9 9

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9 8.gif

9_8

9 10.gif

9_10

9 9.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 9's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 9 at Knotilus!

[edit 9 9 Quick Notes]
[edit 9 9 Further Notes and Views]


Knot presentations

Planar diagram presentation X1627 X3,12,4,13 X7,16,8,17 X9,18,10,1 X17,8,18,9 X15,10,16,11 X5,14,6,15 X11,2,12,3 X13,4,14,5
Gauss code -1, 8, -2, 9, -7, 1, -3, 5, -4, 6, -8, 2, -9, 7, -6, 3, -5, 4
Dowker-Thistlethwaite code 6 12 14 16 18 2 4 10 8
Conway Notation [423]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif

Length is 10, width is 3,

Braid index is 3

9 9 ML.gif 9 9 AP.gif
[{6, 1}, {11, 2}, {1, 3}, {2, 5}, {3, 7}, {4, 6}, {5, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 4}]

[edit Notes on presentations of 9 9]


Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["9 9"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1627 X3,12,4,13 X7,16,8,17 X9,18,10,1 X17,8,18,9 X15,10,16,11 X5,14,6,15 X11,2,12,3 X13,4,14,5
In[5]:= GaussCode[K]
Out[5]= -1, 8, -2, 9, -7, 1, -3, 5, -4, 6, -8, 2, -9, 7, -6, 3, -5, 4
In[6]:= DTCode[K]
Out[6]= 6 12 14 16 18 2 4 10 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [423]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= [math]\displaystyle{ \textrm{BR}(3,\{-1,-1,-1,-1,-1,-2,1,-2,-2,-2\}) }[/math]
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 3, 10, 3 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
9 9 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{6, 1}, {11, 2}, {1, 3}, {2, 5}, {3, 7}, {4, 6}, {5, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 4}]
In[14]:= Draw[ap]
9 9 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-16][5]
Hyperbolic Volume 8.01682
A-Polynomial See Data:9 9/A-polynomial

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

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 3 }[/math]
Topological 4 genus [math]\displaystyle{ 3 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -6

