10 20

10_19

10_21

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 20 at Knotilus!

[edit 10 20 Quick Notes]

[edit 10 20 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_11,14,12,15 X_3,13,4,12 X_13,3,14,2 X_5,18,6,19 X_7,20,8,1 X_19,6,20,7 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9
Gauss code	-1, 4, -3, 1, -5, 7, -6, 10, -8, 9, -2, 3, -4, 2, -9, 8, -10, 5, -7, 6
Dowker-Thistlethwaite code	4 12 18 20 16 14 2 10 8 6
Conway Notation	[352]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 12, width is 5,

Braid index is 5

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

[edit Notes on presentations of 10 20]

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 20"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X_11,14,12,15 X_3,13,4,12 X_13,3,14,2 X_5,18,6,19 X_7,20,8,1 X_19,6,20,7 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, -5, 7, -6, 10, -8, 9, -2, 3, -4, 2, -9, 8, -10, 5, -7, 6

In[6]:= DTCode[K]

Out[6]= 4 12 18 20 16 14 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]= [352]

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]= ${\textrm {BR}}(5,\{-1,-1,-1,-1,-2,1,-2,-3,2,4,-3,4\})$

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, 3}, {4, 2}, {3, 11}, {1, 4}, {10, 12}, {11, 9}, {8, 10}, {9, 5}, {2, 6}, {5, 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	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-11][-1]
Hyperbolic Volume	8.31738
A-Polynomial	See Data:10 20/A-polynomial

[edit Notes for 10 20's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$1$
Topological 4 genus	$1$
Concordance genus	$2$
Rasmussen s-Invariant	-2

[edit Notes for 10 20'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+9 t-11+9 t^{-1} -3 t^{-2} }
Conway 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 z^4-3 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	{ 35, -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-1+3 q^{-1} -4 q^{-2} +5 q^{-3} -6 q^{-4} +5 q^{-5} -4 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}+a^{8}-z^{4}a^{6}-2z^{2}a^{6}-a^{6}-z^{4}a^{4}-z^{2}a^{4}-z^{4}a^{2}-2z^{2}a^{2}-a^{2}+z^{2}+2$
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+7 z^3 a^9-z a^9+2 z^8 a^8-8 z^6 a^8+9 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+3 z^5 a^7-4 z^3 a^7+2 z a^7+3 z^8 a^6-12 z^6 a^6+17 z^4 a^6-9 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+9 z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4+3 z^4 a^4+z^7 a^3-z^5 a^3+2 z^3 a^3-z a^3+z^6 a^2-2 z^2 a^2+a^2+z^5 a-z^3 a-z a+z^4-3 z^2+2}
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}-q^{20}-q^{14}-q^{10}-q^4+2 q^2+1+ 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}-2 q^{132}-q^{130}+6 q^{128}-9 q^{126}+10 q^{124}-9 q^{122}+3 q^{120}+4 q^{118}-11 q^{116}+16 q^{114}-13 q^{112}+9 q^{110}+q^{108}-8 q^{106}+12 q^{104}-9 q^{102}+3 q^{100}+3 q^{98}-7 q^{96}+7 q^{94}-2 q^{92}-4 q^{90}+11 q^{88}-12 q^{86}+9 q^{84}-4 q^{82}-5 q^{80}+10 q^{78}-15 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+5 q^{68}-12 q^{66}+13 q^{64}-10 q^{62}+4 q^{60}+q^{58}-7 q^{56}+7 q^{54}-3 q^{52}-2 q^{50}+5 q^{48}-6 q^{46}+q^{44}+2 q^{42}-7 q^{40}+7 q^{38}-5 q^{36}+3 q^{34}-q^{32}-3 q^{30}+4 q^{28}-5 q^{26}+5 q^{24}-5 q^{22}+4 q^{20}-3 q^{18}-q^{16}+4 q^{14}-6 q^{12}+8 q^{10}-4 q^8+3 q^6+q^4-q^2+4-3 q^{-2} +4 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} +2 q^{-14} + q^{-18} }

Further Quantum Invariants

Further quantum knot invariants for 10_20.

