9 23

9_22

9_24

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 23's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 23 at Knotilus!

[edit 9 23 Quick Notes]

[edit 9 23 Further Notes and Views]

Symmetrical decorative knot	With crossings on 3x3 grid	Depiction with two axes of symmetry	Mongolian ornament (two crossings are unnecessary)
Mongolian ornament, sum of two 9.23	Logo of the ICMC-USP, Brazil	Other depicture with central symmetry by Alain Esculier

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_3,10,4,11 X_5,12,6,13 X_7,16,8,17 X_13,18,14,1 X_17,14,18,15 X_15,6,16,7 X_11,8,12,9 X_9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 7, -4, 8, -9, 2, -8, 3, -5, 6, -7, 4, -6, 5
Dowker-Thistlethwaite code	4 10 12 16 2 8 18 6 14
Conway Notation	[22122]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 4,

Braid index is 4

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

[edit Notes on presentations of 9 23]

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 23"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X_3,10,4,11 X_5,12,6,13 X_7,16,8,17 X_13,18,14,1 X_17,14,18,15 X_15,6,16,7 X_11,8,12,9 X_9,2,10,3

In[5]:= GaussCode[K]

Out[5]= -1, 9, -2, 1, -3, 7, -4, 8, -9, 2, -8, 3, -5, 6, -7, 4, -6, 5

In[6]:= DTCode[K]

Out[6]= 4 10 12 16 2 8 18 6 14

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [22122]

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}(4,\{-1,-1,-1,-2,1,-2,-2,-3,2,-3,-3\}) }[/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]= { 4, 11, 4 }

In[11]:= Show[BraidPlot[br]]

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."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{11, 4}, {3, 9}, {8, 10}, {9, 11}, {10, 5}, {4, 6}, {5, 2}, {1, 3}, {2, 7}, {6, 8}, {7, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index	2
Super bridge index	[math]\displaystyle{ \{4,7\} }[/math]
Nakanishi index	1
Maximal Thurston-Bennequin number	[-14][3]
Hyperbolic Volume	10.6113
A-Polynomial	See Data:9 23/A-polynomial

