z^6 a^6-8 z^4 a^6+3 z^2 a^6+4 z^7 a^5+z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4+11 z^6 a^4-23 z^4 a^4+13 z^2 a^4-2 a^4+7 z^7 a^3-7 z^5 a^3-5 z^3 a^3+5 z a^3+z^8 a^2+8 z^6 a^2-21 z^4 a^2+15 z^2 a^2-4 a^2+3 z^7 a-3 z^5 a-3 z^3 a+3 z a+3 z^6-7 z^4+5 z^2-1+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} }[/math]
The A2 invariant	[math]\displaystyle{ q^{22}-q^{20}-2 q^{18}+q^{16}-2 q^{14}+q^{12}+q^{10}+3 q^6-q^4+3 q^2- q^{-2} + q^{-4} - q^{-6} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+8 q^{106}-4 q^{104}-5 q^{102}+22 q^{100}-33 q^{98}+45 q^{96}-41 q^{94}+16 q^{92}+17 q^{90}-54 q^{88}+80 q^{86}-86 q^{84}+65 q^{82}-20 q^{80}-33 q^{78}+75 q^{76}-90 q^{74}+70 q^{72}-28 q^{70}-23 q^{68}+48 q^{66}-52 q^{64}+24 q^{62}+26 q^{60}-67 q^{58}+82 q^{56}-60 q^{54}+2 q^{52}+65 q^{50}-121 q^{48}+136 q^{46}-108 q^{44}+48 q^{42}+32 q^{40}-96 q^{38}+129 q^{36}-115 q^{34}+68 q^{32}-4 q^{30}-51 q^{28}+73 q^{26}-55 q^{24}+22 q^{22}+29 q^{20}-57 q^{18}+59 q^{16}-26 q^{14}-24 q^{12}+72 q^{10}-94 q^8+83 q^6-43 q^4-9 q^2+55-80 q^{-2} +81 q^{-4} -55 q^{-6} +18 q^{-8} +13 q^{-10} -35 q^{-12} +38 q^{-14} -31 q^{-16} +19 q^{-18} -5 q^{-20} -5 q^{-22} +7 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_31.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{15}-3 q^{13}+2 q^{11}-2 q^9+2 q^7+q^5-q^3+3 q-2 q^{-1} +2 q^{-3} - q^{-5} }[/math]
2	[math]\displaystyle{ q^{42}-3 q^{40}-q^{38}+10 q^{36}-7 q^{34}-8 q^{32}+18 q^{30}-7 q^{28}-15 q^{26}+18 q^{24}-q^{22}-15 q^{20}+7 q^{18}+6 q^{16}-5 q^{14}-7 q^{12}+11 q^{10}+8 q^8-17 q^6+7 q^4+15 q^2-18+14 q^{-4} -9 q^{-6} -3 q^{-8} +7 q^{-10} -2 q^{-12} -2 q^{-14} + q^{-16} }[/math]
3	[math]\displaystyle{ q^{81}-3 q^{79}-q^{77}+7 q^{75}+5 q^{73}-11 q^{71}-18 q^{69}+20 q^{67}+29 q^{65}-22 q^{63}-49 q^{61}+21 q^{59}+71 q^{57}-13 q^{55}-89 q^{53}-q^{51}+101 q^{49}+18 q^{47}-96 q^{45}-38 q^{43}+87 q^{41}+48 q^{39}-63 q^{37}-60 q^{35}+35 q^{33}+56 q^{31}-4 q^{29}-56 q^{27}-27 q^{25}+50 q^{23}+54 q^{21}-36 q^{19}-76 q^{17}+23 q^{15}+94 q^{13}-3 q^{11}-99 q^9-15 q^7+96 q^5+36 q^3-84 q-47 q^{-1} +62 q^{-3} +54 q^{-5} -41 q^{-7} -49 q^{-9} +20 q^{-11} +39 q^{-13} -6 q^{-15} -26 q^{-17} -2 q^{-19} +16 q^{-21} +3 q^{-23} -7 q^{-25} -3 q^{-27} +2 q^{-29} +2 q^{-31} - q^{-33} }[/math]
