10 7

10_6

10_8

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 7 at Knotilus!

[edit 10 7 Quick Notes]

[edit 10 7 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_5,14,6,15 X_3,13,4,12 X_13,3,14,2 X_11,20,12,1 X_19,6,20,7 X_7,18,8,19 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9
Gauss code	-1, 4, -3, 1, -2, 6, -7, 10, -8, 9, -5, 3, -4, 2, -9, 8, -10, 7, -6, 5
Dowker-Thistlethwaite code	4 12 14 18 16 20 2 10 8 6
Conway Notation	[5212]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 12, width is 5,

Braid index is 5

[{12, 7}, {1, 10}, {11, 8}, {7, 9}, {10, 12}, {6, 11}, {8, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}]

[edit Notes on presentations of 10 7]

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["10 7"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X_5,14,6,15 X_3,13,4,12 X_13,3,14,2 X_11,20,12,1 X_19,6,20,7 X_7,18,8,19 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9

In[5]:= GaussCode[K]

Out[5]= -1, 4, -3, 1, -2, 6, -7, 10, -8, 9, -5, 3, -4, 2, -9, 8, -10, 7, -6, 5

In[6]:= DTCode[K]

Out[6]= 4 12 14 18 16 20 2 10 8 6

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [5212]

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}(5,\{-1,-1,-2,1,-2,-3,2,-3,-3,4,-3,4\})}

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]= { 5, 12, 5 }

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[{12, 7}, {1, 10}, {11, 8}, {7, 9}, {10, 12}, {6, 11}, {8, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	2
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-11][-1]
Hyperbolic Volume	9.11591
A-Polynomial	See Data:10 7/A-polynomial

[edit Notes for 10 7'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 1}
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 1}
Concordance 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 2}
Rasmussen s-Invariant	-2

[edit Notes for 10 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial	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 t^2+11 t-15+11 t^{-1} -3 t^{-2} }
Conway polynomial	$-3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	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 \{1\}}
Determinant and Signature	{ 43, -2 }
Jones polynomial	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-2+4 q^{-1} -5 q^{-2} +7 q^{-3} -7 q^{-4} +6 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }
HOMFLY-PT polynomial (db, data sources)	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 z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-2 a^6-z^4 a^4+a^4-z^4 a^2-z^2 a^2+z^2+1}
Kauffman polynomial (db, data sources)	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 z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+8 z^3 a^9-3 z a^9+2 z^8 a^8-7 z^6 a^8+6 z^4 a^8-3 z^2 a^8+a^8+z^9 a^7-z^7 a^7-6 z^5 a^7+10 z^3 a^7-5 z a^7+4 z^8 a^6-15 z^6 a^6+20 z^4 a^6-10 z^2 a^6+2 a^6+z^9 a^5-z^7 a^5-2 z^5 a^5+6 z^3 a^5-2 z a^5+2 z^8 a^4-5 z^6 a^4+8 z^4 a^4-4 z^2 a^4+a^4+2 z^7 a^3-2 z^5 a^3+z^3 a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1}
The A2 invariant	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^{28}+q^{22}-2 q^{20}-q^{18}-q^{14}+q^{12}+q^8+q^6-q^4+2 q^2+ q^{-4} }
The G2 invariant	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^{142}-q^{140}+2 q^{138}-3 q^{136}+2 q^{134}-3 q^{132}-q^{130}+6 q^{128}-10 q^{126}+12 q^{124}-12 q^{122}+8 q^{120}+2 q^{118}-12 q^{116}+22 q^{114}-24 q^{112}+21 q^{110}-8 q^{108}-7 q^{106}+20 q^{104}-23 q^{102}+22 q^{100}-9 q^{98}-4 q^{96}+14 q^{94}-16 q^{92}+8 q^{90}+2 q^{88}-14 q^{86}+19 q^{84}-16 q^{82}+q^{80}+10 q^{78}-23 q^{76}+27 q^{74}-25 q^{72}+11 q^{70}+2 q^{68}-19 q^{66}+29 q^{64}-30 q^{62}+21 q^{60}-6 q^{58}-8 q^{56}+18 q^{54}-20 q^{52}+15 q^{50}-3 q^{48}-6 q^{46}+11 q^{44}-8 q^{42}+q^{40}+10 q^{38}-14 q^{36}+14 q^{34}-7 q^{32}-2 q^{30}+9 q^{28}-14 q^{26}+16 q^{24}-13 q^{22}+9 q^{20}-2 q^{18}-5 q^{16}+10 q^{14}-12 q^{12}+14 q^{10}-10 q^8+6 q^6-6 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }

Further Quantum Invariants

Further quantum knot invariants for 10_7.

A1 Invariants.

Weight	Invariant
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 q^{19}-q^{17}+q^{15}-2 q^{13}+q^{11}-q^9+2 q^5-q^3+2 q- q^{-1} + q^{-3} }
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^{54}-q^{52}-q^{50}+3 q^{48}-2 q^{46}-4 q^{44}+5 q^{42}+q^{40}-6 q^{38}+4 q^{36}+4 q^{34}-6 q^{32}+q^{30}+5 q^{28}-4 q^{26}-q^{24}+3 q^{22}-4 q^{18}-q^{16}+6 q^{14}-4 q^{12}-3 q^{10}+7 q^8-2 q^6-3 q^4+5 q^2-1- q^{-2} +3 q^{-4} - q^{-6} - q^{-8} + q^{-10} }
3	$q^{105}-q^{103}-q^{101}+q^{99}+2q^{97}-2q^{95}-5q^{93}+2q^{91}+8q^{89}-10q^{85}-3q^{83}+12q^{81}+9q^{79}-11q^{77}-13q^{75}+4q^{73}+16q^{71}+q^{69}-15q^{67}-8q^{65}+12q^{63}+14q^{61}-7q^{59}-17q^{57}+3q^{55}+17q^{53}-16q^{49}-3q^{47}+13q^{45}+4q^{43}-9q^{41}-7q^{39}+9q^{37}+8q^{35}-13q^{31}-4q^{29}+13q^{27}+9q^{25}-12q^{23}-13q^{21}+7q^{19}+13q^{17}-2q^{15}-11q^{13}-q^{11}+6q^{9}+4q^{7}-q^{5}-q^{3}-2q+q^{-1}+3q^{-3}+2q^{-5}-2q^{-7}-3q^{-9}+q^{-11}+3q^{-13}-q^{-17}-q^{-19}+q^{-21}$
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^{172}-q^{170}-q^{168}+q^{166}+2 q^{162}-4 q^{160}-3 q^{158}+4 q^{156}+3 q^{154}+8 q^{152}-8 q^{150}-13 q^{148}+2 q^{146}+8 q^{144}+23 q^{142}-4 q^{140}-26 q^{138}-16 q^{136}-2 q^{134}+39 q^{132}+20 q^{130}-15 q^{128}-29 q^{126}-32 q^{124}+24 q^{122}+34 q^{120}+21 q^{118}-5 q^{116}-47 q^{114}-18 q^{112}+3 q^{110}+36 q^{108}+44 q^{106}-16 q^{104}-42 q^{102}-48 q^{100}+13 q^{98}+72 q^{96}+31 q^{94}-28 q^{92}-73 q^{90}-22 q^{88}+62 q^{86}+55 q^{84}-4 q^{82}-66 q^{80}-36 q^{78}+38 q^{76}+47 q^{74}+7 q^{72}-43 q^{70}-32 q^{68}+17 q^{66}+36 q^{64}+12 q^{62}-22 q^{60}-30 q^{58}-10 q^{56}+29 q^{54}+30 q^{52}+12 q^{50}-34 q^{48}-53 q^{46}+8 q^{44}+46 q^{42}+58 q^{40}-11 q^{38}-82 q^{36}-38 q^{34}+27 q^{32}+85 q^{30}+34 q^{28}-63 q^{26}-62 q^{24}-20 q^{22}+62 q^{20}+60 q^{18}-14 q^{16}-42 q^{14}-46 q^{12}+18 q^{10}+45 q^8+15 q^6-6 q^4-38 q^2-8+18 q^{-2} +15 q^{-4} +11 q^{-6} -18 q^{-8} -10 q^{-10} + q^{-12} +5 q^{-14} +12 q^{-16} -4 q^{-18} -4 q^{-20} -3 q^{-22} - q^{-24} +5 q^{-26} - q^{-32} - q^{-34} + q^{-36} }
