9 22

9_21

9_23

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 22's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 22 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17
Gauss code	1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -9, 8, -6, 7, -5, 9, -8
Dowker-Thistlethwaite code	4 8 10 14 2 16 18 6 12
Conway Notation	[211,3,2]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{6, 12}, {1, 9}, {11, 4}, {12, 10}, {8, 3}, {9, 7}, {5, 8}, {7, 11}, {4, 2}, {3, 6}, {2, 5}, {10, 1}]

[edit Notes on presentations of 9 22]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 22"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -9, 8, -6, 7, -5, 9, -8

In[6]:=

DTCode[K]

Out[6]=

4 8 10 14 2 16 18 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[211,3,2]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-8] Hyperbolic Volume 10.6207 A-Polynomial See Data:9 22/A-polynomial

[edit Notes for 9 22'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 3} Rasmussen s-Invariant -2

[edit Notes for 9 22'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 t^3-5 t^2+10 t-11+10 t^{-1} -5 t^{-2} + t^{-3} }
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 z^6+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	{ 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^6+3 q^5-5 q^4+7 q^3-7 q^2+7 q-6+4 q^{-1} -2 q^{-2} + q^{-3} }
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^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+6 z^2 a^{-2} -2 z^2 a^{-4} -6 z^2+2 a^2+4 a^{-2} - a^{-4} -4}
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}+2az^{7}+6z^{7}a^{-1}+4z^{7}a^{-3}+a^{2}z^{6}+7z^{6}a^{-2}+6z^{6}a^{-4}+2z^{6}-7az^{5}-16z^{5}a^{-1}-4z^{5}a^{-3}+5z^{5}a^{-5}-4a^{2}z^{4}-23z^{4}a^{-2}-9z^{4}a^{-4}+3z^{4}a^{-6}-15z^{4}+7az^{3}+10z^{3}a^{-1}-2z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+5a^{2}z^{2}+17z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}+16z^{2}-2az-2za^{-1}+za^{-3}+za^{-5}-2a^{2}-4a^{-2}-a^{-4}-4$
The A2 invariant	$q^{10}+q^{8}+q^{4}-2q^{2}-1-q^{-4}+3q^{-6}+2q^{-10}-q^{-14}+q^{-16}-q^{-18}$
The G2 invariant	$q^{46}-q^{44}+4q^{42}-5q^{40}+5q^{38}-3q^{36}-3q^{34}+14q^{32}-20q^{30}+25q^{28}-17q^{26}+2q^{24}+19q^{22}-35q^{20}+43q^{18}-35q^{16}+14q^{14}+11q^{12}-35q^{10}+41q^{8}-32q^{6}+10q^{4}+12q^{2}-28+24q^{-2}-14q^{-4}-11q^{-6}+29q^{-8}-38q^{-10}+31q^{-12}-9q^{-14}-20q^{-16}+46q^{-18}-56q^{-20}+50q^{-22}-25q^{-24}-5q^{-26}+37q^{-28}-52q^{-30}+54q^{-32}-30q^{-34}+5q^{-36}+23q^{-38}-34q^{-40}+28q^{-42}-9q^{-44}-11q^{-46}+25q^{-48}-25q^{-50}+12q^{-52}+8q^{-54}-28q^{-56}+36q^{-58}-33q^{-60}+17q^{-62}+2q^{-64}-21q^{-66}+28q^{-68}-29q^{-70}+24q^{-72}-11q^{-74}+8q^{-78}-14q^{-80}+14q^{-82}-11q^{-84}+8q^{-86}-2q^{-88}-q^{-90}+2q^{-92}-4q^{-94}+3q^{-96}-2q^{-98}+q^{-100}$