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}-q^{13}+q^{11}-q^9-q^7+q^5-q^3+2 q+ 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}-q^{46}-4 q^{44}+3 q^{42}+2 q^{40}-4 q^{38}+q^{36}+3 q^{34}-3 q^{32}-q^{30}+3 q^{28}-q^{26}-q^{24}+2 q^{22}+2 q^{20}-2 q^{18}-2 q^{16}+4 q^{14}-q^{12}-3 q^{10}+3 q^8-q^6-2 q^4+2 q^2-1+2 q^{-4} + q^{-10} }
3	$q^{105}-q^{103}-q^{101}+q^{99}+2q^{97}-q^{95}-5q^{93}+7q^{89}+3q^{87}-6q^{85}-7q^{83}+4q^{81}+9q^{79}-q^{77}-8q^{75}-4q^{73}+6q^{71}+7q^{69}-2q^{67}-8q^{65}-2q^{63}+8q^{61}+6q^{59}-7q^{57}-6q^{55}+6q^{53}+7q^{51}-6q^{49}-7q^{47}+2q^{45}+6q^{43}-2q^{41}-6q^{39}+5q^{35}+6q^{33}-4q^{31}-8q^{29}+10q^{25}+3q^{23}-7q^{21}-7q^{19}+6q^{17}+6q^{15}+q^{13}-5q^{11}-2q^{9}+3q^{7}+3q^{5}-4q-q^{-1}+2q^{-3}+2q^{-5}-2q^{-7}-q^{-9}+2q^{-13}+q^{-21}$
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}-3 q^{160}-3 q^{158}+2 q^{156}+2 q^{154}+9 q^{152}-3 q^{150}-11 q^{148}-5 q^{146}-q^{144}+18 q^{142}+9 q^{140}-6 q^{138}-13 q^{136}-18 q^{134}+10 q^{132}+16 q^{130}+10 q^{128}-20 q^{124}-8 q^{122}-2 q^{120}+11 q^{118}+21 q^{116}+2 q^{114}-8 q^{112}-25 q^{110}-14 q^{108}+20 q^{106}+27 q^{104}+13 q^{102}-28 q^{100}-36 q^{98}+4 q^{96}+33 q^{94}+30 q^{92}-15 q^{90}-40 q^{88}-10 q^{86}+24 q^{84}+29 q^{82}-5 q^{80}-30 q^{78}-11 q^{76}+16 q^{74}+22 q^{72}+q^{70}-19 q^{68}-11 q^{66}+10 q^{64}+14 q^{62}+8 q^{60}-8 q^{58}-20 q^{56}-8 q^{54}+8 q^{52}+29 q^{50}+14 q^{48}-26 q^{46}-34 q^{44}-13 q^{42}+37 q^{40}+45 q^{38}-7 q^{36}-43 q^{34}-41 q^{32}+18 q^{30}+51 q^{28}+17 q^{26}-20 q^{24}-43 q^{22}-6 q^{20}+30 q^{18}+22 q^{16}+3 q^{14}-26 q^{12}-13 q^{10}+10 q^8+13 q^6+11 q^4-11 q^2-10+ q^{-2} +5 q^{-4} +9 q^{-6} -4 q^{-8} -5 q^{-10} -2 q^{-12} +5 q^{-16} - q^{-18} - q^{-20} - q^{-22} - q^{-24} +2 q^{-26} + 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}-q^{241}-2 q^{239}+2 q^{237}+5 q^{235}+2 q^{233}-q^{231}-6 q^{229}-9 q^{227}-5 q^{225}+9 q^{223}+17 q^{221}+11 q^{219}-q^{217}-19 q^{215}-25 q^{213}-13 q^{211}+14 q^{209}+31 q^{207}+26 q^{205}+5 q^{203}-24 q^{201}-36 q^{199}-23 q^{197}+8 q^{195}+29 q^{193}+29 q^{191}+16 q^{189}-6 q^{187}-22 q^{185}-28 q^{183}-21 q^{181}-6 q^{179}+18 q^{177}+43 q^{175}+44 q^{173}+10 q^{171}-45 q^{169}-75 q^{167}-54 q^{165}+19 q^{163}+95 q^{161}+99 q^{159}+19 q^{157}-88 q^{155}-130 q^{153}-66 q^{151}+60 q^{149}+144 q^{147}+108 q^{145}-23 q^{143}-139 q^{141}-133 q^{139}-15 q^{137}+115 q^{135}+146 q^{133}+48 q^{131}-91 q^{129}-141 q^{127}-62 q^{125}+62 q^{123}+123 q^{121}+72 q^{119}-42 q^{117}-103 q^{115}-63 q^{113}+28 q^{111}+79 q^{109}+53 q^{107}-20 q^{105}-65 q^{103}-42 q^{101}+16 q^{99}+48 q^{97}+35 q^{95}-6 q^{93}-42 q^{91}-42 q^{89}-8 q^{87}+38 q^{85}+54 q^{83}+36 q^{81}-20 q^{79}-78 q^{77}-75 q^{75}-3 q^{73}+89 q^{71}+121 q^{69}+48 q^{67}-83 q^{65}-158 q^{63}-99 q^{61}+55 q^{59}+181 q^{57}+146 q^{55}-18 q^{53}-164 q^{51}-174 q^{49}-33 q^{47}+134 q^{45}+174 q^{43}+58 q^{41}-87 q^{39}-144 q^{37}-81 q^{35}+46 q^{33}+110 q^{31}+71 q^{29}-17 q^{27}-72 q^{25}-55 q^{23}+45 q^{19}+41 q^{17}+4 q^{15}-28 q^{13}-25 q^{11}-3 q^9+18 q^7+20 q^5+3 q^3-13 q-13 q^{-1} -4 q^{-3} +10 q^{-5} +13 q^{-7} +3 q^{-9} -6 q^{-11} -7 q^{-13} -6 q^{-15} +2 q^{-17} +9 q^{-19} +3 q^{-21} - q^{-23} -2 q^{-25} -4 q^{-27} -2 q^{-29} +3 q^{-31} + q^{-33} - q^{-39} - q^{-41} + q^{-43} + 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}-q^{20}-q^{14}-q^{10}-q^4+2 q^2+1+ 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}+15 q^{68}-22 q^{66}+30 q^{64}-38 q^{62}+43 q^{60}-44 q^{58}+38 q^{56}-28 q^{54}+11 q^{52}+8 q^{50}-26 q^{48}+44 q^{46}-59 q^{44}+64 q^{42}-68 q^{40}+64 q^{38}-55 q^{36}+46 q^{34}-28 q^{32}+18 q^{30}-3 q^{28}-4 q^{26}+12 q^{24}-12 q^{22}+13 q^{20}-12 q^{18}+10 q^{16}-14 q^{14}+11 q^{12}-18 q^{10}+14 q^8-18 q^6+16 q^4-12 q^2+12-6 q^{-2} +8 q^{-4} -2 q^{-6} +4 q^{-8} + q^{-12} }
2,0	$q^{72}+q^{64}-3q^{60}-q^{58}+q^{56}+q^{54}-q^{52}-q^{50}+2q^{48}+q^{46}-2q^{44}-2q^{42}+q^{40}+q^{38}-q^{36}+2q^{32}+2q^{30}+3q^{26}+q^{24}-q^{22}+2q^{20}+q^{18}-3q^{16}-3q^{14}+q^{12}-5q^{8}-3q^{6}+2q^{4}+2q^{-2}+3q^{-4}+q^{-6}+q^{-8}+q^{-10}+q^{-12}$

