8 11

8_10

8_12

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 11's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 11 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X9,16,10,1 X15,6,16,7 X7,14,8,15 X13,8,14,9
Gauss code	-1, 4, -3, 1, -2, 6, -7, 8, -5, 3, -4, 2, -8, 7, -6, 5
Dowker-Thistlethwaite code	4 10 12 14 16 2 8 6
Conway Notation	[3212]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{10, 5}, {1, 8}, {9, 6}, {5, 7}, {8, 10}, {4, 9}, {6, 3}, {2, 4}, {3, 1}, {7, 2}]

[edit Notes on presentations of 8 11]

Knot 8_11.

A graph, knot 8_11.

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["8 11"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X9,16,10,1 X15,6,16,7 X7,14,8,15 X13,8,14,9

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -2, 6, -7, 8, -5, 3, -4, 2, -8, 7, -6, 5

In[6]:=

DTCode[K]

Out[6]=

4 10 12 14 16 2 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3212]

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

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 4, 9, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{10, 5}, {1, 8}, {9, 6}, {5, 7}, {8, 10}, {4, 9}, {6, 3}, {2, 4}, {3, 1}, {7, 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 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 \{4,5\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-1] Hyperbolic Volume 8.28632 A-Polynomial See Data:8 11/A-polynomial

[edit Notes for 8 11's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ 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 1} Rasmussen s-Invariant -2

[edit Notes for 8 11'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 -2 t^2+7 t-9+7 t^{-1} -2 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 -2 z^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	{ 27, -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} -4 q^{-2} +5 q^{-3} -5 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7} }
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^6+a^6-z^4 a^4-2 z^2 a^4-2 a^4-z^4 a^2-z^2 a^2+a^2+z^2+1}
Kauffman polynomial (db, data sources)	$z^{4}a^{8}-2z^{2}a^{8}+2z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+2z^{6}a^{6}-3z^{4}a^{6}+2z^{2}a^{6}-a^{6}+z^{7}a^{5}+z^{5}a^{5}-3z^{3}a^{5}+3za^{5}+4z^{6}a^{4}-7z^{4}a^{4}+6z^{2}a^{4}-2a^{4}+z^{7}a^{3}+z^{5}a^{3}-2z^{3}a^{3}+za^{3}+2z^{6}a^{2}-2z^{4}a^{2}-a^{2}+2z^{5}a-3z^{3}a+z^{4}-2z^{2}+1$
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^{22}+q^{16}-2 q^{14}-q^{12}-q^{10}+2 q^6+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^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-q^{104}-3 q^{102}+7 q^{100}-8 q^{98}+8 q^{96}-6 q^{94}+2 q^{92}+7 q^{90}-13 q^{88}+16 q^{86}-13 q^{84}+7 q^{82}+3 q^{80}-10 q^{78}+13 q^{76}-9 q^{74}+8 q^{72}+4 q^{70}-10 q^{68}+7 q^{66}-2 q^{64}-8 q^{62}+14 q^{60}-18 q^{58}+10 q^{56}-3 q^{54}-9 q^{52}+18 q^{50}-25 q^{48}+19 q^{46}-14 q^{44}-q^{42}+11 q^{40}-17 q^{38}+16 q^{36}-9 q^{34}+4 q^{32}+6 q^{30}-9 q^{28}+7 q^{26}-7 q^{22}+14 q^{20}-11 q^{18}+5 q^{16}+6 q^{14}-11 q^{12}+17 q^{10}-14 q^8+9 q^6-2 q^4-7 q^2+10-9 q^{-2} +8 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 8_11.

