9 9

9_8

9_10

(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!

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
Length is 10, width is 3, Braid index is 3		[{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]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(3,\{-1,-1,-1,-1,-1,-2,1,-2,-2,-2\})}

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]]

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[{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]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 2 Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,6\}} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} Concordance genus $3$ Rasmussen s-Invariant -6

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

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 9_9.

A1 Invariants.

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

A2 Invariants.

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

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{82}+q^{80}-2q^{78}+3q^{76}-4q^{74}+4q^{72}-5q^{70}+4q^{68}-3q^{66}+2q^{64}+q^{62}-3q^{60}+5q^{58}-7q^{56}+7q^{54}-9q^{52}+8q^{50}-8q^{48}+5q^{46}-3q^{44}+q^{42}+q^{40}-2q^{38}+5q^{36}-3q^{34}+5q^{32}-3q^{30}+4q^{28}-2q^{26}+2q^{24}+q^{20}$
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}

.

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]=

$2t^{3}-4t^{2}+6t-7+6t^{-1}-4t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+8z^{4}+8z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 31, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-3}-q^{-4}+3q^{-5}-4q^{-6}+5q^{-7}-5q^{-8}+5q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{10}-3z^{2}a^{10}-2a^{10}+z^{6}a^{8}+4z^{4}a^{8}+4z^{2}a^{8}+a^{8}+z^{6}a^{6}+5z^{4}a^{6}+7z^{2}a^{6}+2a^{6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-z^{2}a^{14}+3z^{5}a^{13}-3z^{3}a^{13}+2za^{13}+3z^{6}a^{12}-4z^{4}a^{12}+3z^{2}a^{12}+2z^{7}a^{11}-2z^{5}a^{11}+z^{8}a^{10}-z^{6}a^{10}+2z^{4}a^{10}-6z^{2}a^{10}+2a^{10}+3z^{7}a^{9}-8z^{5}a^{9}+5z^{3}a^{9}-2za^{9}+z^{8}a^{8}-3z^{6}a^{8}+3z^{4}a^{8}-3z^{2}a^{8}+a^{8}+z^{7}a^{7}-3z^{5}a^{7}+z^{3}a^{7}+za^{7}+z^{6}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-2a^{6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}

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]=

{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^3-4 t^2+6 t-7+6 t^{-1} -4 t^{-2} +2 t^{-3} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} } }

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 $32$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -176} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 512} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3760}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{560}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -5632} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{29696}{3}} $-{\frac {5216}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1264} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{16384}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 15488} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{120320}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{17920}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1196284}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{40544}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1354096}{45}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4628}{9}} ${\frac {57964}{15}}$

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -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 -1 0 χ -5                   1 1 -7                 1 1 0 -9               2     2 -11             2 1     -1 -13           3 2       1 -15         2 2         0 -17       3 3           0 -19     1 2             1 -21   1 3               -2 -23   1                 1 -25 1                   -1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-7} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-9} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-7} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} $r=-4$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }

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 9_10