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^{255}-q^{253}-q^{251}+q^{249}-2 q^{241}-2 q^{239}+4 q^{237}+6 q^{235}+q^{233}-4 q^{231}-9 q^{229}-8 q^{227}+4 q^{225}+19 q^{223}+17 q^{221}-3 q^{219}-23 q^{217}-33 q^{215}-14 q^{213}+25 q^{211}+52 q^{209}+35 q^{207}-14 q^{205}-59 q^{203}-65 q^{201}-15 q^{199}+54 q^{197}+88 q^{195}+53 q^{193}-24 q^{191}-85 q^{189}-85 q^{187}-22 q^{185}+52 q^{183}+90 q^{181}+65 q^{179}-56 q^{175}-80 q^{173}-64 q^{171}-14 q^{169}+60 q^{167}+104 q^{165}+94 q^{163}+16 q^{161}-107 q^{159}-172 q^{157}-106 q^{155}+66 q^{153}+210 q^{151}+200 q^{149}+16 q^{147}-212 q^{145}-274 q^{143}-104 q^{141}+171 q^{139}+310 q^{137}+184 q^{135}-104 q^{133}-311 q^{131}-244 q^{129}+40 q^{127}+281 q^{125}+263 q^{123}+17 q^{121}-230 q^{119}-259 q^{117}-54 q^{115}+179 q^{113}+228 q^{111}+69 q^{109}-131 q^{107}-184 q^{105}-70 q^{103}+92 q^{101}+152 q^{99}+64 q^{97}-71 q^{95}-116 q^{93}-63 q^{91}+38 q^{89}+108 q^{87}+84 q^{85}-18 q^{83}-98 q^{81}-113 q^{79}-46 q^{77}+93 q^{75}+169 q^{73}+110 q^{71}-60 q^{69}-216 q^{67}-205 q^{65}+242 q^{61}+300 q^{59}+89 q^{57}-230 q^{55}-371 q^{53}-187 q^{51}+171 q^{49}+397 q^{47}+276 q^{45}-85 q^{43}-369 q^{41}-326 q^{39}-9 q^{37}+294 q^{35}+329 q^{33}+89 q^{31}-199 q^{29}-294 q^{27}-132 q^{25}+115 q^{23}+225 q^{21}+143 q^{19}-40 q^{17}-163 q^{15}-131 q^{13}+4 q^{11}+105 q^9+103 q^7+23 q^5-63 q^3-80 q-30 q^{-1} +37 q^{-3} +58 q^{-5} +30 q^{-7} -16 q^{-9} -40 q^{-11} -27 q^{-13} +3 q^{-15} +27 q^{-17} +24 q^{-19} + q^{-21} -13 q^{-23} -16 q^{-25} -7 q^{-27} +5 q^{-29} +12 q^{-31} +5 q^{-33} -2 q^{-35} -3 q^{-37} -5 q^{-39} - q^{-41} +3 q^{-43} +2 q^{-45} - q^{-51} - q^{-53} + q^{-55} }