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

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-4 t^2+6 t-7+6 t^{-1} -4 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+8 z^4+8 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 31, -6 }
Jones polynomial [math]\displaystyle{ q^{-3} - q^{-4} +3 q^{-5} -4 q^{-6} +5 q^{-7} -5 q^{-8} +5 q^{-9} -4 q^{-10} +2 q^{-11} - q^{-12} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^4 a^{10}-3 z^2 a^{10}-2 a^{10}+z^6 a^8+4 z^4 a^8+4 z^2 a^8+a^8+z^6 a^6+5 z^4 a^6+7 z^2 a^6+2 a^6 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^3 a^{15}-z a^{15}+2 z^4 a^{14}-z^2 a^{14}+3 z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-2 z^5 a^{11}+z^8 a^{10}-z^6 a^{10}+2 z^4 a^{10}-6 z^2 a^{10}+2 a^{10}+3 z^7 a^9-8 z^5 a^9+5 z^3 a^9-2 z a^9+z^8 a^8-3 z^6 a^8+3 z^4 a^8-3 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z a^7+z^6 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6 }[/math]
The A2 invariant [math]\displaystyle{ -q^{36}-q^{32}-q^{30}-q^{26}+2 q^{24}+q^{20}+q^{18}+2 q^{14}+q^{10} }[/math]
The G2 invariant [math]\displaystyle{ q^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{188}-2 q^{184}+5 q^{182}-6 q^{180}+6 q^{178}-6 q^{176}+2 q^{174}+3 q^{172}-7 q^{170}+12 q^{168}-11 q^{166}+9 q^{164}-5 q^{162}-2 q^{160}+6 q^{158}-11 q^{156}+11 q^{154}-7 q^{152}+3 q^{148}-7 q^{146}+5 q^{144}-q^{142}-8 q^{140}+9 q^{138}-12 q^{136}+5 q^{134}+5 q^{132}-15 q^{130}+20 q^{128}-18 q^{126}+10 q^{124}+q^{122}-12 q^{120}+18 q^{118}-18 q^{116}+14 q^{114}-4 q^{112}-4 q^{110}+11 q^{108}-10 q^{106}+7 q^{104}-2 q^{102}-5 q^{100}+8 q^{98}-8 q^{96}+2 q^{94}+6 q^{92}-11 q^{90}+16 q^{88}-11 q^{86}+q^{84}+7 q^{82}-11 q^{80}+16 q^{78}-12 q^{76}+6 q^{74}+2 q^{72}-5 q^{70}+10 q^{68}-7 q^{66}+5 q^{64}+2 q^{58}-2 q^{56}+2 q^{54}+q^{50} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_9.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{25}+q^{23}-2 q^{21}+q^{19}+q^{13}-q^{11}+2 q^9+q^5 }[/math]
2 [math]\displaystyle{ q^{68}-q^{66}+3 q^{62}-3 q^{60}-q^{58}+5 q^{56}-5 q^{54}-2 q^{52}+5 q^{50}-2 q^{48}-2 q^{46}+q^{44}+q^{42}-3 q^{40}-2 q^{38}+4 q^{36}-4 q^{32}+4 q^{30}+2 q^{28}-5 q^{26}+2 q^{24}+3 q^{22}-3 q^{20}+q^{18}+3 q^{16}+q^{10} }[/math]
3 [math]\displaystyle{ -q^{129}+q^{127}-q^{123}-q^{121}+2 q^{119}+2 q^{117}-3 q^{115}-q^{113}+5 q^{111}-7 q^{107}-q^{105}+11 q^{103}+2 q^{101}-13 q^{99}-2 q^{97}+12 q^{95}+6 q^{93}-10 q^{91}-6 q^{89}+5 q^{87}+8 q^{85}-q^{83}-8 q^{81}-5 q^{79}+6 q^{77}+8 q^{75}-6 q^{73}-11 q^{71}+4 q^{69}+10 q^{67}-2 q^{65}-12 q^{63}-q^{61}+11 q^{59}+3 q^{57}-11 q^{55}-6 q^{53}+9 q^{51}+10 q^{49}-6 q^{47}-10 q^{45}+3 q^{43}+10 q^{41}+q^{39}-9 q^{37}-3 q^{35}+5 q^{33}+5 q^{31}-2 q^{29}-2 q^{27}+q^{25}+3 q^{23}+q^{21}+q^{15} }[/math]