Further Quantum Invariants

Further quantum knot invariants for 9_22.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+2q^{3}-2q+q^{-1}+2q^{-7}-2q^{-9}+2q^{-11}-q^{-13}$
2	$q^{22}-q^{20}-2q^{18}+4q^{16}-7q^{12}+6q^{10}+5q^{8}-10q^{6}+3q^{4}+9q^{2}-8-2q^{-2}+8q^{-4}-2q^{-6}-5q^{-8}+3q^{-10}+5q^{-12}-5q^{-14}-5q^{-16}+10q^{-18}-2q^{-20}-9q^{-22}+10q^{-24}+q^{-26}-7q^{-28}+4q^{-30}-2q^{-34}+q^{-36}$
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^{45}-q^{43}-2 q^{41}+5 q^{37}+2 q^{35}-8 q^{33}-7 q^{31}+10 q^{29}+15 q^{27}-7 q^{25}-24 q^{23}+29 q^{19}+12 q^{17}-29 q^{15}-25 q^{13}+25 q^{11}+33 q^9-14 q^7-39 q^5+4 q^3+41 q+6 q^{-1} -37 q^{-3} -14 q^{-5} +33 q^{-7} +21 q^{-9} -26 q^{-11} -25 q^{-13} +19 q^{-15} +27 q^{-17} -6 q^{-19} -29 q^{-21} -7 q^{-23} +26 q^{-25} +21 q^{-27} -21 q^{-29} -34 q^{-31} +12 q^{-33} +41 q^{-35} - q^{-37} -42 q^{-39} -5 q^{-41} +35 q^{-43} +10 q^{-45} -25 q^{-47} -10 q^{-49} +16 q^{-51} +7 q^{-53} -10 q^{-55} -2 q^{-57} +3 q^{-59} + q^{-61} -2 q^{-63} +2 q^{-67} - q^{-69} }
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^{76}-q^{74}-2 q^{72}+q^{68}+7 q^{66}-8 q^{62}-8 q^{60}-5 q^{58}+22 q^{56}+18 q^{54}-5 q^{52}-28 q^{50}-41 q^{48}+19 q^{46}+51 q^{44}+44 q^{42}-13 q^{40}-95 q^{38}-46 q^{36}+32 q^{34}+108 q^{32}+78 q^{30}-79 q^{28}-121 q^{26}-74 q^{24}+89 q^{22}+172 q^{20}+30 q^{18}-106 q^{16}-177 q^{14}-20 q^{12}+172 q^{10}+138 q^8-13 q^6-194 q^4-119 q^2+99+172 q^{-2} +70 q^{-4} -149 q^{-6} -156 q^{-8} +29 q^{-10} +160 q^{-12} +102 q^{-14} -102 q^{-16} -152 q^{-18} -16 q^{-20} +133 q^{-22} +114 q^{-24} -44 q^{-26} -137 q^{-28} -75 q^{-30} +81 q^{-32} +126 q^{-34} +53 q^{-36} -87 q^{-38} -148 q^{-40} -30 q^{-42} +100 q^{-44} +171 q^{-46} +25 q^{-48} -171 q^{-50} -152 q^{-52} +4 q^{-54} +212 q^{-56} +142 q^{-58} -100 q^{-60} -184 q^{-62} -95 q^{-64} +142 q^{-66} +159 q^{-68} -8 q^{-70} -106 q^{-72} -107 q^{-74} +48 q^{-76} +89 q^{-78} +18 q^{-80} -28 q^{-82} -56 q^{-84} +10 q^{-86} +26 q^{-88} +4 q^{-90} +2 q^{-92} -17 q^{-94} +3 q^{-96} +5 q^{-98} -2 q^{-100} +3 q^{-102} -3 q^{-104} +2 q^{-106} -2 q^{-110} + q^{-112} }
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^{115}-q^{113}-2 q^{111}+q^{107}+3 q^{105}+5 q^{103}-10 q^{99}-10 q^{97}-3 q^{95}+9 q^{93}+24 q^{91}+21 q^{89}-8 q^{87}-41 q^{85}-47 q^{83}-16 q^{81}+44 q^{79}+90 q^{77}+69 q^{75}-20 q^{73}-120 q^{71}-146 q^{69}-57 q^{67}+105 q^{65}+222 q^{63}+186 q^{61}-16 q^{59}-251 q^{57}-324 q^{55}-155 q^{53}+173 q^{51}+423 q^{49}+375 q^{47}+12 q^{45}-413 q^{43}-563 q^{41}-286 q^{39}+257 q^{37}+662 q^{35}+579 q^{33}+12 q^{31}-616 q^{29}-795 q^{27}-352 q^{25}+419 q^{23}+903 q^{21}+668 q^{19}-136 q^{17}-856 q^{15}-894 q^{13}-189 q^{11}+700 q^9+1010 q^7+464 q^5-482 q^3-1003 q-655 q^{-1} +255 q^{-3} +925 q^{-5} +753 q^{-7} -78 q^{-9} -803 q^{-11} -765 q^{-13} -37 q^{-15} +679 q^{-17} +733 q^{-19} +97 q^{-21} -587 q^{-23} -679 q^{-25} -126 q^{-27} +520 q^{-29} +642 q^{-31} +153 q^{-33} -461 q^{-35} -629 q^{-37} -222 q^{-39} +383 q^{-41} +638 q^{-43} +332 q^{-45} -242 q^{-47} -624 q^{-49} -507 q^{-51} +28 q^{-53} +568 q^{-55} +687 q^{-57} +264 q^{-59} -413 q^{-61} -829 q^{-63} -605 q^{-65} +162 q^{-67} +876 q^{-69} +912 q^{-71} +168 q^{-73} -782 q^{-75} -1123 q^{-77} -518 q^{-79} +571 q^{-81} +1183 q^{-83} +780 q^{-85} -275 q^{-87} -1073 q^{-89} -926 q^{-91} -8 q^{-93} +845 q^{-95} +911 q^{-97} +213 q^{-99} -565 q^{-101} -768 q^{-103} -311 q^{-105} +314 q^{-107} +568 q^{-109} +309 q^{-111} -146 q^{-113} -362 q^{-115} -237 q^{-117} +40 q^{-119} +203 q^{-121} +162 q^{-123} -2 q^{-125} -110 q^{-127} -82 q^{-129} -9 q^{-131} +43 q^{-133} +45 q^{-135} +8 q^{-137} -21 q^{-139} -19 q^{-141} +8 q^{-145} +4 q^{-147} +2 q^{-149} - q^{-151} -4 q^{-153} - q^{-155} +3 q^{-157} -2 q^{-159} +2 q^{-163} - q^{-165} }