A2 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^{28}+q^{22}-2 q^{20}-q^{18}-q^{14}+q^{12}+q^8+q^6-q^4+2 q^2+ q^{-4} }
1,1	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 q^{76}-2 q^{74}+4 q^{72}-8 q^{70}+17 q^{68}-26 q^{66}+36 q^{64}-54 q^{62}+67 q^{60}-78 q^{58}+82 q^{56}-80 q^{54}+68 q^{52}-38 q^{50}+10 q^{48}+26 q^{46}-65 q^{44}+96 q^{42}-122 q^{40}+132 q^{38}-135 q^{36}+134 q^{34}-112 q^{32}+92 q^{30}-66 q^{28}+36 q^{26}-14 q^{24}-10 q^{22}+22 q^{20}-36 q^{18}+40 q^{16}-38 q^{14}+39 q^{12}-42 q^{10}+42 q^8-36 q^6+38 q^4-32 q^2+30-20 q^{-2} +15 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }
2,0	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^{72}+q^{64}-q^{62}-4 q^{60}-q^{58}+2 q^{56}+q^{54}-3 q^{52}+5 q^{48}+4 q^{46}-3 q^{44}-2 q^{42}+3 q^{40}+q^{38}-3 q^{36}-2 q^{34}+3 q^{32}+q^{30}-2 q^{28}+q^{26}-q^{24}-3 q^{22}+2 q^{20}+q^{18}-4 q^{16}-2 q^{14}+5 q^{12}+2 q^{10}-5 q^8-q^6+7 q^4+2 q^2-2+ q^{-2} +2 q^{-4} - q^{-8} + q^{-12} }

A3 Invariants.

Weight

Invariant

0,1,0

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^{60}-q^{58}+q^{54}-3 q^{52}+q^{48}-3 q^{46}+4 q^{44}+4 q^{42}-3 q^{40}+5 q^{38}+3 q^{36}-6 q^{34}-q^{32}+q^{30}-5 q^{28}-2 q^{26}+q^{24}+q^{22}-2 q^{20}-q^{18}+6 q^{16}-3 q^{14}-2 q^{12}+7 q^{10}-q^8-4 q^6+5 q^4+q^2-2+3 q^{-2} + q^{-4} - q^{-6} + q^{-8} }

1,0,0

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^{37}+q^{33}+q^{29}-2 q^{27}-q^{25}-2 q^{23}-q^{19}+q^{17}+q^{15}+q^{11}+q^7-q^5+2 q^3+ q^{-1} + q^{-5} }

B2 Invariants.

Weight

Invariant

0,1

q^{60}-q^{58}+2q^{56}-3q^{54}+5q^{52}-6q^{50}+7q^{48}-7q^{46}+6q^{44}-6q^{42}+3q^{40}-q^{38}-3q^{36}+6q^{34}-9q^{32}+11q^{30}-13q^{28}+14q^{26}-13q^{24}+11q^{22}-8q^{20}+5q^{18}-2q^{16}-q^{14}+4q^{12}-5q^{10}+7q^{8}-6q^{6}+7q^{4}-5q^{2}+4-3q^{-2}+3q^{-4}-q^{-6}+q^{-8}

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^{98}-q^{94}-q^{92}+q^{90}+2 q^{88}-q^{86}-4 q^{84}-q^{82}+4 q^{80}+3 q^{78}-4 q^{76}-6 q^{74}+2 q^{72}+8 q^{70}+4 q^{68}-6 q^{66}-5 q^{64}+4 q^{62}+8 q^{60}+q^{58}-6 q^{56}-3 q^{54}+3 q^{52}+2 q^{50}-3 q^{48}-4 q^{46}+q^{44}+3 q^{42}-2 q^{40}-5 q^{38}+5 q^{34}+q^{32}-5 q^{30}-3 q^{28}+5 q^{26}+5 q^{24}-3 q^{22}-6 q^{20}+q^{18}+7 q^{16}+4 q^{14}-4 q^{12}-5 q^{10}+6 q^6+3 q^4-2 q^2-3+3 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }

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^{142}-q^{140}+2 q^{138}-3 q^{136}+2 q^{134}-3 q^{132}-q^{130}+6 q^{128}-10 q^{126}+12 q^{124}-12 q^{122}+8 q^{120}+2 q^{118}-12 q^{116}+22 q^{114}-24 q^{112}+21 q^{110}-8 q^{108}-7 q^{106}+20 q^{104}-23 q^{102}+22 q^{100}-9 q^{98}-4 q^{96}+14 q^{94}-16 q^{92}+8 q^{90}+2 q^{88}-14 q^{86}+19 q^{84}-16 q^{82}+q^{80}+10 q^{78}-23 q^{76}+27 q^{74}-25 q^{72}+11 q^{70}+2 q^{68}-19 q^{66}+29 q^{64}-30 q^{62}+21 q^{60}-6 q^{58}-8 q^{56}+18 q^{54}-20 q^{52}+15 q^{50}-3 q^{48}-6 q^{46}+11 q^{44}-8 q^{42}+q^{40}+10 q^{38}-14 q^{36}+14 q^{34}-7 q^{32}-2 q^{30}+9 q^{28}-14 q^{26}+16 q^{24}-13 q^{22}+9 q^{20}-2 q^{18}-5 q^{16}+10 q^{14}-12 q^{12}+14 q^{10}-10 q^8+6 q^6-6 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }

.

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["10 7"];

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

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

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 -3 t^2+11 t-15+11 t^{-1} -3 t^{-2} }

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

Out[5]= $-3z^{4}-z^{2}+1$

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]= 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 \{1\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 43, -2 }

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

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

Out[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 q-2+4 q^{-1} -5 q^{-2} +7 q^{-3} -7 q^{-4} +6 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }

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

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

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 z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-2 a^6-z^4 a^4+a^4-z^4 a^2-z^2 a^2+z^2+1}

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

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

Out[10]= 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 z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+8 z^3 a^9-3 z a^9+2 z^8 a^8-7 z^6 a^8+6 z^4 a^8-3 z^2 a^8+a^8+z^9 a^7-z^7 a^7-6 z^5 a^7+10 z^3 a^7-5 z a^7+4 z^8 a^6-15 z^6 a^6+20 z^4 a^6-10 z^2 a^6+2 a^6+z^9 a^5-z^7 a^5-2 z^5 a^5+6 z^3 a^5-2 z a^5+2 z^8 a^4-5 z^6 a^4+8 z^4 a^4-4 z^2 a^4+a^4+2 z^7 a^3-2 z^5 a^3+z^3 a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a59, K11n3,}

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.

In[3]:= K = Knot["10 7"];

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 -3 t^2+11 t-15+11 t^{-1} -3 t^{-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-2+4 q^{-1} -5 q^{-2} +7 q^{-3} -7 q^{-4} +6 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} } }

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]= {K11a59, K11n3,}

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:

(-1, 3)

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

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}

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 24}

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 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 -\frac{110}{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{86}{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 -96}

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 -64}

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 -168}

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{32}{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 288}

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{440}{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{344}{3}}

{\frac {36929}{30}}

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{5498}{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{47578}{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{3967}{18}}

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{2849}{30}}

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

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=} -2 is the signature of 10 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