4 [math]\displaystyle{ q^{208}-q^{206}+q^{202}-q^{200}+2 q^{198}-3 q^{196}-q^{194}+2 q^{192}-2 q^{190}+5 q^{188}-4 q^{186}+q^{184}+4 q^{182}-9 q^{180}+q^{178}-3 q^{176}+13 q^{174}+12 q^{172}-19 q^{170}-13 q^{168}-10 q^{166}+26 q^{164}+32 q^{162}-18 q^{160}-31 q^{158}-28 q^{156}+25 q^{154}+46 q^{152}+2 q^{150}-24 q^{148}-37 q^{146}+2 q^{144}+34 q^{142}+21 q^{140}+q^{138}-26 q^{136}-20 q^{134}+2 q^{132}+21 q^{130}+23 q^{128}-5 q^{126}-26 q^{124}-19 q^{122}+15 q^{120}+29 q^{118}+8 q^{116}-26 q^{114}-29 q^{112}+14 q^{110}+29 q^{108}+13 q^{106}-27 q^{104}-33 q^{102}+11 q^{100}+27 q^{98}+25 q^{96}-19 q^{94}-39 q^{92}-5 q^{90}+18 q^{88}+37 q^{86}+q^{84}-33 q^{82}-22 q^{80}-6 q^{78}+35 q^{76}+24 q^{74}-10 q^{72}-23 q^{70}-26 q^{68}+14 q^{66}+26 q^{64}+13 q^{62}-5 q^{60}-26 q^{58}-7 q^{56}+8 q^{54}+14 q^{52}+9 q^{50}-10 q^{48}-8 q^{46}-4 q^{44}+4 q^{42}+8 q^{40}-2 q^{34}+3 q^{30}+q^{28}+q^{26}+q^{20} }[/math]
5 [math]\displaystyle{ -q^{305}+q^{303}-q^{299}+q^{297}-q^{293}+2 q^{291}+2 q^{289}-2 q^{287}-q^{285}-q^{283}-3 q^{281}+2 q^{279}+4 q^{277}+4 q^{275}+q^{273}-3 q^{271}-8 q^{269}-12 q^{267}+16 q^{263}+24 q^{261}+9 q^{259}-22 q^{257}-43 q^{255}-27 q^{253}+27 q^{251}+72 q^{249}+52 q^{247}-26 q^{245}-94 q^{243}-86 q^{241}+6 q^{239}+113 q^{237}+124 q^{235}+21 q^{233}-114 q^{231}-154 q^{229}-57 q^{227}+88 q^{225}+165 q^{223}+98 q^{221}-56 q^{219}-151 q^{217}-117 q^{215}+2 q^{213}+112 q^{211}+125 q^{209}+39 q^{207}-63 q^{205}-102 q^{203}-68 q^{201}+7 q^{199}+75 q^{197}+80 q^{195}+32 q^{193}-36 q^{191}-78 q^{189}-60 q^{187}+8 q^{185}+73 q^{183}+75 q^{181}+9 q^{179}-67 q^{177}-78 q^{175}-13 q^{173}+67 q^{171}+82 q^{169}+14 q^{167}-73 q^{165}-88 q^{163}-11 q^{161}+82 q^{159}+95 q^{157}+19 q^{155}-88 q^{153}-112 q^{151}-28 q^{149}+84 q^{147}+124 q^{145}+49 q^{143}-71 q^{141}-129 q^{139}-77 q^{137}+41 q^{135}+124 q^{133}+100 q^{131}-6 q^{129}-102 q^{127}-115 q^{125}-38 q^{123}+67 q^{121}+115 q^{119}+70 q^{117}-22 q^{115}-92 q^{113}-94 q^{111}-27 q^{109}+57 q^{107}+93 q^{105}+59 q^{103}-14 q^{101}-71 q^{99}-78 q^{97}-27 q^{95}+40 q^{93}+70 q^{91}+47 q^{89}-q^{87}-46 q^{85}-55 q^{83}-21 q^{81}+21 q^{79}+41 q^{77}+33 q^{75}+4 q^{73}-24 q^{71}-28 q^{69}-14 q^{67}+5 q^{65}+18 q^{63}+15 q^{61}+2 q^{59}-6 q^{57}-10 q^{55}-6 q^{53}+2 q^{51}+6 q^{49}+3 q^{47}+3 q^{45}-2 q^{41}+2 q^{37}+q^{35}+q^{33}+q^{31}+q^{25} }[/math]
6 [math]\displaystyle{ q^{420}-q^{418}+q^{414}-q^{412}-q^{408}+2 q^{406}-3 q^{404}-2 q^{402}+5 q^{400}+q^{396}+3 q^{392}-8 q^{390}-8 q^{388}+6 q^{386}+q^{384}+6 q^{382}+9 q^{380}+14 q^{378}-14 q^{376}-24 q^{374}-8 q^{372}-13 q^{370}+13 q^{368}+37 q^{366}+50 q^{364}-6 q^{362}-56 q^{360}-58 q^{358}-59 q^{356}+18 q^{354}+106 q^{352}+143 q^{350}+38 q^{348}-103 q^{346}-176 q^{344}-183 q^{342}-13 q^{340}+207 q^{338}+326 q^{336}+176 q^{334}-108 q^{332}-327 q^{330}-405 q^{328}-157 q^{326}+232 q^{324}+516 q^{322}+422 q^{320}+45 q^{318}-352 q^{316}-602 q^{314}-417 q^{312}+50 q^{310}+508 q^{308}+601 q^{306}+322 q^{304}-124 q^{302}-538 q^{300}-570 q^{298}-253 q^{296}+219 q^{294}+491 q^{292}+462 q^{290}+190 q^{288}-207 q^{286}-421 q^{284}-384 q^{282}-117 q^{280}+152 q^{278}+318 q^{276}+312 q^{274}+117 q^{272}-106 q^{270}-264 q^{268}-250 q^{266}-127 q^{264}+75 q^{262}+229 q^{260}+240 q^{258}+105 q^{256}-102 q^{254}-222 q^{252}-213 q^{250}-36 q^{248}+147 q^{246}+239 q^{244}+141 q^{242}-73 q^{240}-216 