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^{10}+q^8+q^4-2 q^2-1- q^{-4} +3 q^{-6} +2 q^{-10} - q^{-14} + q^{-16} - q^{-18} }
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^{28}-2 q^{26}+8 q^{24}-16 q^{22}+31 q^{20}-54 q^{18}+78 q^{16}-106 q^{14}+126 q^{12}-140 q^{10}+138 q^8-110 q^6+76 q^4-16 q^2-44+112 q^{-2} -171 q^{-4} +214 q^{-6} -248 q^{-8} +250 q^{-10} -236 q^{-12} +200 q^{-14} -148 q^{-16} +90 q^{-18} -24 q^{-20} -26 q^{-22} +70 q^{-24} -96 q^{-26} +108 q^{-28} -108 q^{-30} +98 q^{-32} -86 q^{-34} +70 q^{-36} -54 q^{-38} +42 q^{-40} -32 q^{-42} +20 q^{-44} -12 q^{-46} +8 q^{-48} -4 q^{-50} + q^{-52} }
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^{28}+q^{26}-2 q^{22}+2 q^{18}-q^{16}-5 q^{14}-q^{12}+4 q^{10}+2 q^8-q^6+4 q^4+7 q^2-1-3 q^{-2} + q^{-4} -2 q^{-6} -5 q^{-8} + q^{-12} -2 q^{-14} +6 q^{-18} + q^{-20} -5 q^{-22} +3 q^{-24} +5 q^{-26} -2 q^{-28} -3 q^{-30} +3 q^{-32} + q^{-34} -2 q^{-36} -2 q^{-38} + q^{-40} - q^{-44} + q^{-46} }

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^{20}-q^{18}+2 q^{16}+2 q^{14}-3 q^{12}+4 q^{10}+q^8-7 q^6+5 q^4-8+6 q^{-2} +2 q^{-4} -6 q^{-6} +3 q^{-8} +4 q^{-10} - q^{-14} +2 q^{-16} +5 q^{-18} -5 q^{-20} -2 q^{-22} +8 q^{-24} -7 q^{-26} -3 q^{-28} +8 q^{-30} -4 q^{-32} -3 q^{-34} +5 q^{-36} - q^{-38} -2 q^{-40} + q^{-42} }
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^{13}+q^{11}+2 q^9+q^5-3 q^3-q-3 q^{-1} +2 q^{-7} +3 q^{-9} + q^{-11} +2 q^{-13} - q^{-15} + q^{-17} -2 q^{-19} + q^{-21} - q^{-23} }