4	[math]\displaystyle{ q^{132}-3 q^{130}-q^{128}+7 q^{126}+2 q^{124}+q^{122}-21 q^{120}-10 q^{118}+29 q^{116}+23 q^{114}+19 q^{112}-72 q^{110}-63 q^{108}+58 q^{106}+96 q^{104}+89 q^{102}-148 q^{100}-197 q^{98}+31 q^{96}+212 q^{94}+268 q^{92}-164 q^{90}-396 q^{88}-127 q^{86}+273 q^{84}+507 q^{82}-34 q^{80}-504 q^{78}-358 q^{76}+169 q^{74}+631 q^{72}+184 q^{70}-404 q^{68}-484 q^{66}-45 q^{64}+522 q^{62}+330 q^{60}-156 q^{58}-424 q^{56}-226 q^{54}+256 q^{52}+343 q^{50}+102 q^{48}-247 q^{46}-320 q^{44}-43 q^{42}+286 q^{40}+318 q^{38}-53 q^{36}-359 q^{34}-306 q^{32}+198 q^{30}+474 q^{28}+150 q^{26}-327 q^{24}-521 q^{22}+46 q^{20}+524 q^{18}+348 q^{16}-187 q^{14}-617 q^{12}-162 q^{10}+402 q^8+461 q^6+49 q^4-515 q^2-312+152 q^{-2} +395 q^{-4} +229 q^{-6} -269 q^{-8} -292 q^{-10} -57 q^{-12} +201 q^{-14} +235 q^{-16} -58 q^{-18} -149 q^{-20} -106 q^{-22} +39 q^{-24} +128 q^{-26} +19 q^{-28} -34 q^{-30} -59 q^{-32} -13 q^{-34} +41 q^{-36} +14 q^{-38} +2 q^{-40} -16 q^{-42} -10 q^{-44} +7 q^{-46} +3 q^{-48} +3 q^{-50} -2 q^{-52} -2 q^{-54} + q^{-56} }[/math]
5	[math]\displaystyle{ q^{195}-3 q^{193}-q^{191}+7 q^{189}+2 q^{187}-2 q^{185}-9 q^{183}-13 q^{181}-q^{179}+29 q^{177}+37 q^{175}-3 q^{173}-55 q^{171}-74 q^{169}-16 q^{167}+86 q^{165}+165 q^{163}+74 q^{161}-160 q^{159}-290 q^{157}-171 q^{155}+177 q^{153}+496 q^{151}+398 q^{149}-184 q^{147}-756 q^{145}-715 q^{143}+53 q^{141}+1009 q^{139}+1206 q^{137}+231 q^{135}-1204 q^{133}-1773 q^{131}-710 q^{129}+1230 q^{127}+2331 q^{125}+1378 q^{123}-1030 q^{121}-2774 q^{119}-2105 q^{117}+580 q^{115}+2947 q^{113}+2787 q^{111}+59 q^{109}-2838 q^{107}-3243 q^{105}-771 q^{103}+2405 q^{101}+3437 q^{99}+1405 q^{97}-1779 q^{95}-3268 q^{93}-1888 q^{91}+1030 q^{89}+2882 q^{87}+2136 q^{85}-324 q^{83}-2268 q^{81}-2189 q^{79}-336 q^{77}+1640 q^{75}+2098 q^{73}+849 q^{71}-986 q^{69}-1933 q^{67}-1298 q^{65}+391 q^{63}+1769 q^{61}+1674 q^{59}+133 q^{57}-1609 q^{55}-2038 q^{53}-646 q^{51}+1454 q^{49}+2397 q^{47}+1152 q^{45}-1274 q^{43}-2698 q^{41}-1691 q^{39}+994 q^{37}+2926 q^{35}+2239 q^{33}-586 q^{31}-2993 q^{29}-2738 q^{27}+37 q^{25}+2839 q^{23}+3121 q^{21}+621 q^{19}-2429 q^{17}-3304 q^{15}-1265 q^{13}+1794 q^{11}+3190 q^9+1818 q^7-1007 q^5-2819 q^3-2140 q+240 q^{-1} +2189 q^{-3} +2180 q^{-5} +426 q^{-7} -1472 q^{-9} -1953 q^{-11} -829 q^{-13} +769 q^{-15} +1529 q^{-17} +981 q^{-19} -222 q^{-21} -1034 q^{-23} -908 q^{-25} -123 q^{-27} +590 q^{-29} +702 q^{-31} +260 q^{-33} -254 q^{-35} -458 q^{-37} -277 q^{-39} +66 q^{-41} +261 q^{-43} +205 q^{-45} +20 q^{-47} -117 q^{-49} -132 q^{-51} -46 q^{-53} +49 q^{-55} +71 q^{-57} +33 q^{-59} -13 q^{-61} -29 q^{-63} -23 q^{-65} -2 q^{-67} +16 q^{-69} +10 q^{-71} -3 q^{-75} -3 q^{-77} -3 q^{-79} +2 q^{-81} +2 q^{-83} - q^{-85} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{22}-q^{20}-2 q^{18}+q^{16}-2 q^{14}+q^{12}+q^{10}+3 q^6-q^4+3 q^2- q^{-2} + q^{-4} - q^{-6} }[/math]