q^{238}-204 q^{236}-6 q^{234}+181 q^{232}+262 q^{230}+118 q^{228}-139 q^{226}-297 q^{224}-236 q^{222}+27 q^{220}+262 q^{218}+360 q^{216}+168 q^{214}-165 q^{212}-396 q^{210}-344 q^{208}-43 q^{206}+271 q^{204}+463 q^{202}+310 q^{200}-59 q^{198}-400 q^{196}-459 q^{194}-225 q^{192}+129 q^{190}+456 q^{188}+453 q^{186}+166 q^{184}-233 q^{182}-447 q^{180}-400 q^{178}-137 q^{176}+257 q^{174}+447 q^{172}+372 q^{170}+62 q^{168}-229 q^{166}-396 q^{164}-354 q^{162}-64 q^{160}+214 q^{158}+365 q^{156}+278 q^{154}+94 q^{152}-154 q^{150}-324 q^{148}-268 q^{146}-99 q^{144}+121 q^{142}+229 q^{140}+261 q^{138}+127 q^{136}-76 q^{134}-194 q^{132}-221 q^{130}-122 q^{128}+156 q^{124}+192 q^{122}+121 q^{120}+14 q^{118}-99 q^{116}-142 q^{114}-135 q^{112}-24 q^{110}+62 q^{108}+106 q^{106}+101 q^{104}+43 q^{102}-23 q^{100}-86 q^{98}-69 q^{96}-38 q^{94}+7 q^{92}+44 q^{90}+55 q^{88}+36 q^{86}-7 q^{84}-19 q^{82}-31 q^{80}-24 q^{78}-8 q^{76}+13 q^{74}+20 q^{72}+8 q^{70}+7 q^{68}-3 q^{66}-8 q^{64}-9 q^{62}-q^{60}+4 q^{58}+q^{56}+5 q^{54}+3 q^{52}+q^{50}-2 q^{48}+2 q^{44}+q^{40}+q^{38}+q^{36}+q^{30} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{36}-q^{32}-q^{30}-q^{26}+2 q^{24}+q^{20}+q^{18}+2 q^{14}+q^{10} }[/math]
1,1 [math]\displaystyle{ q^{100}-2 q^{98}+4 q^{96}-6 q^{94}+11 q^{92}-14 q^{90}+18 q^{88}-24 q^{86}+28 q^{84}-30 q^{82}+28 q^{80}-30 q^{78}+27 q^{76}-16 q^{74}+8 q^{72}+4 q^{70}-20 q^{68}+36 q^{66}-52 q^{64}+64 q^{62}-70 q^{60}+72 q^{58}-68 q^{56}+54 q^{54}-47 q^{52}+22 q^{50}-10 q^{48}-10 q^{46}+21 q^{44}-34 q^{42}+42 q^{40}-38 q^{38}+38 q^{36}-28 q^{34}+28 q^{32}-12 q^{30}+12 q^{28}-4 q^{26}+4 q^{24}+q^{20} }[/math]
2,0 [math]\displaystyle{ q^{90}+q^{86}+q^{84}+q^{82}-2 q^{74}-2 q^{72}+q^{70}-3 q^{66}+q^{62}-2 q^{60}-4 q^{58}-q^{56}-q^{54}-3 q^{52}+3 q^{48}+3 q^{42}+q^{40}-q^{38}+2 q^{36}+3 q^{34}+q^{32}+3 q^{28}+2 q^{26}+q^{20} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{82}-q^{80}+3 q^{76}-2 q^{74}-2 q^{72}+5 q^{70}-2 q^{68}-3 q^{66}+4 q^{64}-q^{62}-3 q^{60}+q^{58}-q^{56}-3 q^{54}-3 q^{52}-5 q^{46}+q^{44}+3 q^{42}-3 q^{40}+2 q^{38}+5 q^{36}-q^{34}+3 q^{32}+3 q^{30}+2 q^{26}+2 q^{24}+q^{20} }[/math]
1,0,0 [math]\displaystyle{ -q^{47}-2 q^{43}-2 q^{39}-q^{35}+q^{33}+q^{31}+q^{29}+2 q^{27}+2 q^{23}+2 q^{19}+q^{15} }[/math]
1,0,1 [math]\displaystyle{ q^{132}-2 q^{130}+3 q^{128}-q^{126}-3 q^{124}+9 q^{122}-9 q^{120}+5 q^{118}+5 q^{116}-17 q^{114}+19 q^{112}-11 q^{110}-6 q^{108}+19 q^{106}-26 q^{104}+16 q^{102}+5 q^{100}-18 q^{98}+31 q^{96}-23 q^{94}+3 q^{92}+14 q^{90}-31 q^{88}+27 q^{86}-8 q^{84}+2 q^{82}+17 q^{80}-2 q^{78}-4 q^{76}+3 q^{74}-14 q^{72}-19 q^{70}+10 q^{68}-39 q^{66}+21 q^{64}-20 q^{62}-8 q^{60}+22 q^{58}-30 q^{56}+32 q^{54}-11 q^{52}+9 q^{50}+16 q^{48}-5 q^{46}+20 q^{44}-2 q^{42}+6 q^{40}+4 q^{38}+4 q^{34}+q^{30} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{104}-q^{100}+2 q^{98}+3 q^{96}-q^{94}+3 q^{90}+q^{88}-2 q^{86}+q^{84}+2 q^{82}-3 q^{80}-4 q^{78}-q^{76}-5 q^{74}-9 q^{72}-2 q^{70}-2 q^{68}-6 q^{66}-q^{64}+3 q^{62}-2 q^{58}+3 q^{56}+3 q^{54}+q^{52}+2 q^{50}+5 q^{48}+3 q^{46}+2 