[edit Notes for 9 23's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 23's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 4 t^2-11 t+15-11 t^{-1} +4 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 4 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 45, -4 }
Jones polynomial	[math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +8 q^{-6} -8 q^{-7} +6 q^{-8} -5 q^{-9} +3 q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{10}+z^4 a^8-2 a^8+2 z^4 a^6+4 z^2 a^6+2 a^6+z^4 a^4+2 z^2 a^4+a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-7 z^4 a^{12}+3 z^2 a^{12}+3 z^7 a^{11}-5 z^5 a^{11}+z a^{11}+z^8 a^{10}+4 z^6 a^{10}-10 z^4 a^{10}+3 z^2 a^{10}+5 z^7 a^9-6 z^5 a^9-2 z^3 a^9+4 z a^9+z^8 a^8+4 z^6 a^8-8 z^4 a^8+6 z^2 a^8-2 a^8+2 z^7 a^7+2 z^5 a^7-6 z^3 a^7+4 z a^7+3 z^6 a^6-4 z^4 a^6+4 z^2 a^6-2 a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{34}+q^{32}+q^{30}-2 q^{28}-2 q^{24}-q^{22}+q^{20}+3 q^{16}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{176}-2 q^{174}+5 q^{172}-8 q^{170}+7 q^{168}-3 q^{166}-6 q^{164}+20 q^{162}-27 q^{160}+30 q^{158}-24 q^{156}+q^{154}+23 q^{152}-47 q^{150}+55 q^{148}-43 q^{146}+18 q^{144}+14 q^{142}-38 q^{140}+48 q^{138}-38 q^{136}+13 q^{134}+12 q^{132}-30 q^{130}+30 q^{128}-10 q^{126}-16 q^{124}+40 q^{122}-43 q^{120}+36 q^{118}-13 q^{116}-28 q^{114}+56 q^{112}-73 q^{110}+66 q^{108}-39 q^{106}-6 q^{104}+43 q^{102}-64 q^{100}+61 q^{98}-44 q^{96}+7 q^{94}+20 q^{92}-36 q^{90}+30 q^{88}-7 q^{86}-16 q^{84}+35 q^{82}-29 q^{80}+11 q^{78}+13 q^{76}-34 q^{74}+47 q^{72}-41 q^{70}+28 q^{68}-2 q^{66}-19 q^{64}+35 q^{62}-35 q^{60}+31 q^{58}-17 q^{56}+4 q^{54}+7 q^{52}-15 q^{50}+16 q^{48}-12 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_23.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{23}+2 q^{21}-2 q^{19}+q^{17}-2 q^{15}+2 q^{11}-q^9+3 q^7-q^5+q^3 }[/math]
2	[math]\displaystyle{ q^{64}-2 q^{62}-2 q^{60}+7 q^{58}-q^{56}-9 q^{54}+9 q^{52}+3 q^{50}-13 q^{48}+6 q^{46}+7 q^{44}-10 q^{42}+2 q^{40}+7 q^{38}-3 q^{36}-6 q^{34}+q^{32}+7 q^{30}-10 q^{28}-4 q^{26}+14 q^{24}-6 q^{22}-6 q^{20}+11 q^{18}-2 q^{16}-3 q^{14}+5 q^{12}-q^8+q^6 }[/math]
3	[math]\displaystyle{ -q^{123}+2 q^{121}+2 q^{119}-3 q^{117}-7 q^{115}+q^{113}+16 q^{111}+4 q^{109}-21 q^{107}-15 q^{105}+22 q^{103}+29 q^{101}-20 q^{99}-41 q^{97}+10 q^{95}+49 q^{93}+3 q^{91}-50 q^{89}-16 q^{87}+50 q^{85}+24 q^{83}-41 q^{81}-30 q^{79}+32 q^{77}+32 q^{75}-23 q^{73}-32 q^{71}+8 q^{69}+33 q^{67}+5 q^{65}-24 q^{63}-25 q^{61}+21 q^{59}+38 q^{57}-10 q^{55}-51 q^{53}-3 q^{51}+53 q^{49}+14 q^{47}-52 q^{45}-23 q^{43}+40 q^{41}+24 q^{39}-28 q^{37}-23 q^{35}+20 q^{33}+16 q^{31}-9 q^{29}-10 q^{27}+8 q^{25}+7 q^{23}-3 q^{21}-3 q^{19}+3 q^{17}+2 q^{15}-q^{11}+q^9 }[/math]
4	[math]\displaystyle{ q^{200}-2 q^{198}-2 q^{196}+3 q^{194}+3 q^{192}+7 q^{190}-8 q^{188}-16 q^{186}-4 q^{184}+8 q^{182}+42 q^{180}+8 q^{178}-38 q^{176}-47 q^{174}-26 q^{172}+80 q^{170}+74 q^{168}-2 q^{166}-91 q^{164}-129 q^{162}+48 q^{160}+141 q^{158}+111 q^{156}-53 q^{154}-222 q^{152}-72 q^{150}+115 q^{148}+222 q^{146}+68 q^{144}-223 q^{142}-187 q^{140}+12 q^{138}+241 q^{136}+171 q^{134}-143 q^{132}-218 q^{130}-80 q^{128}+186 q^{126}+200 q^{124}-53 q^{122}-179 q^{120}-116 q^{118}+108 q^{116}+175 q^{114}+23 q^{112}-124 q^{110}-133 q^{108}+16 q^{106}+128 q^{104}+114 q^{102}-40 q^{100}-145 q^{98}-109 q^{96}+55 q^{94}+212 q^{92}+80 q^{90}-120 q^{88}-226 q^{86}-64 q^{84}+234 q^{82}+193 q^{80}-21 q^{78}-254 q^{76}-175 q^{74}+153 q^{72}+208 q^{70}+81 q^{68}-167 q^{66}-188 q^{64}+43 q^{62}+122 q^{60}+103 q^{58}-58 q^{56}-115 q^{54}-3 q^{52}+36 q^{50}+61 q^{48}-8 q^{46}-44 q^{44}+4 q^{40}+22 q^{38}-2 q^{36}-13 q^{34}+6 q^{32}+7 q^{28}-q^{26}-4 q^{24}+3 q^{22}+2 q^{18}-q^{14}+q^{12} }[/math]