1,1	[math]\displaystyle{ q^{60}-6 q^{58}+18 q^{56}-38 q^{54}+69 q^{52}-120 q^{50}+186 q^{48}-254 q^{46}+324 q^{44}-382 q^{42}+414 q^{40}-396 q^{38}+326 q^{36}-208 q^{34}+40 q^{32}+156 q^{30}-368 q^{28}+554 q^{26}-708 q^{24}+792 q^{22}-811 q^{20}+756 q^{18}-632 q^{16}+466 q^{14}-253 q^{12}+60 q^{10}+126 q^8-270 q^6+367 q^4-406 q^2+396-354 q^{-2} +292 q^{-4} -218 q^{-6} +150 q^{-8} -96 q^{-10} +54 q^{-12} -28 q^{-14} +12 q^{-16} -4 q^{-18} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{56}-q^{54}-3 q^{52}+5 q^{48}+3 q^{46}-6 q^{44}+8 q^{40}-9 q^{36}+7 q^{32}-5 q^{30}-10 q^{28}+2 q^{26}+2 q^{24}-7 q^{22}+3 q^{20}+7 q^{18}-q^{16}+q^{14}+10 q^{12}+2 q^{10}-8 q^8+2 q^6+10 q^4-4 q^2-8+5 q^{-2} +5 q^{-4} -4 q^{-6} -3 q^{-8} +2 q^{-10} +2 q^{-12} -2 q^{-14} - q^{-16} + q^{-18} }[/math]

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{62}-q^{60}-3 q^{58}+q^{56}+4 q^{54}-q^{52}-4 q^{50}+6 q^{48}+8 q^{46}-6 q^{44}-8 q^{42}+9 q^{40}+2 q^{38}-13 q^{36}-2 q^{34}+7 q^{32}-8 q^{30}-12 q^{28}+4 q^{26}+q^{24}-10 q^{22}+4 q^{20}+17 q^{18}-2 q^{16}+19 q^{12}+9 q^{10}-9 q^8+2 q^6+9 q^4-4 q^2-10+ q^{-2} +3 q^{-4} -5 q^{-6} -2 q^{-8} +3 q^{-10} - q^{-14} + q^{-16} }[/math]
1,0,0,0	[math]\displaystyle{ q^{36}-q^{34}-2 q^{28}+q^{26}-3 q^{24}-q^{20}-q^{18}+q^{16}+q^{14}+3 q^{12}+2 q^{10}+4 q^8+3 q^4-q^2-2 q^{-4} + q^{-6} - q^{-8} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{48}-3 q^{46}+6 q^{44}-9 q^{42}+12 q^{40}-15 q^{38}+15 q^{36}-15 q^{34}+11 q^{32}-8 q^{30}+8 q^{26}-16 q^{24}+23 q^{22}-26 q^{20}+30 q^{18}-28 q^{16}+26 q^{14}-18 q^{12}+11 q^{10}-3 q^8-3 q^6+10 q^4-13 q^2+16-14 q^{-2} +13 q^{-4} -10 q^{-6} +7 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-14} }[/math]
1,0	[math]\displaystyle{ q^{78}-3 q^{74}-3 q^{72}+3 q^{70}+8 q^{68}+q^{66}-10 q^{64}-7 q^{62}+10 q^{60}+14 q^{58}-2 q^{56}-17 q^{54}-7 q^{52}+12 q^{50}+12 q^{48}-8 q^{46}-15 q^{44}+q^{42}+12 q^{40}+2 q^{38}-12 q^{36}-5 q^{34}+9 q^{32}+7 q^{30}-8 q^{28}-8 q^{26}+6 q^{24}+11 q^{22}-3 q^{20}-10 q^{18}+3 q^{16}+16 q^{14}+5 q^{12}-13 q^{10}-10 q^8+11 q^6+16 q^4-2 q^2-16-6 q^{-2} +10 q^{-4} +10 q^{-6} -4 q^{-8} -9 q^{-10} -2 q^{-12} +5 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-24} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{66}-3 q^{64}+3 q^{62}-4 q^{60}+8 q^{58}-10 