q^{44}+5 q^{42}+2 q^{40}+2 q^{38}+2 q^{36}+2 q^{34}+q^{30} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{58}-2 q^{54}-q^{52}-q^{50}-2 q^{48}-q^{44}+q^{42}+2 q^{38}+q^{36}+2 q^{34}+q^{32}+q^{30}+2 q^{28}+2 q^{24}+q^{20} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{82}+q^{80}-2 q^{78}+3 q^{76}-4 q^{74}+4 q^{72}-5 q^{70}+4 q^{68}-3 q^{66}+2 q^{64}+q^{62}-3 q^{60}+5 q^{58}-7 q^{56}+7 q^{54}-9 q^{52}+8 q^{50}-8 q^{48}+5 q^{46}-3 q^{44}+q^{42}+q^{40}-2 q^{38}+5 q^{36}-3 q^{34}+5 q^{32}-3 q^{30}+4 q^{28}-2 q^{26}+2 q^{24}+q^{20} }[/math]
1,0 [math]\displaystyle{ q^{132}-q^{128}-q^{126}+q^{124}+3 q^{122}+q^{120}-3 q^{118}-3 q^{116}+q^{114}+5 q^{112}+q^{110}-4 q^{108}-3 q^{106}+2 q^{104}+4 q^{102}-q^{100}-4 q^{98}-q^{96}+3 q^{94}+q^{92}-4 q^{90}-3 q^{88}+q^{86}+2 q^{84}-2 q^{82}-3 q^{80}+2 q^{76}-q^{74}-4 q^{72}-q^{70}+4 q^{68}+3 q^{66}-3 q^{64}-4 q^{62}+2 q^{60}+6 q^{58}+2 q^{56}-2 q^{54}-3 q^{52}+3 q^{50}+4 q^{48}+q^{46}-2 q^{44}+2 q^{40}+2 q^{38}+q^{30} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{114}-q^{112}+q^{110}-q^{108}+3 q^{106}-3 q^{104}+2 q^{102}-3 q^{100}+5 q^{98}-3 q^{96}+2 q^{94}-3 q^{92}+2 q^{90}-q^{86}+q^{84}-4 q^{82}+4 q^{80}-6 q^{78}+4 q^{76}-9 q^{74}+4 q^{72}-8 q^{70}+4 q^{68}-7 q^{66}+3 q^{64}-3 q^{62}+2 q^{60}+4 q^{54}-q^{52}+4 q^{50}-q^{48}+6 q^{46}-q^{44}+5 q^{42}-q^{40}+4 q^{38}+2 q^{34}+q^{30} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{188}-2 q^{184}+5 q^{182}-6 q^{180}+6 q^{178}-6 q^{176}+2 q^{174}+3 q^{172}-7 q^{170}+12 q^{168}-11 q^{166}+9 q^{164}-5 q^{162}-2 q^{160}+6 q^{158}-11 q^{156}+11 q^{154}-7 q^{152}+3 q^{148}-7 q^{146}+5 q^{144}-q^{142}-8 q^{140}+9 q^{138}-12 q^{136}+5 q^{134}+5 q^{132}-15 q^{130}+20 q^{128}-18 q^{126}+10 q^{124}+q^{122}-12 q^{120}+18 q^{118}-18 q^{116}+14 q^{114}-4 q^{112}-4 q^{110}+11 q^{108}-10 q^{106}+7 q^{104}-2 q^{102}-5 q^{100}+8 q^{98}-8 q^{96}+2 q^{94}+6 q^{92}-11 q^{90}+16 q^{88}-11 q^{86}+q^{84}+7 q^{82}-11 q^{80}+16 q^{78}-12 q^{76}+6 q^{74}+2 q^{72}-5 q^{70}+10 q^{68}-7 q^{66}+5 q^{64}+2 q^{58}-2 q^{56}+2 q^{54}+q^{50} }[/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 9"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-4 t^2+6 t-7+6 t^{-1} -4 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+8 z^4+8 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]= { 31, -6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{-3} - q^{-4} +3 q^{-5} -4 q^{-6} +5 q^{-7} -5 q^{-8} +5 q^{-9} -4 q^{-10} +2 q^{-11} - q^{-12} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^4 a^{10}-3 z^2 a^{10}-2 a^{10}+z^6 a^8+4 z^4 a^8+4 z^2 a^8+a^8+z^6 a^6+5 z^4 a^6+7 z^2 a^6+2 a^6 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^3 a^{15}-z a^{15}+2 z^4 a^{14}-z^2 a^{14}+3 z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-2 z^5 a^{11}+z^8 a^{10}-z^6 a^{10}+2 z^4 a^{10}-6 z^2 a^{10}+2 a^{10}+3 z^7 a^9-8 z^5 a^9+5 z^3 a^9-2 z a^9+z^8 a^8-3 z^6 a^8+3 z^4 a^8-3 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z a^7+z^6 a^6-5 z^4 a^6+7 z^2 a^6-2 a^6 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