5	[math]\displaystyle{ -q^{295}+2 q^{293}+2 q^{291}-3 q^{289}-3 q^{287}-3 q^{285}+8 q^{281}+16 q^{279}+4 q^{277}-17 q^{275}-29 q^{273}-26 q^{271}+8 q^{269}+54 q^{267}+74 q^{265}+19 q^{263}-70 q^{261}-124 q^{259}-94 q^{257}+38 q^{255}+183 q^{253}+209 q^{251}+46 q^{249}-195 q^{247}-327 q^{245}-217 q^{243}+118 q^{241}+429 q^{239}+433 q^{237}+53 q^{235}-435 q^{233}-637 q^{231}-334 q^{229}+320 q^{227}+792 q^{225}+648 q^{223}-82 q^{221}-822 q^{219}-937 q^{217}-253 q^{215}+725 q^{213}+1141 q^{211}+595 q^{209}-506 q^{207}-1226 q^{205}-894 q^{203}+234 q^{201}+1175 q^{199}+1097 q^{197}+55 q^{195}-1039 q^{193}-1188 q^{191}-273 q^{189}+841 q^{187}+1168 q^{185}+438 q^{183}-642 q^{181}-1082 q^{179}-522 q^{177}+464 q^{175}+946 q^{173}+545 q^{171}-302 q^{169}-814 q^{167}-563 q^{165}+174 q^{163}+686 q^{161}+565 q^{159}-13 q^{157}-553 q^{155}-629 q^{153}-153 q^{151}+432 q^{149}+669 q^{147}+394 q^{145}-249 q^{143}-752 q^{141}-648 q^{139}+33 q^{137}+765 q^{135}+919 q^{133}+263 q^{131}-717 q^{129}-1145 q^{127}-588 q^{125}+552 q^{123}+1285 q^{121}+894 q^{119}-303 q^{117}-1273 q^{115}-1131 q^{113}-4 q^{111}+1127 q^{109}+1242 q^{107}+280 q^{105}-859 q^{103}-1192 q^{101}-493 q^{99}+546 q^{97}+1019 q^{95}+593 q^{93}-269 q^{91}-766 q^{89}-564 q^{87}+43 q^{85}+502 q^{83}+475 q^{81}+66 q^{79}-295 q^{77}-329 q^{75}-112 q^{73}+136 q^{71}+212 q^{69}+96 q^{67}-57 q^{65}-112 q^{63}-66 q^{61}+16 q^{59}+55 q^{57}+34 q^{55}-2 q^{53}-19 q^{51}-16 q^{49}+3 q^{47}+10 q^{45}+3 q^{43}-2 q^{41}+2 q^{39}-2 q^{37}+2 q^{35}+5 q^{33}-q^{31}-2 q^{29}+2 q^{27}+2 q^{21}-q^{17}+q^{15} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{34}+q^{32}+q^{30}-2 q^{28}-2 q^{24}-q^{22}+q^{20}+3 q^{16}+q^{12}+2 q^{10}-q^8+q^6 }[/math]
1,1	[math]\displaystyle{ q^{92}-4 q^{90}+12 q^{88}-28 q^{86}+50 q^{84}-80 q^{82}+118 q^{80}-154 q^{78}+184 q^{76}-194 q^{74}+186 q^{72}-156 q^{70}+93 q^{68}-18 q^{66}-74 q^{64}+166 q^{62}-244 q^{60}+316 q^{58}-352 q^{56}+364 q^{54}-340 q^{52}+284 q^{50}-212 q^{48}+120 q^{46}-36 q^{44}-54 q^{42}+108 q^{40}-152 q^{38}+165 q^{36}-166 q^{34}+154 q^{32}-124 q^{30}+105 q^{28}-72 q^{26}+56 q^{24}-32 q^{22}+23 q^{20}-10 q^{18}+6 q^{16}-2 q^{14}+q^{12} }[/math]
2,0	[math]\displaystyle{ q^{86}-q^{84}-2 q^{82}+4 q^{78}+3 q^{76}-5 q^{74}-2 q^{72}+4 q^{70}+2 q^{68}-5 q^{66}-3 q^{64}+7 q^{62}+2 q^{60}-4 q^{58}+5 q^{54}-2 q^{52}-4 q^{50}-q^{48}-4 q^{46}-6 q^{44}-q^{42}+3 q^{40}-6 q^{38}-q^{36}+10 q^{34}+4 q^{32}-5 q^{30}+2 q^{28}+8 q^{26}+2 q^{24}-4 q^{22}+2 q^{20}+4 q^{18}-q^{16}-q^{14}+q^{12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{74}-2 q^{72}+q^{70}+3 q^{68}-6 q^{66}+4 q^{64}+4 q^{62}-10 q^{60}+5 q^{58}+5 q^{56}-9 q^{54}+3 q^{52}+8 q^{50}-5 q^{48}-2 q^{46}-2 q^{42}-7 q^{40}-6 q^{38}+8 q^{36}-4 q^{34}-4 q^{32}+13 q^{30}-5 q^{26}+10 q^{24}+q^{22}-3 q^{20}+4 q^{18}+q^{16}-q^{14}+q^{12} }[/math]
1,0,0	[math]\displaystyle{ -q^{45}+q^{43}+q^{39}-2 q^{37}-3 q^{33}-q^{31}-q^{29}+q^{27}+q^{25}+q^{23}+3 q^{21}+2 q^{17}+2 q^{13}-q^{11}+q^9 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{96}-q^{94}-2 q^{92}+3 q^{90}+3 q^{88}-4 q^{86}-2 q^{84}+6 q^{82}+q^{80}-8 q^{78}-q^{76}+9 q^{74}-6 q^{70}+7 q^{68}+6 q^{66}-6 q^{64}-3 q^{62}+2 q^{60}-8 q^{58}-11 q^{56}-2 q^{54}-2 q^{52}-10 q^{50}-2 q^{48}+11 q^{46}+2 q^{44}-3 q^{42}+8 q^{40}+10 q^{38}-q^{34}+5 q^{32}+5 q^{30}-q^{28}+3 q^{24}+q^{22}-q^{20}+q^{18} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{56}+q^{54}+q^{48}-2 q^{46}-3 q^{42}-2 q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+2 q^{30}+q^{28}+3 q^{26}+2 q^{22}+q^{20}+2 q^{16}-q^{14}+q^{12} }[/math]