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^{26}+q^{24}+q^{22}+2 q^{20}+2 q^{18}-q^{14}-2 q^{10}-6 q^8-q^6+4 q^4-2 q^2-1+9 q^{-2} +6 q^{-4} -5 q^{-6} -2 q^{-8} +4 q^{-10} -4 q^{-12} -8 q^{-14} +3 q^{-16} +6 q^{-18} -4 q^{-20} +4 q^{-22} +9 q^{-24} -2 q^{-26} -4 q^{-28} +5 q^{-30} + q^{-32} -7 q^{-34} - q^{-36} +5 q^{-38} - q^{-40} -5 q^{-42} +2 q^{-44} +3 q^{-46} -2 q^{-48} - q^{-50} + q^{-52} }
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^{16}+q^{14}+2 q^{12}+2 q^{10}+q^6-3 q^4-2 q^2-3-3 q^{-2} +3 q^{-8} +2 q^{-10} +4 q^{-12} + q^{-14} +2 q^{-16} - q^{-18} -2 q^{-24} + q^{-26} - q^{-28} }

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-q^{18}+4q^{16}-4q^{14}+7q^{12}-8q^{10}+9q^{8}-9q^{6}+7q^{4}-6q^{2}+2q^{-2}-8q^{-4}+12q^{-6}-15q^{-8}+18q^{-10}-16q^{-12}+17q^{-14}-12q^{-16}+9q^{-18}-3q^{-20}+4q^{-24}-7q^{-26}+9q^{-28}-10q^{-30}+8q^{-32}-7q^{-34}+5q^{-36}-3q^{-38}+2q^{-40}-q^{-42}$
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^{34}-q^{30}-q^{28}+3 q^{26}+3 q^{24}-2 q^{22}-5 q^{20}+7 q^{16}+4 q^{14}-7 q^{12}-8 q^{10}+3 q^8+10 q^6+2 q^4-10 q^2-5+6 q^{-2} +8 q^{-4} -2 q^{-6} -7 q^{-8} +6 q^{-12} + q^{-14} -6 q^{-16} - q^{-18} +6 q^{-20} +4 q^{-22} -5 q^{-24} -5 q^{-26} +4 q^{-28} +8 q^{-30} -2 q^{-32} -8 q^{-34} - q^{-36} +9 q^{-38} +4 q^{-40} -7 q^{-42} -8 q^{-44} +3 q^{-46} +9 q^{-48} + q^{-50} -6 q^{-52} -5 q^{-54} +2 q^{-56} +5 q^{-58} + q^{-60} -2 q^{-62} -2 q^{-64} + q^{-68} }

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{26}-q^{24}+3 q^{22}-2 q^{20}+6 q^{18}-4 q^{16}+7 q^{14}-6 q^{12}+8 q^{10}-8 q^8+4 q^6-7 q^4+2 q^2-3-4 q^{-2} +3 q^{-4} -6 q^{-6} +9 q^{-8} -10 q^{-10} +15 q^{-12} -10 q^{-14} +16 q^{-16} -11 q^{-18} +13 q^{-20} -8 q^{-22} +9 q^{-24} -6 q^{-26} + q^{-28} - q^{-30} -2 q^{-32} +4 q^{-34} -7 q^{-36} +5 q^{-38} -7 q^{-40} +9 q^{-42} -6 q^{-44} +4 q^{-46} -5 q^{-48} +5 q^{-50} -2 q^{-52} + q^{-54} -2 q^{-56} + q^{-58} }

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^{46}-q^{44}+4 q^{42}-5 q^{40}+5 q^{38}-3 q^{36}-3 q^{34}+14 q^{32}-20 q^{30}+25 q^{28}-17 q^{26}+2 q^{24}+19 q^{22}-35 q^{20}+43 q^{18}-35 q^{16}+14 q^{14}+11 q^{12}-35 q^{10}+41 q^8-32 q^6+10 q^4+12 q^2-28+24 q^{-2} -14 q^{-4} -11 q^{-6} +29 q^{-8} -38 q^{-10} +31 q^{-12} -9 q^{-14} -20 q^{-16} +46 q^{-18} -56 q^{-20} +50 q^{-22} -25 q^{-24} -5 q^{-26} +37 q^{-28} -52 q^{-30} +54 q^{-32} -30 q^{-34} +5 q^{-36} +23 q^{-38} -34 q^{-40} +28 q^{-42} -9 q^{-44} -11 q^{-46} +25 q^{-48} -25 q^{-50} +12 q^{-52} +8 q^{-54} -28 q^{-56} +36 q^{-58} -33 q^{-60} +17 q^{-62} +2 q^{-64} -21 q^{-66} +28 q^{-68} -29 q^{-70} +24 q^{-72} -11 q^{-74} +8 q^{-78} -14 q^{-80} +14 q^{-82} -11 q^{-84} +8 q^{-86} -2 q^{-88} - q^{-90} +2 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 22"];

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 t^3-5 t^2+10 t-11+10 t^{-1} -5 t^{-2} + t^{-3} }