q^{56}+10 q^{54}-10 q^{52}+14 q^{50}-12 q^{48}+8 q^{46}-9 q^{44}+8 q^{42}-2 q^{40}-5 q^{38}+4 q^{36}-11 q^{34}+15 q^{32}-21 q^{30}+16 q^{28}-25 q^{26}+23 q^{24}-20 q^{22}+20 q^{20}-15 q^{18}+18 q^{16}-q^{14}+9 q^{12}+2 q^{10}-2 q^8+10 q^6-9 q^4+9 q^2-14+11 q^{-2} -11 q^{-4} +8 q^{-6} -9 q^{-8} +6 q^{-10} -4 q^{-12} +3 q^{-14} -2 q^{-16} + q^{-18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+8 q^{106}-4 q^{104}-5 q^{102}+22 q^{100}-33 q^{98}+45 q^{96}-41 q^{94}+16 q^{92}+17 q^{90}-54 q^{88}+80 q^{86}-86 q^{84}+65 q^{82}-20 q^{80}-33 q^{78}+75 q^{76}-90 q^{74}+70 q^{72}-28 q^{70}-23 q^{68}+48 q^{66}-52 q^{64}+24 q^{62}+26 q^{60}-67 q^{58}+82 q^{56}-60 q^{54}+2 q^{52}+65 q^{50}-121 q^{48}+136 q^{46}-108 q^{44}+48 q^{42}+32 q^{40}-96 q^{38}+129 q^{36}-115 q^{34}+68 q^{32}-4 q^{30}-51 q^{28}+73 q^{26}-55 q^{24}+22 q^{22}+29 q^{20}-57 q^{18}+59 q^{16}-26 q^{14}-24 q^{12}+72 q^{10}-94 q^8+83 q^6-43 q^4-9 q^2+55-80 q^{-2} +81 q^{-4} -55 q^{-6} +18 q^{-8} +13 q^{-10} -35 q^{-12} +38 q^{-14} -31 q^{-16} +19 q^{-18} -5 q^{-20} -5 q^{-22} +7 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32} }[/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["9 31"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+z^4+2 z^2+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]=

{ 55, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^2+3 q-5+8 q^{-1} -9 q^{-2} +10 q^{-3} -8 q^{-4} +6 q^{-5} -4 q^{-6} + q^{-7} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^2 a^6-2 z^4 a^4-4 z^2 a^4-2 a^4+z^6 a^2+4 z^4 a^2+7 z^2 a^2+4 a^2-z^4-2 z^2-1 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^4 a^8+4 z^5 a^7-4 z^3 a^7+6 z^6 a^6-8 z^4 a^6+3 z^2 a^6+4 z^7 a^5+z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4+11 z^6 a^4-23 z^4 a^4+13 z^2 a^4-2 a^4+7 z^7 a^3-7 z^5 a^3-5 z^3 a^3+5 z a^3+z^8 a^2+8 z^6 a^2-21 z^4 a^2+15 z^2 a^2-4 a^2+3 z^7 a-3 z^5 a-3 z^3 a+3 z a+3 z^6-7 z^4+5 z^2-1+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} }[/math]

Vassiliev invariants

V2 and V3:

(2, -2)

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{ 8 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{172}{3} }[/math] [math]\displaystyle{ \frac{20}{3} }[/math] [math]\displaystyle{ -128 }[/math] [math]\displaystyle{ -\frac{640}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1376}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ \frac{11911}{15} }[/math] [math]\displaystyle{ \frac{916}{15} }[/math] [math]\displaystyle{ \frac{12604}{45} }[/math] [math]\displaystyle{ -\frac{55}{9} }[/math] [math]\displaystyle{ \frac{631}{15} }[/math]