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

In[3]:= K = Knot["9 9"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { [math]\displaystyle{ 2 t^3-4 t^2+6 t-7+6 t^{-1} -4 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ q^{-3} - q^{-4} +3 q^{-5} -4 q^{-6} +5 q^{-7} -5 q^{-8} +5 q^{-9} -4 q^{-10} +2 q^{-11} - q^{-12} }[/math] }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

Vassiliev invariants

V2 and V3: (8, -22)
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{ 32 }[/math] [math]\displaystyle{ -176 }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{3760}{3} }[/math] [math]\displaystyle{ \frac{560}{3} }[/math] [math]\displaystyle{ -5632 }[/math] [math]\displaystyle{ -\frac{29696}{3} }[/math] [math]\displaystyle{ -\frac{5216}{3} }[/math] [math]\displaystyle{ -1264 }[/math] [math]\displaystyle{ \frac{16384}{3} }[/math] [math]\displaystyle{ 15488 }[/math] [math]\displaystyle{ \frac{120320}{3} }[/math] [math]\displaystyle{ \frac{17920}{3} }[/math] [math]\displaystyle{ \frac{1196284}{15} }[/math] [math]\displaystyle{ \frac{40544}{15} }[/math] [math]\displaystyle{ \frac{1354096}{45} }[/math] [math]\displaystyle{ \frac{4628}{9} }[/math] [math]\displaystyle{ \frac{57964}{15} }[/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]-6 is the signature of 9 9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -9-8-7-6-5-4-3-2-10χ
-5         11
-7        110
-9       2  2
-11      21  -1
-13     32   1
-15    22    0
-17   33     0
-19  12      1
-21 13       -2
-23 1        1
-251         -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math]
[math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