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 z^6+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]=

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

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-2}+z^{8}+2az^{7}+6z^{7}a^{-1}+4z^{7}a^{-3}+a^{2}z^{6}+7z^{6}a^{-2}+6z^{6}a^{-4}+2z^{6}-7az^{5}-16z^{5}a^{-1}-4z^{5}a^{-3}+5z^{5}a^{-5}-4a^{2}z^{4}-23z^{4}a^{-2}-9z^{4}a^{-4}+3z^{4}a^{-6}-15z^{4}+7az^{3}+10z^{3}a^{-1}-2z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+5a^{2}z^{2}+17z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}+16z^{2}-2az-2za^{-1}+za^{-3}+za^{-5}-2a^{2}-4a^{-2}-a^{-4}-4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n128,}

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}} ): {K11n3,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 22"];

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 t^3-5 t^2+10 t-11+10 t^{-1} -5 t^{-2} + t^{-3} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+3 q^5-5 q^4+7 q^3-7 q^2+7 q-6+4 q^{-1} -2 q^{-2} + q^{-3} } }

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]=

{K11n128,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n3,}

Vassiliev invariants

V2 and V3:

(-1, 1)

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} $8$ 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 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{34}{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{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 -32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{208}{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{160}{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 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{32}{3}} $32$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{136}{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{40}{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{751}{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{766}{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 \frac{7018}{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{305}{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{1169}{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 9 22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 13                   1 -1 11                 2   2 9               3 1   -2 7             4 2     2 5           3 3       0 3         4 4         0 1       3 4           1 -1     1 3             -2 -3   1 3               2 -5   1                 -1 -7 1                   1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=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 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 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}} Failed to parse (SVG (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}\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}^{3}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-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}^{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=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}^{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}^{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}^{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=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{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=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}\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=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} Failed to parse (SVG (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^{17}-3 q^{16}+2 q^{15}+5 q^{14}-14 q^{13}+10 q^{12}+14 q^{11}-33 q^{10}+17 q^9+26 q^8-48 q^7+17 q^6+36 q^5-50 q^4+9 q^3+39 q^2-40 q-1+33 q^{-1} -23 q^{-2} -7 q^{-3} +20 q^{-4} -8 q^{-5} -6 q^{-6} +7 q^{-7} - q^{-8} -2 q^{-9} + 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^{33}+3 q^{32}-2 q^{31}-2 q^{30}+2 q^{29}+5 q^{28}-7 q^{27}-10 q^{26}+19 q^{25}+14 q^{24}-33 q^{23}-25 q^{22}+54 q^{21}+39 q^{20}-73 q^{19}-62 q^{18}+95 q^{17}+81 q^{16}-102 q^{15}-108 q^{14}+108 q^{13}+123 q^{12}-97 q^{11}-141 q^{10}+86 q^9+146 q^8-64 q^7-149 q^6+42 q^5+145 q^4-17 q^3-137 q^2-5 q+122+26 q^{-1} -102 q^{-2} -42 q^{-3} +79 q^{-4} +51 q^{-5} -55 q^{-6} -50 q^{-7} +29 q^{-8} +47 q^{-9} -14 q^{-10} -33 q^{-11} +23 q^{-13} +3 q^{-14} -11 q^{-15} -5 q^{-16} +6 q^{-17} +2 q^{-18} - q^{-19} -2 q^{-20} + 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^{54}-3 q^{53}+2 q^{52}+2 q^{51}-5 q^{50}+7 q^{49}-8 q^{48}+9 q^{47}-25 q^{45}+26 q^{44}-6 q^{43}+31 q^{42}-16 q^{41}-91 q^{40}+54 q^{39}+40 q^{38}+102 q^{37}-57 q^{36}-246 q^{35}+55 q^{34}+138 q^{33}+269 q^{32}-74 q^{31}-483 q^{30}-34 q^{29}+222 q^{28}+511 q^{27}-4 q^{26}-691 q^{25}-190 q^{24}+203 q^{23}+707 q^{22}+142 q^{21}-762 q^{20}-320 q^{19}+85 q^{18}+768 q^{17}+282 q^{16}-689 q^{15}-365 q^{14}-71 q^{13}+706 q^{12}+375 q^{11}-531 q^{10}-346 q^9-220 q^8+570 q^7+425 q^6-327 q^5-288 q^4-351 q^3+385 q^2+432 q-108-186 q^{-1} -424 q^{-2} +167 q^{-3} +357 q^{-4} +73 q^{-5} -35 q^{-6} -390 q^{-7} -25 q^{-8} +200 q^{-9} +144 q^{-10} +101 q^{-11} -248 q^{-12} -108 q^{-13} +37 q^{-14} +97 q^{-15} +143 q^{-16} -91 q^{-17} -78 q^{-18} -39 q^{-19} +19 q^{-20} +94 q^{-21} -9 q^{-22} -21 q^{-23} -32 q^{-24} -13 q^{-25} +34 q^{-26} +4 q^{-27} +2 q^{-28} -9 q^{-29} -9 q^{-30} +7 q^{-31} + q^{-32} +2 q^{-33} - q^{-34} -2 q^{-35} + 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^{80}+3 q^{79}-2 q^{78}-2 q^{77}+5 q^{76}-4 q^{75}-4 q^{74}+6 q^{73}+q^{72}+9 q^{70}-12 q^{69}-23 q^{68}+4 q^{67}+30 q^{66}+37 q^{65}+7 q^{64}-64 q^{63}-96 q^{62}-24 q^{61}+138 q^{60}+201 q^{59}+48 q^{58}-227 q^{57}-373 q^{56}-149 q^{55}+354 q^{54}+656 q^{53}+307 q^{52}-481 q^{51}-998 q^{50}-606 q^{49}+557 q^{48}+1434 q^{47}+1005 q^{46}-547 q^{45}-1851 q^{44}-1524 q^{43}+410 q^{42}+2232 q^{41}+2060 q^{40}-144 q^{39}-2463 q^{38}-2613 q^{37}-195 q^{36}+2573 q^{35}+3010 q^{34}+600 q^{33}-2499 q^{32}-3327 q^{31}-962 q^{30}+2349 q^{29}+3426 q^{28}+1277 q^{27}-2076 q^{26}-3446 q^{25}-1502 q^{24}+1814 q^{23}+3309 q^{22}+1659 q^{21}-1502 q^{20}-3140 q^{19}-1757 q^{18}+1209 q^{17}+2902 q^{16}+1827 q^{15}-888 q^{14}-2651 q^{13}-1879 q^{12}+563 q^{11}+2349 q^{10}+1919 q^9-204 q^8-2015 q^7-1933 q^6-153 q^5+1621 q^4+1881 q^3+521 q^2-1184 q-1761-823 q^{-1} +711 