B2 Invariants.

Weight

Invariant

0,1

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

1,0

[math]\displaystyle{ q^{120}-2 q^{116}-2 q^{114}+3 q^{112}+5 q^{110}-2 q^{108}-7 q^{106}-2 q^{104}+9 q^{102}+7 q^{100}-7 q^{98}-11 q^{96}+q^{94}+11 q^{92}+4 q^{90}-8 q^{88}-7 q^{86}+5 q^{84}+8 q^{82}-7 q^{78}+6 q^{74}+q^{72}-9 q^{70}-5 q^{68}+5 q^{66}+3 q^{64}-7 q^{62}-8 q^{60}+4 q^{58}+8 q^{56}-q^{54}-10 q^{52}-q^{50}+11 q^{48}+8 q^{46}-5 q^{44}-8 q^{42}+q^{40}+10 q^{38}+5 q^{36}-3 q^{34}-5 q^{32}+q^{30}+4 q^{28}+2 q^{26}-q^{24}-q^{22}+q^{18} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{102}-2 q^{100}+3 q^{98}-4 q^{96}+6 q^{94}-7 q^{92}+7 q^{90}-8 q^{88}+9 q^{86}-8 q^{84}+5 q^{82}-4 q^{80}+3 q^{78}-4 q^{74}+7 q^{72}-6 q^{70}+12 q^{68}-14 q^{66}+13 q^{64}-16 q^{62}+13 q^{60}-18 q^{58}+6 q^{56}-13 q^{54}+6 q^{52}-4 q^{50}+q^{48}+2 q^{46}-q^{44}+11 q^{42}-4 q^{40}+8 q^{38}-6 q^{36}+10 q^{34}-4 q^{32}+6 q^{30}-4 q^{28}+5 q^{26}-q^{24}+2 q^{22}-q^{20}+q^{18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{176}-2 q^{174}+5 q^{172}-8 q^{170}+7 q^{168}-3 q^{166}-6 q^{164}+20 q^{162}-27 q^{160}+30 q^{158}-24 q^{156}+q^{154}+23 q^{152}-47 q^{150}+55 q^{148}-43 q^{146}+18 q^{144}+14 q^{142}-38 q^{140}+48 q^{138}-38 q^{136}+13 q^{134}+12 q^{132}-30 q^{130}+30 q^{128}-10 q^{126}-16 q^{124}+40 q^{122}-43 q^{120}+36 q^{118}-13 q^{116}-28 q^{114}+56 q^{112}-73 q^{110}+66 q^{108}-39 q^{106}-6 q^{104}+43 q^{102}-64 q^{100}+61 q^{98}-44 q^{96}+7 q^{94}+20 q^{92}-36 q^{90}+30 q^{88}-7 q^{86}-16 q^{84}+35 q^{82}-29 q^{80}+11 q^{78}+13 q^{76}-34 q^{74}+47 q^{72}-41 q^{70}+28 q^{68}-2 q^{66}-19 q^{64}+35 q^{62}-35 q^{60}+31 q^{58}-17 q^{56}+4 q^{54}+7 q^{52}-15 q^{50}+16 q^{48}-12 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30} }[/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.