V_2,1 through V_6,9 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]-2 is the signature of 9 31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                   1 -1 3                 2   2 1               3 1   -2 -1             5 2     3 -3           5 4       -1 -5         5 4         1 -7       3 5           2 -9     3 5             -2 -11   1 3               2 -13   3                 -3 -15 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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[9, 31]]
Out[2]=	9
In[3]:=	PD[Knot[9, 31]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 18], X[5, 13, 6, 12], X[17, 7, 18, 6], X[7, 14, 8, 15], X[13, 16, 14, 17], X[15, 8, 16, 9], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[9, 31]]
Out[4]=	GaussCode[-1, 9, -2, 1, -4, 5, -6, 8, -9, 2, -3, 4, -7, 6, -8, 7, -5, 3]
In[5]:=	BR[Knot[9, 31]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[9, 31]][t]
Out[6]=	-3 5 13 2 3 -17 + t - -- + -- + 13 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[9, 31]][z]
Out[7]=	2 4 6 1 + 2 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 31], Knot[11, NonAlternating, 11], Knot[11, NonAlternating, 22], Knot[11, NonAlternating, 112], Knot[11, NonAlternating, 127]}
In[9]:=	{KnotDet[Knot[9, 31]], KnotSignature[Knot[9, 31]]}
Out[9]=	{55, -2}
In[10]:=	J=Jones[Knot[9, 31]][q]
Out[10]=	-7 4 6 8 10 9 8 2 -5 + q - -- + -- - -- + -- - -- + - + 3 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 31]}
In[12]:=	A2Invariant[Knot[9, 31]][q]
Out[12]=	-22 -20 2 -16 2 -12 -10 3 -4 3 2 q - q - --- + q - --- + q + q + -- - q + -- - q + 18 14 6 2 q q q q 4 6 q - q
In[13]:=	Kauffman[Knot[9, 31]][a, z]
Out[13]=	2 4 z 3 5 2 2 2 -1 - 4 a - 2 a + - + 3 a z + 5 a z + 3 a z + 5 z + 15 a z + a 3 4 2 6 2 2 z 3 3 3 5 3 7 3 13 a z + 3 a z - ---- - 3 a z - 5 a z - 8 a z - 4 a z - a 5 4 2 4 4 4 6 4 8 4 z 5 7 z - 21 a z - 23 a z - 8 a z + a z + -- - 3 a z - a 3 5 5 5 7 5 6 2 6 4 6 6 6 7 a z + a z + 4 a z + 3 z + 8 a z + 11 a z + 6 a z + 7 3 7 5 7 2 8 4 8 3 a z + 7 a z + 4 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 31]], Vassiliev[3][Knot[9, 31]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[9, 31]][q, t]
Out[15]=	4 5 1 3 1 3 3 5 3 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 5 5 4 5 2 t 2 3 2 5 3 ----- + ----- + ---- + ---- + --- + 3 q t + q t + 2 q t + q t 7 2 5 2 5 3 q q t q t q t q t