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^{15}-q^{13}+q^{11}-2 q^9+q^5+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^{42}-q^{40}-q^{38}+3 q^{36}-2 q^{34}-4 q^{32}+5 q^{30}-5 q^{26}+6 q^{24}+2 q^{22}-4 q^{20}+q^{16}-4 q^{12}+2 q^{10}+4 q^8-5 q^6+q^4+6 q^2-4- q^{-2} +4 q^{-4} - q^{-6} - q^{-8} + q^{-10} }
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^{81}-q^{79}-q^{77}+q^{75}+2 q^{73}-2 q^{71}-5 q^{69}+2 q^{67}+8 q^{65}-11 q^{61}-2 q^{59}+14 q^{57}+8 q^{55}-15 q^{53}-11 q^{51}+12 q^{49}+14 q^{47}-13 q^{45}-15 q^{43}+8 q^{41}+13 q^{39}-q^{37}-11 q^{35}+6 q^{31}+6 q^{29}-4 q^{27}-9 q^{25}-q^{23}+14 q^{21}+3 q^{19}-16 q^{17}-6 q^{15}+16 q^{13}+10 q^{11}-14 q^9-12 q^7+11 q^5+14 q^3-7 q-11 q^{-1} +2 q^{-3} +10 q^{-5} + q^{-7} -6 q^{-9} - q^{-11} +3 q^{-13} + q^{-15} - 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^{132}-q^{130}-q^{128}+q^{126}+2 q^{122}-4 q^{120}-3 q^{118}+4 q^{116}+3 q^{114}+8 q^{112}-8 q^{110}-13 q^{108}+q^{106}+9 q^{104}+25 q^{102}-5 q^{100}-28 q^{98}-20 q^{96}+3 q^{94}+49 q^{92}+19 q^{90}-28 q^{88}-47 q^{86}-23 q^{84}+56 q^{82}+47 q^{80}-7 q^{78}-54 q^{76}-47 q^{74}+43 q^{72}+53 q^{70}+12 q^{68}-39 q^{66}-49 q^{64}+15 q^{62}+36 q^{60}+22 q^{58}-16 q^{56}-33 q^{54}-7 q^{52}+17 q^{50}+24 q^{48}+6 q^{46}-15 q^{44}-31 q^{42}-5 q^{40}+30 q^{38}+29 q^{36}-2 q^{34}-52 q^{32}-25 q^{30}+32 q^{28}+51 q^{26}+18 q^{24}-60 q^{22}-43 q^{20}+16 q^{18}+54 q^{16}+40 q^{14}-43 q^{12}-48 q^{10}-11 q^8+35 q^6+49 q^4-13 q^2-29-23 q^{-2} +6 q^{-4} +32 q^{-6} +5 q^{-8} -6 q^{-10} -16 q^{-12} -6 q^{-14} +12 q^{-16} +3 q^{-18} +2 q^{-20} -4 q^{-22} -3 q^{-24} +3 q^{-26} + q^{-30} - 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^{195}-q^{193}-q^{191}+q^{189}-2 q^{181}-2 q^{179}+4 q^{177}+6 q^{175}+q^{173}-4 q^{171}-9 q^{169}-8 q^{167}+4 q^{165}+18 q^{163}+18 q^{161}-q^{159}-24 q^{157}-35 q^{155}-16 q^{153}+25 q^{151}+58 q^{149}+43 q^{147}-15 q^{145}-74 q^{143}-84 q^{141}-17 q^{139}+82 q^{137}+126 q^{135}+65 q^{133}-65 q^{131}-161 q^{129}-120 q^{127}+33 q^{125}+172 q^{123}+175 q^{121}+18 q^{119}-170 q^{117}-209 q^{115}-64 q^{113}+138 q^{111}+220 q^{109}+109 q^{107}-107 q^{105}-212 q^{103}-124 q^{101}+62 q^{99}+182 q^{97}+130 q^{95}-23 q^{93}-146 q^{91}-124 q^{89}+3 q^{87}+103 q^{85}+101 q^{83}+25 q^{81}-67 q^{79}-93 q^{77}-33 q^{75}+37 q^{73}+71 q^{71}+61 q^{69}-5 q^{67}-69 q^{65}-76 q^{63}-24 q^{61}+61 q^{59}+107 q^{57}+56 q^{55}-63 q^{53}-136 q^{51}-92 q^{49}+49 q^{47}+164 q^{45}+130 q^{43}-34 q^{41}-179 q^{39}-170 q^{37}+5 q^{35}+181 q^{33}+199 q^{31}+37 q^{29}-161 q^{27}-211 q^{25}-79 q^{23}+118 q^{21}+208 q^{19}+115 q^{17}-67 q^{15}-176 q^{13}-131 q^{11}+9 q^9+131 q^7+134 q^5+28 q^3-77 q-106 q^{-1} -52 q^{-3} +28 q^{-5} +76 q^{-7} +55 q^{-9} - q^{-11} -40 q^{-13} -43 q^{-15} -13 q^{-17} +16 q^{-19} +27 q^{-21} +16 q^{-23} -5 q^{-25} -13 q^{-27} -9 q^{-29} +3 q^{-33} +5 q^{-35} +2 q^{-37} -3 q^{-39} - q^{-41} + q^{-43} + q^{-49} - 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^{22}+q^{16}-2 q^{14}-q^{12}-q^{10}+2 q^6+2 q^2+ q^{-4} }
1,1	$q^{60}-2q^{58}+4q^{56}-8q^{54}+15q^{52}-20q^{50}+26q^{48}-38q^{46}+43q^{44}-44q^{42}+40q^{40}-34q^{38}+20q^{36}+4q^{34}-26q^{32}+50q^{30}-66q^{28}+84q^{26}-88q^{24}+86q^{22}-79q^{20}+56q^{18}-38q^{16}+14q^{14}+6q^{12}-26q^{10}+42q^{8}-44q^{6}+44q^{4}-40q^{2}+36-22q^{-2}+17q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
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^{56}+q^{48}-q^{46}-4 q^{44}-q^{42}+2 q^{40}-3 q^{36}+2 q^{34}+5 q^{32}+3 q^{30}-q^{28}+2 q^{26}-4 q^{22}-q^{20}-q^{18}-2 q^{16}-q^{14}+3 q^{12}-2 q^8+q^6+4 q^4-q^2-2+3 q^{-2} +3 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^{48}-q^{46}+q^{42}-4 q^{40}+4 q^{36}-3 q^{34}+2 q^{32}+7 q^{30}-2 q^{28}+q^{24}-3 q^{22}-4 q^{20}-3 q^{18}+q^{16}-q^{14}-2 q^{12}+5 q^{10}+2 q^8-4 q^6+5 q^4+2 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^{29}+q^{25}+q^{21}-2 q^{19}-q^{17}-2 q^{15}-q^{13}+q^9+2 q^7+2 q^3+ q^{-1} + q^{-5} }