In[3]:= K = Knot["9 23"];

In[4]:= Alexander[K][t]

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

Out[4]= [math]\displaystyle{ 4 t^2-11 t+15-11 t^{-1} +4 t^{-2} }[/math]

In[5]:= Conway[K][z]

Out[5]= [math]\displaystyle{ 4 z^4+5 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]= { 45, -4 }

In[8]:= Jones[K][q]

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

Out[8]= [math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +8 q^{-6} -8 q^{-7} +6 q^{-8} -5 q^{-9} +3 q^{-10} - q^{-11} }[/math]

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= [math]\displaystyle{ -z^2 a^{10}+z^4 a^8-2 a^8+2 z^4 a^6+4 z^2 a^6+2 a^6+z^4 a^4+2 z^2 a^4+a^4 }[/math]

In[10]:= Kauffman[K][a, z]

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

Out[10]= [math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-7 z^4 a^{12}+3 z^2 a^{12}+3 z^7 a^{11}-5 z^5 a^{11}+z a^{11}+z^8 a^{10}+4 z^6 a^{10}-10 z^4 a^{10}+3 z^2 a^{10}+5 z^7 a^9-6 z^5 a^9-2 z^3 a^9+4 z a^9+z^8 a^8+4 z^6 a^8-8 z^4 a^8+6 z^2 a^8-2 a^8+2 z^7 a^7+2 z^5 a^7-6 z^3 a^7+4 z a^7+3 z^6 a^6-4 z^4 a^6+4 z^2 a^6-2 a^6+2 z^5 a^5-2 z^3 a^5+z^4 a^4-2 z^2 a^4+a^4 }[/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.

In[3]:= K = Knot["9 23"];

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{ 4 t^2-11 t+15-11 t^{-1} +4 t^{-2} }[/math], [math]\displaystyle{ q^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +8 q^{-6} -8 q^{-7} +6 q^{-8} -5 q^{-9} +3 q^{-10} - q^{-11} }[/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:

(5, -11)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

[math]\displaystyle{ 20 }[/math]

[math]\displaystyle{ -88 }[/math]

[math]\displaystyle{ 200 }[/math]

[math]\displaystyle{ \frac{1462}{3} }[/math]

[math]\displaystyle{ \frac{194}{3} }[/math]

[math]\displaystyle{ -1760 }[/math]

[math]\displaystyle{ -\frac{9040}{3} }[/math]

[math]\displaystyle{ -\frac{1504}{3} }[/math]

[math]\displaystyle{ -344 }[/math]

[math]\displaystyle{ \frac{4000}{3} }[/math]

[math]\displaystyle{ 3872 }[/math]

[math]\displaystyle{ \frac{29240}{3} }[/math]

[math]\displaystyle{ \frac{3880}{3} }[/math]

[math]\displaystyle{ \frac{115087}{6} }[/math]

[math]\displaystyle{ \frac{2986}{3} }[/math]

[math]\displaystyle{ \frac{58214}{9} }[/math]

[math]\displaystyle{ \frac{2389}{18} }[/math]

[math]\displaystyle{ \frac{4687}{6} }[/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]-4 is the signature of 9 23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