-1

3

1

2

-3

3

2

-1

-5

4

2

-7

3

0

-9

3

4

-1

-11

2

3

1

-13

1

3

-2

-15

1

2

1

-17

1

-1

-19

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=-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 i=-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 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}}
$r=-7$	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}\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}	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=-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}}
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=-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}^{3}\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}^{4}\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=-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}^{3}\oplus{\mathbb Z}_2^{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}^{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 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}\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=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}^{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}^{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=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}\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=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 {\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^4-2 q^3+5 q-6+11 q^{-2} -14 q^{-3} + q^{-4} +20 q^{-5} -24 q^{-6} +30 q^{-8} -31 q^{-9} -3 q^{-10} +34 q^{-11} -28 q^{-12} -7 q^{-13} +31 q^{-14} -19 q^{-15} -11 q^{-16} +24 q^{-17} -9 q^{-18} -11 q^{-19} +14 q^{-20} -2 q^{-21} -7 q^{-22} +5 q^{-23} -2 q^{-25} + q^{-26} }
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^9-2 q^8+q^6+4 q^5-4 q^4-4 q^3+2 q^2+8 q-3-6 q^{-1} - q^{-2} +9 q^{-3} -3 q^{-4} - q^{-5} + q^{-6} +2 q^{-7} -13 q^{-8} +8 q^{-9} +16 q^{-10} -4 q^{-11} -33 q^{-12} +9 q^{-13} +37 q^{-14} -50 q^{-16} +50 q^{-18} +8 q^{-19} -49 q^{-20} -16 q^{-21} +48 q^{-22} +21 q^{-23} -40 q^{-24} -32 q^{-25} +35 q^{-26} +37 q^{-27} -23 q^{-28} -46 q^{-29} +15 q^{-30} +47 q^{-31} -2 q^{-32} -48 q^{-33} -5 q^{-34} +40 q^{-35} +14 q^{-36} -33 q^{-37} -17 q^{-38} +23 q^{-39} +16 q^{-40} -13 q^{-41} -14 q^{-42} +8 q^{-43} +9 q^{-44} -3 q^{-45} -6 q^{-46} +2 q^{-47} +2 q^{-48} -2 q^{-50} + q^{-51} }
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^{16}-2 q^{15}+q^{13}+6 q^{11}-8 q^{10}-2 q^9+22 q^6-15 q^5-6 q^4-11 q^3-8 q^2+51 q-11-3 q^{-1} -37 q^{-2} -38 q^{-3} +83 q^{-4} +10 q^{-5} +27 q^{-6} -64 q^{-7} -102 q^{-8} +87 q^{-9} +38 q^{-10} +101 q^{-11} -62 q^{-12} -184 q^{-13} +45 q^{-14} +37 q^{-15} +198 q^{-16} -11 q^{-17} -242 q^{-18} -20 q^{-19} -7 q^{-20} +269 q^{-21} +58 q^{-22} -254 q^{-23} -58 q^{-24} -68 q^{-25} +288 q^{-26} +104 q^{-27} -236 q^{-28} -59 q^{-29} -107 q^{-30} +268 q^{-31} +112 q^{-32} -202 q^{-33} -35 q^{-34} -126 q^{-35} +219 q^{-36} +101 q^{-37} -152 q^{-38} +5 q^{-39} -135 q^{-40} +145 q^{-41} +71 q^{-42} -90 q^{-43} +64 q^{-44} -128 q^{-45} +61 q^{-46} +20 q^{-47} -45 q^{-48} +123 q^{-49} -87 q^{-50} +2 q^{-51} -41 q^{-52} -39 q^{-53} +149 q^{-54} -27 q^{-55} -6 q^{-56} -74 q^{-57} -60 q^{-58} +120 q^{-59} +15 q^{-60} +20 q^{-61} -61 q^{-62} -70 q^{-63} +64 q^{-64} +18 q^{-65} +34 q^{-66} -26 q^{-67} -51 q^{-68} +23 q^{-69} +4 q^{-70} +24 q^{-71} -4 q^{-72} -24 q^{-73} +8 q^{-74} -2 q^{-75} +9 q^{-76} + q^{-77} -8 q^{-78} +3 q^{-79} - q^{-80} +2 q^{-81} -2 q^{-83} + q^{-84} }
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 \textrm{NotAvailable}(q)}
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 \textrm{NotAvailable}(q)}
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 \textrm{NotAvailable}(q)}

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.

10_6

10_8

10 7

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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