A4 Invariants.

Weight	Invariant
0,1,0,0	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^{62}-q^{58}+q^{56}+q^{54}-3 q^{52}-3 q^{50}+q^{48}+q^{46}-3 q^{44}+7 q^{40}+4 q^{38}+q^{36}+5 q^{34}+4 q^{32}-4 q^{30}-4 q^{28}-3 q^{26}-6 q^{24}-7 q^{22}-q^{20}+2 q^{18}-3 q^{16}+q^{14}+6 q^{12}+2 q^{10}-2 q^8+3 q^6+4 q^4+q^2+2 q^{-2} +2 q^{-4} + q^{-10} }
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^{36}+q^{32}+q^{30}+q^{26}-2 q^{24}-q^{22}-2 q^{20}-2 q^{18}-q^{16}+q^{12}+q^{10}+2 q^8+2 q^4+1+ q^{-2} + q^{-6} }

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^{48}-q^{46}+2 q^{44}-3 q^{42}+4 q^{40}-4 q^{38}+4 q^{36}-3 q^{34}+2 q^{32}-q^{30}-2 q^{28}+4 q^{26}-7 q^{24}+7 q^{22}-8 q^{20}+7 q^{18}-7 q^{16}+5 q^{14}-2 q^{12}+q^{10}+2 q^8-2 q^6+5 q^4-4 q^2+4-3 q^{-2} +3 q^{-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^{78}-q^{74}-q^{72}+q^{70}+2 q^{68}-q^{66}-4 q^{64}-2 q^{62}+3 q^{60}+4 q^{58}-q^{56}-4 q^{54}+5 q^{50}+4 q^{48}-2 q^{46}-2 q^{44}+2 q^{42}+3 q^{40}-2 q^{38}-4 q^{36}-q^{34}+2 q^{32}-q^{30}-4 q^{28}-2 q^{26}+2 q^{24}+2 q^{22}-2 q^{20}-2 q^{18}+2 q^{16}+5 q^{14}-4 q^{10}-2 q^8+5 q^6+4 q^4-q^2-3+3 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} }