 [math]\displaystyle{ n }[/math]  [math]\displaystyle{ J_n }[/math]
2 [math]\displaystyle{ q^{-6} - q^{-7} +4 q^{-9} -3 q^{-10} -4 q^{-11} +10 q^{-12} -4 q^{-13} -11 q^{-14} +17 q^{-15} -2 q^{-16} -19 q^{-17} +21 q^{-18} +2 q^{-19} -25 q^{-20} +20 q^{-21} +6 q^{-22} -25 q^{-23} +17 q^{-24} +6 q^{-25} -18 q^{-26} +10 q^{-27} +3 q^{-28} -8 q^{-29} +4 q^{-30} + q^{-31} -2 q^{-32} + q^{-33} }[/math]
3 [math]\displaystyle{ q^{-9} - q^{-10} + q^{-12} +3 q^{-13} -3 q^{-14} -3 q^{-15} + q^{-16} +10 q^{-17} -3 q^{-18} -11 q^{-19} -5 q^{-20} +20 q^{-21} +6 q^{-22} -18 q^{-23} -18 q^{-24} +24 q^{-25} +22 q^{-26} -19 q^{-27} -33 q^{-28} +19 q^{-29} +36 q^{-30} -11 q^{-31} -45 q^{-32} +8 q^{-33} +46 q^{-34} + q^{-35} -51 q^{-36} -7 q^{-37} +51 q^{-38} +15 q^{-39} -53 q^{-40} -18 q^{-41} +48 q^{-42} +22 q^{-43} -44 q^{-44} -21 q^{-45} +37 q^{-46} +18 q^{-47} -28 q^{-48} -15 q^{-49} +23 q^{-50} +7 q^{-51} -13 q^{-52} -6 q^{-53} +11 q^{-54} + q^{-55} -6 q^{-56} - q^{-57} +5 q^{-58} - q^{-59} - q^{-60} - q^{-61} +2 q^{-62} - q^{-63} }[/math]
4 [math]\displaystyle{ q^{-12} - q^{-13} + q^{-15} +3 q^{-17} -4 q^{-18} -2 q^{-19} +3 q^{-20} +11 q^{-22} -8 q^{-23} -10 q^{-24} - q^{-25} -2 q^{-26} +30 q^{-27} -3 q^{-28} -16 q^{-29} -16 q^{-30} -21 q^{-31} +51 q^{-32} +15 q^{-33} -3 q^{-34} -28 q^{-35} -61 q^{-36} +54 q^{-37} +28 q^{-38} +31 q^{-39} -17 q^{-40} -102 q^{-41} +38 q^{-42} +17 q^{-43} +65 q^{-44} +19 q^{-45} -121 q^{-46} +15 q^{-47} -17 q^{-48} +85 q^{-49} +63 q^{-50} -119 q^{-51} - q^{-52} -61 q^{-53} +91 q^{-54} +103 q^{-55} -103 q^{-56} -16 q^{-57} -104 q^{-58} +94 q^{-59} +137 q^{-60} -82 q^{-61} -30 q^{-62} -138 q^{-63} +87 q^{-64} +158 q^{-65} -54 q^{-66} -32 q^{-67} -157 q^{-68} +65 q^{-69} +152 q^{-70} -27 q^{-71} -12 q^{-72} -144 q^{-73} +33 q^{-74} +113 q^{-75} -14 q^{-76} +14 q^{-77} -100 q^{-78} +12 q^{-79} +60 q^{-80} -17 q^{-81} +27 q^{-82} -50 q^{-83} +6 q^{-84} +24 q^{-85} -20 q^{-86} +21 q^{-87} -19 q^{-88} +7 q^{-89} +8 q^{-90} -16 q^{-91} +11 q^{-92} -6 q^{-93} +4 q^{-94} +3 q^{-95} -7 q^{-96} +4 q^{-97} -2 q^{-98} + q^{-99} + q^{-100} -2 q^{-101} + q^{-102} }[/math]
5 [math]\displaystyle{ q^{-15} - q^{-16} + q^{-18} +2 q^{-21} -3 q^{-22} -2 q^{-23} +3 q^{-24} +3 q^{-25} +5 q^{-27} -7 q^{-28} -10 q^{-29} - q^{-30} +7 q^{-31} +8 q^{-32} +18 q^{-33} -4 q^{-34} -23 q^{-35} -20 q^{-36} -7 q^{-37} +12 q^{-38} +46 q^{-39} +25 q^{-40} -15 q^{-41} -40 q^{-42} -49 q^{-43} -22 q^{-44} +55 q^{-45} +70 q^{-46} +33 q^{-47} -17 q^{-48} -79 q^{-49} -89 q^{-50} +4 q^{-51} +77 q^{-52} +90 q^{-53} +56 q^{-54} -45 q^{-55} -125 q^{-56} -80 q^{-57} +10 q^{-58} +92 q^{-59} +126 q^{-60} +47 q^{-61} -80 q^{-62} -128 q^{-63} -95 q^{-64} +15 q^{-65} +139 q^{-66} +143 q^{-67} +26 q^{-68} -104 q^{-69} -178 q^{-70} -103 q^{-71} +87 q^{-72} +201 q^{-73} +146 q^{-74} -29 q^{-75} -218 q^{-76} -215 q^{-77} +3 q^{-78} +225 q^{-79} +253 q^{-80} +47 q^{-81} -231 q^{-82} -308 q^{-83} -74 q^{-84} +240 q^{-85} +340 q^{-86} +115 q^{-87} -246 q^{-88} -388 q^{-89} -139 q^{-90} +251 q^{-91} +416 q^{-92} +181 q^{-93} -248 q^{-94} -453 q^{-95} -207 q^{-96} +233 q^{-97} +458 q^{-98} +249 q^{-99} -200 q^{-100} -458 q^{-101} -275 q^{-102} +158 q^{-103} +424 