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}-2 q^{52}+q^{50}+q^{48}-2 q^{46}+2 q^{44}-3 q^{40}+2 q^{38}+2 q^{36}-2 q^{34}+2 q^{32}+3 q^{30}-q^{28}+q^{22}-3 q^{20}-q^{18}+2 q^{16}-4 q^{14}-3 q^{12}+2 q^{10}-2 q^8-3 q^6+4 q^4+2 q^2+1+3 q^{-2} +2 q^{-4} + 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}-q^{27}-q^{23}-q^{19}-q^{13}-q^9-q^5+2 q^3+q+2 q^{-1} + q^{-3} + q^{-5} }

B2 Invariants.

Weight

Invariant

0,1

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^{60}-q^{58}+2 q^{56}-3 q^{54}+4 q^{52}-5 q^{50}+5 q^{48}-4 q^{46}+4 q^{44}-2 q^{42}+q^{40}+2 q^{38}-4 q^{36}+6 q^{34}-8 q^{32}+7 q^{30}-9 q^{28}+8 q^{26}-8 q^{24}+5 q^{22}-3 q^{20}+q^{18}-2 q^{14}+3 q^{12}-4 q^{10}+4 q^8-3 q^6+4 q^4-2 q^2+3- q^{-2} +2 q^{-4} + 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}-3 q^{84}+4 q^{80}+2 q^{78}-4 q^{76}-4 q^{74}+2 q^{72}+5 q^{70}-5 q^{66}-3 q^{64}+3 q^{62}+4 q^{60}-3 q^{56}+3 q^{52}+q^{50}-q^{48}-q^{46}+3 q^{44}+2 q^{42}-2 q^{40}-3 q^{38}+2 q^{36}+3 q^{34}-q^{32}-4 q^{30}-q^{28}+3 q^{26}+q^{24}-4 q^{22}-4 q^{20}+3 q^{16}+q^{14}-3 q^{12}-3 q^{10}+4 q^6+2 q^4-1+ q^{-2} +2 q^{-4} +2 q^{-6} + q^{-14} }

G2 Invariants.