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-q^{64}+q^{62}-2q^{60}+2q^{58}-4q^{56}+2q^{54}-3q^{52}+4q^{50}-2q^{48}+3q^{46}+q^{44}+4q^{42}+3q^{40}-2q^{38}+3q^{36}-5q^{34}+4q^{32}-9q^{30}+2q^{28}-9q^{26}+4q^{24}-5q^{22}+3q^{20}-3q^{18}+3q^{16}+2q^{14}+q^{12}+2q^{10}-2q^{8}+5q^{6}-2q^{4}+4q^{2}-2+4q^{-2}-q^{-4}+2q^{-6}-q^{-8}+q^{-10}$

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^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{106}-q^{104}-3 q^{102}+7 q^{100}-8 q^{98}+8 q^{96}-6 q^{94}+2 q^{92}+7 q^{90}-13 q^{88}+16 q^{86}-13 q^{84}+7 q^{82}+3 q^{80}-10 q^{78}+13 q^{76}-9 q^{74}+8 q^{72}+4 q^{70}-10 q^{68}+7 q^{66}-2 q^{64}-8 q^{62}+14 q^{60}-18 q^{58}+10 q^{56}-3 q^{54}-9 q^{52}+18 q^{50}-25 q^{48}+19 q^{46}-14 q^{44}-q^{42}+11 q^{40}-17 q^{38}+16 q^{36}-9 q^{34}+4 q^{32}+6 q^{30}-9 q^{28}+7 q^{26}-7 q^{22}+14 q^{20}-11 q^{18}+5 q^{16}+6 q^{14}-11 q^{12}+17 q^{10}-14 q^8+9 q^6-2 q^4-7 q^2+10-9 q^{-2} +8 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 11"];

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 -2 t^2+7 t-9+7 t^{-1} -2 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 -2 z^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]=

{ 27, -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} -4 q^{-2} +5 q^{-3} -5 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7} }

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{8}-2z^{2}a^{8}+2z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+2z^{6}a^{6}-3z^{4}a^{6}+2z^{2}a^{6}-a^{6}+z^{7}a^{5}+z^{5}a^{5}-3z^{3}a^{5}+3za^{5}+4z^{6}a^{4}-7z^{4}a^{4}+6z^{2}a^{4}-2a^{4}+z^{7}a^{3}+z^{5}a^{3}-2z^{3}a^{3}+za^{3}+2z^{6}a^{2}-2z^{4}a^{2}-a^{2}+2z^{5}a-3z^{3}a+z^{4}-2z^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_147, K11n122,}

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["8 11"];

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

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_147, K11n122,}

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, 2)