q^{-2} +1533 q^{-3} +1042 q^{-4} -238 q^{-5} -1215 q^{-6} -1133 q^{-7} -178 q^{-8} +828 q^{-9} +1080 q^{-10} +482 q^{-11} -411 q^{-12} -898 q^{-13} -662 q^{-14} +57 q^{-15} +637 q^{-16} +661 q^{-17} +217 q^{-18} -331 q^{-19} -579 q^{-20} -348 q^{-21} +94 q^{-22} +384 q^{-23} +367 q^{-24} +94 q^{-25} -216 q^{-26} -300 q^{-27} -156 q^{-28} +56 q^{-29} +198 q^{-30} +167 q^{-31} +19 q^{-32} -98 q^{-33} -120 q^{-34} -61 q^{-35} +36 q^{-36} +78 q^{-37} +45 q^{-38} +2 q^{-39} -31 q^{-40} -40 q^{-41} -10 q^{-42} +18 q^{-43} +14 q^{-44} +8 q^{-45} +2 q^{-46} -11 q^{-47} -7 q^{-48} +3 q^{-49} +2 q^{-50} + q^{-51} +2 q^{-52} - q^{-53} -2 q^{-54} + q^{-55} }
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^{111}-3 q^{110}+2 q^{109}+2 q^{108}-5 q^{107}+4 q^{106}+q^{105}+6 q^{104}-16 q^{103}-q^{102}+16 q^{101}-17 q^{100}+17 q^{99}+13 q^{98}+5 q^{97}-59 q^{96}-21 q^{95}+51 q^{94}-5 q^{93}+79 q^{92}+55 q^{91}-43 q^{90}-234 q^{89}-130 q^{88}+138 q^{87}+148 q^{86}+370 q^{85}+216 q^{84}-239 q^{83}-806 q^{82}-626 q^{81}+211 q^{80}+689 q^{79}+1358 q^{78}+895 q^{77}-572 q^{76}-2210 q^{75}-2180 q^{74}-236 q^{73}+1653 q^{72}+3649 q^{71}+2956 q^{70}-391 q^{69}-4448 q^{68}-5493 q^{67}-2268 q^{66}+2177 q^{65}+7078 q^{64}+7070 q^{63}+1559 q^{62}-6289 q^{61}-10064 q^{60}-6488 q^{59}+787 q^{58}+9980 q^{57}+12336 q^{56}+5755 q^{55}-6027 q^{54}-13807 q^{53}-11655 q^{52}-2849 q^{51}+10506 q^{50}+16389 q^{49}+10695 q^{48}-3445 q^{47}-14891 q^{46}-15386 q^{45}-7086 q^{44}+8624 q^{43}+17616 q^{42}+14128 q^{41}-185 q^{40}-13507 q^{39}-16448 q^{38}-9964 q^{37}+5922 q^{36}+16471 q^{35}+15219 q^{34}+2167 q^{33}-11149 q^{32}-15521 q^{31}-11055 q^{30}+3634 q^{29}+14349 q^{28}+14794 q^{27}+3509 q^{26}-8770 q^{25}-13869 q^{24}-11249 q^{23}+1687 q^{22}+11987 q^{21}+13921 q^{20}+4662 q^{19}-6221 q^{18}-11990 q^{17}-11359 q^{16}-578 q^{15}+9173 q^{14}+12821 q^{13}+6111 q^{12}-3022 q^{11}-9519 q^{10}-11259 q^9-3325 q^8+5476 q^7+10899 q^6+7376 q^5+755 q^4-5958 q^3-10084 q^2-5760 q+1096+7547 q^{-1} +7321 q^{-2} +4068 q^{-3} -1572 q^{-4} -7131 q^{-5} -6524 q^{-6} -2724 q^{-7} +3124 q^{-8} +5174 q^{-9} +5400 q^{-10} +2215 q^{-11} -2922 q^{-12} -4862 q^{-13} -4343 q^{-14} -724 q^{-15} +1641 q^{-16} +4096 q^{-17} +3710 q^{-18} +667 q^{-19} -1735 q^{-20} -3270 q^{-21} -2272 q^{-22} -1239 q^{-23} +1390 q^{-24} +2652 q^{-25} +1967 q^{-26} +699 q^{-27} -980 q^{-28} -1493 q^{-29} -1986 q^{-30} -556 q^{-31} +696 q^{-32} +1233 q^{-33} +1215 q^{-34} +485 q^{-35} -97 q^{-36} -1132 q^{-37} -848 q^{-38} -376 q^{-39} +151 q^{-40} +549 q^{-41} +589 q^{-42} +486 q^{-43} -222 q^{-44} -331 q^{-45} -378 q^{-46} -224 q^{-47} -10 q^{-48} +195 q^{-49} +340 q^{-50} +65 q^{-51} +9 q^{-52} -108 q^{-53} -126 q^{-54} -109 q^{-55} -10 q^{-56} +110 q^{-57} +36 q^{-58} +47 q^{-59} +2 q^{-60} -19 q^{-61} -48 q^{-62} -25 q^{-63} +22 q^{-64} + q^{-65} +15 q^{-66} +7 q^{-67} +5 q^{-68} -11 q^{-69} -9 q^{-70} +5 q^{-71} -2 q^{-72} +2 q^{-73} + q^{-74} +2 q^{-75} - q^{-76} -2 q^{-77} + q^{-78} }