Weight

Invariant

1,0

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

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+9 t-11+9 t^{-1} -3 t^{-2} }

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

Out[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 -3 z^4-3 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]= { 35, -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-1+3 q^{-1} -4 q^{-2} +5 q^{-3} -6 q^{-4} +5 q^{-5} -4 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]= $z^{2}a^{8}+a^{8}-z^{4}a^{6}-2z^{2}a^{6}-a^{6}-z^{4}a^{4}-z^{2}a^{4}-z^{4}a^{2}-2z^{2}a^{2}-a^{2}+z^{2}+2$

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+7 z^3 a^9-z a^9+2 z^8 a^8-8 z^6 a^8+9 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+3 z^5 a^7-4 z^3 a^7+2 z a^7+3 z^8 a^6-12 z^6 a^6+17 z^4 a^6-9 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+9 z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4+3 z^4 a^4+z^7 a^3-z^5 a^3+2 z^3 a^3-z a^3+z^6 a^2-2 z^2 a^2+a^2+z^5 a-z^3 a-z a+z^4-3 z^2+2}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_162, K11n117,}

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 20"];

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+9 t-11+9 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-1+3 q^{-1} -4 q^{-2} +5 q^{-3} -6 q^{-4} +5 q^{-5} -4 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]= {10_162, K11n117,}

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:

(-3, 6)

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 -12}

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 48}

72

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 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 62}

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 -576}

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 -896}

-256

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 -272}

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 1152}

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 -216}

-744

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{30689}{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 \frac{6826}{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{7858}{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{2527}{6}}

-{\frac {1951}{10}}

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=} -2 is the signature of 10 20. 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

0

-1

3

1

2

-3

2

1

-1

-5

3

2

1

-7

3

2

-1

-9

2

3

-1

-11

2

3

1

-13

1

2

-1

-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}	${\mathbb {Z} }$
$r=-7$	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}^{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=-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^{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=-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}\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}^{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=-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^{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=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}\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}}
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-q^3+3 q-3- q^{-1} +6 q^{-2} -7 q^{-3} +10 q^{-5} -13 q^{-6} +2 q^{-7} +15 q^{-8} -19 q^{-9} +2 q^{-10} +19 q^{-11} -19 q^{-12} - q^{-13} +19 q^{-14} -15 q^{-15} -5 q^{-16} +17 q^{-17} -9 q^{-18} -7 q^{-19} +12 q^{-20} -3 q^{-21} -6 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-q^8+3 q^5-3 q^4-q^3-q^2+7 q-3-4 q^{-1} -4 q^{-2} +11 q^{-3} -4 q^{-5} -9 q^{-6} +8 q^{-7} +6 q^{-8} + q^{-9} -9 q^{-10} -5 q^{-11} +6 q^{-12} +11 q^{-13} -2 q^{-14} -15 q^{-15} -2 q^{-16} +15 q^{-17} +8 q^{-18} -16 q^{-19} -7 q^{-20} +9 q^{-21} +12 q^{-22} -8 q^{-23} -11 q^{-24} +13 q^{-26} +5 q^{-27} -12 q^{-28} -12 q^{-29} +12 q^{-30} +18 q^{-31} -10 q^{-32} -22 q^{-33} +6 q^{-34} +24 q^{-35} - q^{-36} -23 q^{-37} -4 q^{-38} +20 q^{-39} +6 q^{-40} -13 q^{-41} -9 q^{-42} +9 q^{-43} +7 q^{-44} -4 q^{-45} -5 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}-q^{15}+3 q^{11}-4 q^{10}+9 q^6-9 q^5-2 q^4-3 q^3+q^2+22 q-13-6 q^{-1} -14 q^{-2} +44 q^{-4} -11 q^{-5} -9 q^{-6} -37 q^{-7} -13 q^{-8} +73 q^{-9} +8 q^{-10} - q^{-11} -73 q^{-12} -50 q^{-13} +96 q^{-14} +45 q^{-15} +33 q^{-16} -106 q^{-17} -109 q^{-18} +94 q^{-19} +81 q^{-20} +85 q^{-21} -114 q^{-22} -159 q^{-23} +73 q^{-24} +89 q^{-25} +125 q^{-26} -99 q^{-27} -180 q^{-28} +57 q^{-29} +77 q^{-30} +137 q^{-31} -83 q^{-32} -174 q^{-33} +53 q^{-34} +56 q^{-35} +129 q^{-36} -63 q^{-37} -153 q^{-38} +47 q^{-39} +29 q^{-40} +110 q^{-41} -38 q^{-42} -119 q^{-43} +42 q^{-44} -5 q^{-45} +80 q^{-46} -13 q^{-47} -74 q^{-48} +45 q^{-49} -34 q^{-50} +40 q^{-51} -5 q^{-52} -33 q^{-53} +59 q^{-54} -41 q^{-55} +6 q^{-56} -16 q^{-57} -16 q^{-58} +69 q^{-59} -22 q^{-60} -4 q^{-61} -29 q^{-62} -22 q^{-63} +57 q^{-64} -2 q^{-65} +6 q^{-66} -23 q^{-67} -28 q^{-68} +29 q^{-69} +3 q^{-70} +13 q^{-71} -8 q^{-72} -19 q^{-73} +10 q^{-74} - q^{-75} +7 q^{-76} -7 q^{-78} +3 q^{-79} - q^{-80} +2 q^{-81} -2 q^{-83} + q^{-84} }

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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10_19

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10 20

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

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