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 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 16} 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} $-{\frac {14}{3}}$ 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{62}{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 -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 -\frac{416}{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{128}{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 -80} 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 128} 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{56}{3}} $-{\frac {248}{3}}$ 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{14849}{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{578}{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{11698}{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{1567}{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{31}{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 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 8 11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 3                 1 1 1               1   -1 -1             3 1   2 -3           2 2     0 -5         3 2       1 -7       2 2         0 -9     1 3           -2 -11   1 2             1 -13   1               -1 -15 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}} $i=-3$ 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=-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}} 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}\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=-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} 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=-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^{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}^{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}^{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+6 q-7-3 q^{-1} +16 q^{-2} -12 q^{-3} -9 q^{-4} +25 q^{-5} -14 q^{-6} -15 q^{-7} +29 q^{-8} -13 q^{-9} -16 q^{-10} +25 q^{-11} -7 q^{-12} -12 q^{-13} +14 q^{-14} -2 q^{-15} -7 q^{-16} +5 q^{-17} -2 q^{-19} + q^{-20} }
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+2 q^6+3 q^5-6 q^4-5 q^3+9 q^2+12 q-14-18 q^{-1} +13 q^{-2} +33 q^{-3} -17 q^{-4} -41 q^{-5} +11 q^{-6} +57 q^{-7} -11 q^{-8} -63 q^{-9} + q^{-10} +76 q^{-11} -78 q^{-13} -7 q^{-14} +81 q^{-15} +10 q^{-16} -78 q^{-17} -13 q^{-18} +70 q^{-19} +20 q^{-20} -64 q^{-21} -18 q^{-22} +47 q^{-23} +22 q^{-24} -37 q^{-25} -20 q^{-26} +24 q^{-27} +18 q^{-28} -14 q^{-29} -14 q^{-30} +8 q^{-31} +9 q^{-32} -3 q^{-33} -6 q^{-34} +2 q^{-35} +2 q^{-36} -2 q^{-38} + q^{-39} }
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}+2 q^{13}-q^{12}+4 q^{11}-8 q^{10}-q^9+8 q^8+13 q^6-26 q^5-11 q^4+18 q^3+11 q^2+40 q-52-40 q^{-1} +12 q^{-2} +27 q^{-3} +102 q^{-4} -66 q^{-5} -86 q^{-6} -25 q^{-7} +32 q^{-8} +185 q^{-9} -52 q^{-10} -124 q^{-11} -84 q^{-12} +15 q^{-13} +263 q^{-14} -19 q^{-15} -143 q^{-16} -141 q^{-17} -12 q^{-18} +313 q^{-19} +12 q^{-20} -142 q^{-21} -176 q^{-22} -38 q^{-23} +329 q^{-24} +33 q^{-25} -124 q^{-26} -183 q^{-27} -62 q^{-28} +303 q^{-29} +50 q^{-30} -86 q^{-31} -169 q^{-32} -83 q^{-33} +239 q^{-34} +60 q^{-35} -35 q^{-36} -128 q^{-37} -93 q^{-38} +149 q^{-39} +53 q^{-40} +12 q^{-41} -74 q^{-42} -84 q^{-43} +70 q^{-44} +29 q^{-45} +31 q^{-46} -27 q^{-47} -54 q^{-48} +24 q^{-49} +6 q^{-50} +23 q^{-51} -4 q^{-52} -24 q^{-53} +8 q^{-54} -2 q^{-55} +9 q^{-56} + q^{-57} -8 q^{-58} +3 q^{-59} - q^{-60} +2 q^{-61} -2 q^{-63} + q^{-64} }