q^{-104} +288 q^{-105} -98 q^{-106} -372 q^{-107} -288 q^{-108} +48 q^{-109} +305 q^{-110} +254 q^{-111} -3 q^{-112} -218 q^{-113} -221 q^{-114} -29 q^{-115} +160 q^{-116} +157 q^{-117} +37 q^{-118} -83 q^{-119} -118 q^{-120} -40 q^{-121} +53 q^{-122} +65 q^{-123} +29 q^{-124} -15 q^{-125} -40 q^{-126} -20 q^{-127} +8 q^{-128} +11 q^{-129} +13 q^{-130} +6 q^{-131} -9 q^{-132} -5 q^{-133} -5 q^{-135} + q^{-136} +10 q^{-137} -4 q^{-138} - q^{-139} +3 q^{-140} -5 q^{-141} - q^{-142} +5 q^{-143} -2 q^{-144} - q^{-145} +2 q^{-146} - q^{-147} - q^{-148} +2 q^{-149} - q^{-150} }[/math]
6 [math]\displaystyle{ q^{-18} - q^{-19} + q^{-21} - q^{-24} +3 q^{-25} -3 q^{-26} -2 q^{-27} +4 q^{-28} +2 q^{-29} +2 q^{-30} -5 q^{-31} +6 q^{-32} -8 q^{-33} -10 q^{-34} +5 q^{-35} +7 q^{-36} +12 q^{-37} -4 q^{-38} +18 q^{-39} -15 q^{-40} -31 q^{-41} -11 q^{-42} +24 q^{-44} +8 q^{-45} +61 q^{-46} +4 q^{-47} -42 q^{-48} -48 q^{-49} -45 q^{-50} -7 q^{-51} -9 q^{-52} +124 q^{-53} +70 q^{-54} +16 q^{-55} -43 q^{-56} -89 q^{-57} -93 q^{-58} -120 q^{-59} +117 q^{-60} +113 q^{-61} +129 q^{-62} +64 q^{-63} -18 q^{-64} -129 q^{-65} -276 q^{-66} -5 q^{-67} +14 q^{-68} +156 q^{-69} +182 q^{-70} +185 q^{-71} +5 q^{-72} -308 q^{-73} -113 q^{-74} -206 q^{-75} -13 q^{-76} +126 q^{-77} +355 q^{-78} +253 q^{-79} -124 q^{-80} -26 q^{-81} -357 q^{-82} -291 q^{-83} -164 q^{-84} +313 q^{-85} +420 q^{-86} +167 q^{-87} +284 q^{-88} -282 q^{-89} -481 q^{-90} -558 q^{-91} +50 q^{-92} +373 q^{-93} +381 q^{-94} +683 q^{-95} +5 q^{-96} -478 q^{-97} -885 q^{-98} -304 q^{-99} +139 q^{-100} +440 q^{-101} +1024 q^{-102} +374 q^{-103} -325 q^{-104} -1077 q^{-105} -618 q^{-106} -162 q^{-107} +388 q^{-108} +1255 q^{-109} +707 q^{-110} -133 q^{-111} -1175 q^{-112} -853 q^{-113} -425 q^{-114} +327 q^{-115} +1413 q^{-116} +964 q^{-117} +11 q^{-118} -1256 q^{-119} -1040 q^{-120} -623 q^{-121} +315 q^{-122} +1556 q^{-123} +1178 q^{-124} +109 q^{-125} -1348 q^{-126} -1223 q^{-127} -800 q^{-128} +306 q^{-129} +1676 q^{-130} +1385 q^{-131} +244 q^{-132} -1359 q^{-133} -1377 q^{-134} -1002 q^{-135} +183 q^{-136} +1662 q^{-137} +1543 q^{-138} +467 q^{-139} -1164 q^{-140} -1371 q^{-141} -1168 q^{-142} -86 q^{-143} +1395 q^{-144} +1506 q^{-145} +681 q^{-146} -767 q^{-147} -1099 q^{-148} -1139 q^{-149} -358 q^{-150} +923 q^{-151} +1189 q^{-152} +713 q^{-153} -353 q^{-154} -653 q^{-155} -860 q^{-156} -451 q^{-157} +465 q^{-158} +722 q^{-159} +528 q^{-160} -103 q^{-161} -256 q^{-162} -483 q^{-163} -357 q^{-164} +181 q^{-165} +333 q^{-166} +280 q^{-167} -25 q^{-168} -37 q^{-169} -199 q^{-170} -207 q^{-171} +62 q^{-172} +113 q^{-173} +110 q^{-174} -18 q^{-175} +36 q^{-176} -58 q^{-177} -102 q^{-178} +25 q^{-179} +25 q^{-180} +33 q^{-181} -17 q^{-182} +38 q^{-183} -8 q^{-184} -46 q^{-185} +12 q^{-186} + q^{-187} +7 q^{-188} -12 q^{-189} +22 q^{-190} +2 q^{-191} -18 q^{-192} +7 q^{-193} -2 q^{-194} +2 q^{-195} -7 q^{-196} +8 q^{-197} +2 q^{-198} -7 q^{-199} +4 q^{-200} - q^{-201} + q^{-202} -2 q^{-203} + q^{-204} + q^{-205} -2 q^{-206} + q^{-207} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

9 8.gif

9_8

9 10.gif

9_10

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