1

-5

2

1

-1

-7

3

-9

3

2

-1

-11

5

3

2

-13

3

0

-15

3

5

-2

-17

2

3

1

-19

1

3

-2

-21

2

-23

1

-1

Integral Khovanov Homology

(db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]	[math]\displaystyle{ i=-5 }[/math]	[math]\displaystyle{ i=-3 }[/math]
[math]\displaystyle{ r=-9 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-8 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-7 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-6 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-5 }[/math]	[math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-4 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math]	[math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-3 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math]	[math]\displaystyle{ {\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/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^{-4} -2 q^{-5} + q^{-6} +6 q^{-7} -10 q^{-8} +2 q^{-9} +19 q^{-10} -27 q^{-11} +2 q^{-12} +39 q^{-13} -45 q^{-14} -4 q^{-15} +56 q^{-16} -51 q^{-17} -11 q^{-18} +59 q^{-19} -41 q^{-20} -16 q^{-21} +47 q^{-22} -24 q^{-23} -17 q^{-24} +28 q^{-25} -8 q^{-26} -11 q^{-27} +10 q^{-28} -3 q^{-30} + q^{-31} }[/math]
3	[math]\displaystyle{ q^{-6} -2 q^{-7} + q^{-8} +2 q^{-9} +2 q^{-10} -8 q^{-11} + q^{-12} +12 q^{-13} +3 q^{-14} -26 q^{-15} +2 q^{-16} +37 q^{-17} +7 q^{-18} -69 q^{-19} -3 q^{-20} +89 q^{-21} +23 q^{-22} -132 q^{-23} -32 q^{-24} +155 q^{-25} +62 q^{-26} -188 q^{-27} -80 q^{-28} +196 q^{-29} +110 q^{-30} -205 q^{-31} -126 q^{-32} +197 q^{-33} +139 q^{-34} -177 q^{-35} -151 q^{-36} +157 q^{-37} +148 q^{-38} -122 q^{-39} -151 q^{-40} +95 q^{-41} +137 q^{-42} -57 q^{-43} -125 q^{-44} +29 q^{-45} +103 q^{-46} -4 q^{-47} -79 q^{-48} -10 q^{-49} +52 q^{-50} +17 q^{-51} -30 q^{-52} -17 q^{-53} +15 q^{-54} +11 q^{-55} -5 q^{-56} -5 q^{-57} +3 q^{-59} - q^{-60} }[/math]
4	[math]\displaystyle{ q^{-8} -2 q^{-9} + q^{-10} +2 q^{-11} -2 q^{-12} +4 q^{-13} -9 q^{-14} +4 q^{-15} +10 q^{-16} -9 q^{-17} +10 q^{-18} -28 q^{-19} +15 q^{-20} +34 q^{-21} -27 q^{-22} +6 q^{-23} -72 q^{-24} +51 q^{-25} +103 q^{-26} -52 q^{-27} -33 q^{-28} -184 q^{-29} +108 q^{-30} +264 q^{-31} -33 q^{-32} -112 q^{-33} -415 q^{-34} +129 q^{-35} +512 q^{-36} +94 q^{-37} -167 q^{-38} -743 q^{-39} +50 q^{-40} +745 q^{-41} +308 q^{-42} -126 q^{-43} -1041 q^{-44} -112 q^{-45} +851 q^{-46} +508 q^{-47} +6 q^{-48} -1198 q^{-49} -276 q^{-50} +815 q^{-51} +613 q^{-52} +160 q^{-53} -1184 q^{-54} -388 q^{-55} +666 q^{-56} +622 q^{-57} +307 q^{-58} -1032 q^{-59} -455 q^{-60} +442 q^{-61} +559 q^{-62} +433 q^{-63} -779 q^{-64} -469 q^{-65} +176 q^{-66} +421 q^{-67} +508 q^{-68} -465 q^{-69} -399 q^{-70} -53 q^{-71} +222 q^{-72} +472 q^{-73} -174 q^{-74} -245 q^{-75} -160 q^{-76} +35 q^{-77} +322 q^{-78} -5 q^{-79} -81 q^{-80} -130 q^{-81} -58 q^{-82} +145 q^{-83} +33 q^{-84} +8 q^{-85} -54 q^{-86} -52 q^{-87} +39 q^{-88} +12 q^{-89} +17 q^{-90} -8 q^{-91} -18 q^{-92} +5 q^{-93} +5 q^{-95} -3 q^{-97} + q^{-98} }[/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).

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9 23

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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