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^{147}+3 q^{146}-2 q^{145}-2 q^{144}+5 q^{143}-4 q^{142}-q^{141}-3 q^{140}+4 q^{139}+16 q^{138}-15 q^{137}-8 q^{136}+12 q^{135}-13 q^{134}+2 q^{133}-q^{132}+20 q^{131}+46 q^{130}-49 q^{129}-51 q^{128}-8 q^{127}-28 q^{126}+62 q^{125}+73 q^{124}+91 q^{123}+92 q^{122}-189 q^{121}-268 q^{120}-197 q^{119}-63 q^{118}+376 q^{117}+542 q^{116}+472 q^{115}+158 q^{114}-737 q^{113}-1208 q^{112}-1078 q^{111}-286 q^{110}+1401 q^{109}+2345 q^{108}+2165 q^{107}+688 q^{106}-2240 q^{105}-4243 q^{104}-4291 q^{103}-1697 q^{102}+3385 q^{101}+7198 q^{100}+7661 q^{99}+3690 q^{98}-4322 q^{97}-11066 q^{96}-12903 q^{95}-7506 q^{94}+4628 q^{93}+15832 q^{92}+20120 q^{91}+13474 q^{90}-3445 q^{89}-20589 q^{88}-29035 q^{87}-22218 q^{86}-134 q^{85}+24548 q^{84}+39027 q^{83}+33387 q^{82}+6506 q^{81}-26403 q^{80}-48697 q^{79}-46300 q^{78}-15913 q^{77}+25355 q^{76}+56781 q^{75}+59556 q^{74}+27610 q^{73}-21019 q^{72}-61992 q^{71}-71665 q^{70}-40304 q^{69}+13813 q^{68}+63582 q^{67}+81127 q^{66}+52649 q^{65}-4618 q^{64}-61892 q^{63}-87234 q^{62}-62947 q^{61}-4978 q^{60}+57312 q^{59}+89586 q^{58}+70609 q^{57}+13971 q^{56}-51320 q^{55}-89026 q^{54}-75006 q^{53}-21070 q^{52}+44775 q^{51}+86065 q^{50}+76696 q^{49}+26316 q^{48}-38744 q^{47}-82094 q^{46}-76257 q^{45}-29468 q^{44}+33541 q^{43}+77532 q^{42}+74601 q^{41}+31433 q^{40}-29178 q^{39}-73186 q^{38}-72451 q^{37}-32606 q^{36}+25279 q^{35}+68923 q^{34}+70322 q^{33}+33847 q^{32}-21376 q^{31}-64741 q^{30}-68373 q^{29}-35479 q^{28}+16944 q^{27}+60183 q^{26}+66589 q^{25}+37742 q^{24}-11653 q^{23}-54911 q^{22}-64609 q^{21}-40506 q^{20}+5296 q^{19}+48518 q^{18}+62011 q^{17}+43492 q^{16}+2025 q^{15}-40817 q^{14}-58308 q^{13}-46079 q^{12}-9919 q^{11}+31642 q^{10}+53001 q^9+47681 q^8+17915 q^7-21368 q^6-45901 q^5-47466 q^4-25050 q^3+10375 q^2+36815 q+45006+30590 q^{-1} +395 q^{-2} -26306 q^{-3} -39906 q^{-4} -33501 q^{-5} -9964 q^{-6} +14992 q^{-7} +32305 q^{-8} +33373 q^{-9} +17267 q^{-10} -4072 q^{-11} -22958 q^{-12} -30013 q^{-13} -21349 q^{-14} -5224 q^{-15} +12850 q^{-16} +23874 q^{-17} +21982 q^{-18} +11861 q^{-19} -3461 q^{-20} -16144 q^{-21} -19327 q^{-22} -15020 q^{-23} -3962 q^{-24} +8009 q^{-25} +14268 q^{-26} +14978 q^{-27} +8606 q^{-28} -1119 q^{-29} -8295 q^{-30} -12166 q^{-31} -10059 q^{-32} -3738 q^{-33} +2545 q^{-34} +7993 q^{-35} +9113 q^{-36} +5977 q^{-37} +1579 q^{-38} -3635 q^{-39} -6417 q^{-40} -5987 q^{-41} -3895 q^{-42} +182 q^{-43} +3414 q^{-44} +4533 q^{-45} +4272 q^{-46} +1836 q^{-47} -820 q^{-48} -2521 q^{-49} -3480 q^{-50} -2469 q^{-51} -758 q^{-52} +744 q^{-53} +2154 q^{-54} +2148 q^{-55} +1365 q^{-56} +360 q^{-57} -939 q^{-58} -1363 q^{-59} -1246 q^{-60} -849 q^{-61} +103 q^{-62} +659 q^{-63} +856 q^{-64} +795 q^{-65} +232 q^{-66} -125 q^{-67} -387 q^{-68} -584 q^{-69} -341 q^{-70} -90 q^{-71} +141 q^{-72} +321 q^{-73} +210 q^{-74} +143 q^{-75} +47 q^{-76} -137 q^{-77} -145 q^{-78} -124 q^{-79} -47 q^{-80} +59 q^{-81} +40 q^{-82} +53 q^{-83} +62 q^{-84} +2 q^{-85} -20 q^{-86} -40 q^{-87} -30 q^{-88} +10 q^{-89} -2 q^{-90} +3 q^{-91} +17 q^{-92} +7 q^{-93} +4 q^{-94} -8 q^{-95} -9 q^{-96} +3 q^{-97} -2 q^{-99} +2 q^{-100} + q^{-101} +2 q^{-102} - q^{-103} -2 q^{-104} + q^{-105} }

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.

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