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^{25}-2 q^{24}+2 q^{22}-q^{21}+2 q^{19}-4 q^{18}-2 q^{17}+7 q^{16}+2 q^{15}-2 q^{14}-q^{13}-13 q^{12}-6 q^{11}+15 q^{10}+23 q^9+9 q^8-12 q^7-42 q^6-36 q^5+18 q^4+62 q^3+65 q^2+9 q-90-116 q^{-1} -36 q^{-2} +91 q^{-3} +170 q^{-4} +115 q^{-5} -93 q^{-6} -238 q^{-7} -176 q^{-8} +46 q^{-9} +279 q^{-10} +297 q^{-11} -328 q^{-13} -373 q^{-14} -86 q^{-15} +329 q^{-16} +495 q^{-17} +162 q^{-18} -346 q^{-19} -549 q^{-20} -261 q^{-21} +320 q^{-22} +640 q^{-23} +326 q^{-24} -312 q^{-25} -664 q^{-26} -402 q^{-27} +276 q^{-28} +713 q^{-29} +445 q^{-30} -261 q^{-31} -710 q^{-32} -487 q^{-33} +224 q^{-34} +720 q^{-35} +509 q^{-36} -195 q^{-37} -700 q^{-38} -521 q^{-39} +154 q^{-40} +660 q^{-41} +535 q^{-42} -103 q^{-43} -624 q^{-44} -519 q^{-45} +54 q^{-46} +533 q^{-47} +513 q^{-48} +20 q^{-49} -471 q^{-50} -467 q^{-51} -66 q^{-52} +347 q^{-53} +425 q^{-54} +125 q^{-55} -255 q^{-56} -356 q^{-57} -148 q^{-58} +145 q^{-59} +280 q^{-60} +164 q^{-61} -67 q^{-62} -199 q^{-63} -151 q^{-64} +6 q^{-65} +127 q^{-66} +123 q^{-67} +29 q^{-68} -69 q^{-69} -90 q^{-70} -38 q^{-71} +28 q^{-72} +56 q^{-73} +39 q^{-74} -10 q^{-75} -32 q^{-76} -23 q^{-77} -5 q^{-78} +15 q^{-79} +20 q^{-80} + q^{-81} -9 q^{-82} -4 q^{-83} -5 q^{-84} + q^{-85} +8 q^{-86} -4 q^{-88} + q^{-89} - q^{-91} +2 q^{-92} -2 q^{-94} + q^{-95} }
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^{36}-2 q^{35}+2 q^{33}-q^{32}-2 q^{30}+6 q^{29}-5 q^{28}-3 q^{27}+9 q^{26}-2 q^{25}-2 q^{24}-10 q^{23}+12 q^{22}-12 q^{21}-5 q^{20}+29 q^{19}+9 q^{18}-34 q^{16}+9 q^{15}-48 q^{14}-22 q^{13}+71 q^{12}+59 q^{11}+47 q^{10}-48 q^9-6 q^8-154 q^7-118 q^6+79 q^5+148 q^4+193 q^3+47 q^2+57 q-319-361 q^{-1} -83 q^{-2} +154 q^{-3} +403 q^{-4} +329 q^{-5} +360 q^{-6} -382 q^{-7} -690 q^{-8} -493 q^{-9} -100 q^{-10} +489 q^{-11} +715 q^{-12} +950 q^{-13} -164 q^{-14} -895 q^{-15} -1035 q^{-16} -638 q^{-17} +291 q^{-18} +995 q^{-19} +1667 q^{-20} +322 q^{-21} -837 q^{-22} -1487 q^{-23} -1280 q^{-24} -143 q^{-25} +1056 q^{-26} +2276 q^{-27} +881 q^{-28} -580 q^{-29} -1736 q^{-30} -1805 q^{-31} -618 q^{-32} +955 q^{-33} +2660 q^{-34} +1318 q^{-35} -282 q^{-36} -1812 q^{-37} -2120 q^{-38} -982 q^{-39} +803 q^{-40} +2839 q^{-41} +1576 q^{-42} -48 q^{-43} -1784 q^{-44} -2252 q^{-45} -1213 q^{-46} +652 q^{-47} +2858 q^{-48} +1703 q^{-49} +136 q^{-50} -1676 q^{-51} -2251 q^{-52} -1367 q^{-53} +468 q^{-54} +2720 q^{-55} +1749 q^{-56} +341 q^{-57} -1445 q^{-58} -2120 q^{-59} -1484 q^{-60} +194 q^{-61} +2373 q^{-62} +1698 q^{-63} +598 q^{-64} -1038 q^{-65} -1804 q^{-66} -1526 q^{-67} -180 q^{-68} +1785 q^{-69} +1479 q^{-70} +823 q^{-71} -486 q^{-72} -1275 q^{-73} -1392 q^{-74} -526 q^{-75} +1042 q^{-76} +1050 q^{-77} +870 q^{-78} +29 q^{-79} -629 q^{-80} -1036 q^{-81} -664 q^{-82} +382 q^{-83} +511 q^{-84} +669 q^{-85} +296 q^{-86} -95 q^{-87} -565 q^{-88} -534 q^{-89} +14 q^{-90} +84 q^{-91} +339 q^{-92} +278 q^{-93} +152 q^{-94} -192 q^{-95} -280 q^{-96} -64 q^{-97} -92 q^{-98} +88 q^{-99} +134 q^{-100} +156 q^{-101} -26 q^{-102} -93 q^{-103} -19 q^{-104} -87 q^{-105} -6 q^{-106} +31 q^{-107} +82 q^{-108} +5 q^{-109} -23 q^{-110} +11 q^{-111} -38 q^{-112} -13 q^{-113} - q^{-114} +31 q^{-115} + q^{-116} -8 q^{-117} +11 q^{-118} -11 q^{-119} -4 q^{-120} -3 q^{-121} +10 q^{-122} - q^{-123} -5 q^{-124} +5 q^{-125} -2 q^{-126} - q^{-128} +2 q^{-129} -2 q^{-131} + q^{-132} }
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 q^{49}-2 q^{48}+2 q^{46}-q^{45}-2 q^{43}+2 q^{42}+5 q^{41}-6 q^{40}-q^{39}+5 q^{38}-2 q^{37}-10 q^{35}-q^{34}+17 q^{33}-9 q^{32}+3 q^{31}+15 q^{30}+q^{29}+4 q^{28}-35 q^{27}-27 q^{26}+15 q^{25}-14 q^{24}+23 q^{23}+57 q^{22}+34 q^{21}+46 q^{20}-60 q^{19}-108 q^{18}-55 q^{17}-97 q^{16}+21 q^{15}+134 q^{14}+162 q^{13}+224 q^{12}+25 q^{11}-167 q^{10}-230 q^9-390 q^8-203 q^7+102 q^6+317 q^5+630 q^4+453 q^3+81 q^2-286 q-879-850 q^{-1} -427 q^{-2} +140 q^{-3} +1047 q^{-4} +1282 q^{-5} +971 q^{-6} +282 q^{-7} -1101 q^{-8} -1766 q^{-9} -1642 q^{-10} -875 q^{-11} +899 q^{-12} +2070 q^{-13} +2424 q^{-14} +1790 q^{-15} -448 q^{-16} -2303 q^{-17} -3176 q^{-18} -2750 q^{-19} -270 q^{-20} +2175 q^{-21} +3813 q^{-22} +3919 q^{-23} +1202 q^{-24} -1938 q^{-25} -4306 q^{-26} -4906 q^{-27} -2225 q^{-28} +1351 q^{-29} +4564 q^{-30} +5924 q^{-31} +3301 q^{-32} -782 q^{-33} -4660 q^{-34} -6631 q^{-35} -4276 q^{-36} +20 q^{-37} +4580 q^{-38} +7291 q^{-39} +5133 q^{-40} +589 q^{-41} -4418 q^{-42} -7646 q^{-43} -5818 q^{-44} -1243 q^{-45} +4202 q^{-46} +7969 q^{-47} +6348 q^{-48} +1679 q^{-49} -3984 q^{-50} -8069 q^{-51} -6719 q^{-52} -2117 q^{-53} +3761 q^{-54} +8190 q^{-55} +6990 q^{-56} +2371 q^{-57} -3572 q^{-58} -8156 q^{-59} -7149 q^{-60} -2653 q^{-61} +3361 q^{-62} +8135 q^{-63} +7273 q^{-64} +2847 q^{-65} -3153 q^{-66} -8004 q^{-67} -7327 q^{-68} -3063 q^{-69} +2859 q^{-70} +7798 q^{-71} +7355 q^{-72} +3317 q^{-73} -2509 q^{-74} -7516 q^{-75} -7290 q^{-76} -3535 q^{-77} +2025 q^{-78} +7006 q^{-79} +7159 q^{-80} +3863 q^{-81} -1446 q^{-82} -6434 q^{-83} -6886 q^{-84} -4039 q^{-85} +750 q^{-86} +5551 q^{-87} +6445 q^{-88} +4289 q^{-89} +4 q^{-90} -4645 q^{-91} -5824 q^{-92} -4279 q^{-93} -722 q^{-94} +3488 q^{-95} +5005 q^{-96} +4195 q^{-97} +1379 q^{-98} -2404 q^{-99} -4063 q^{-100} -3827 q^{-101} -1826 q^{-102} +1327 q^{-103} +3008 q^{-104} +3314 q^{-105} +2069 q^{-106} -452 q^{-107} -2001 q^{-108} -2633 q^{-109} -2036 q^{-110} -203 q^{-111} +1099 q^{-112} +1901 q^{-113} +1804 q^{-114} +575 q^{-115} -397 q^{-116} -1198 q^{-117} -1437 q^{-118} -703 q^{-119} -65 q^{-120} +625 q^{-121} +1007 q^{-122} +647 q^{-123} +307 q^{-124} -212 q^{-125} -624 q^{-126} -486 q^{-127} -371 q^{-128} -35 q^{-129} +331 q^{-130} +313 q^{-131} +318 q^{-132} +127 q^{-133} -129 q^{-134} -145 q^{-135} -241 q^{-136} -160 q^{-137} +40 q^{-138} +66 q^{-139} +141 q^{-140} +106 q^{-141} +10 q^{-142} +15 q^{-143} -85 q^{-144} -95 q^{-145} -11 q^{-146} -6 q^{-147} +42 q^{-148} +34 q^{-149} +7 q^{-150} +33 q^{-151} -16 q^{-152} -37 q^{-153} -6 q^{-154} -8 q^{-155} +14 q^{-156} +6 q^{-157} -6 q^{-158} +16 q^{-159} -10 q^{-161} -2 q^{-162} -3 q^{-163} +6 q^{-164} + q^{-165} -6 q^{-166} +4 q^{-167} +2 q^{-168} -2 q^{-169} - q^{-171} +2 q^{-172} -2 q^{-174} + q^{-175} }

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.

8_10 8_12