10 9

10_8

10_10

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 9's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

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[edit 10 9 Quick Notes]

[edit 10 9 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_16,8,17,7 X_12,3,13,4 X_2,15,3,16 X_14,5,15,6 X_4,13,5,14 X_18,10,19,9 X_20,12,1,11 X_8,18,9,17 X_10,20,11,19
Gauss code	1, -4, 3, -6, 5, -1, 2, -9, 7, -10, 8, -3, 6, -5, 4, -2, 9, -7, 10, -8
Dowker-Thistlethwaite code	6 12 14 16 18 20 4 2 8 10
Conway Notation	[5113]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 10, width is 3,

Braid index is 3

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

[edit Notes on presentations of 10 9]

Computer Talk

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

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

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 9"];

In[4]:= PD[K]

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

Out[4]= X₆₂₇₁ X_16,8,17,7 X_12,3,13,4 X_2,15,3,16 X_14,5,15,6 X_4,13,5,14 X_18,10,19,9 X_20,12,1,11 X_8,18,9,17 X_10,20,11,19

In[5]:= GaussCode[K]

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

In[6]:= DTCode[K]

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

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [5113]

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(3,\{1,1,1,1,1,-2,1,-2,-2,-2\})}

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

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

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

Out[10]= { 3, 10, 3 }

In[11]:= Show[BraidPlot[br]]

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

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

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

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{2, 12}, {1, 7}, {11, 6}, {12, 8}, {7, 5}, {6, 4}, {5, 3}, {4, 2}, {3, 9}, {8, 10}, {9, 11}, {10, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	4
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-3][-9]
Hyperbolic Volume	8.2941
A-Polynomial	See Data:10 9/A-polynomial

[edit Notes for 10 9's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 4}
Rasmussen s-Invariant	-2

[edit Notes for 10 9'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^4+3 t^3-5 t^2+7 t-7+7 t^{-1} -5 t^{-2} +3 t^{-3} - t^{-4} }
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^8-5 z^6-7 z^4-2 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	{ 39, 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^7-2 q^6+3 q^5-5 q^4+6 q^3-6 q^2+6 q-4+3 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^8 a^{-2} -7 z^6 a^{-2} +z^6 a^{-4} +z^6-17 z^4 a^{-2} +5 z^4 a^{-4} +5 z^4-16 z^2 a^{-2} +7 z^2 a^{-4} +7 z^2-4 a^{-2} +2 a^{-4} +3}
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+2z^{8}a^{-4}+2z^{8}+2az^{7}-2z^{7}a^{-1}-2z^{7}a^{-3}+2z^{7}a^{-5}+a^{2}z^{6}-18z^{6}a^{-2}-7z^{6}a^{-4}+2z^{6}a^{-6}-8z^{6}-8az^{5}-2z^{5}a^{-1}-4z^{5}a^{-5}+2z^{5}a^{-7}-4a^{2}z^{4}+31z^{4}a^{-2}+13z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+10z^{4}+7az^{3}+4z^{3}a^{-1}+5z^{3}a^{-3}+4z^{3}a^{-5}-4z^{3}a^{-7}+3a^{2}z^{2}-22z^{2}a^{-2}-8z^{2}a^{-4}+z^{2}a^{-6}-2z^{2}a^{-8}-8z^{2}-az-2za^{-1}-2za^{-3}+za^{-7}+4a^{-2}+2a^{-4}+3$
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^8+q^4+ q^{-2} - q^{-4} +2 q^{-6} - q^{-8} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }
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^{46}-q^{44}+2 q^{42}-3 q^{40}+2 q^{38}-2 q^{36}-q^{34}+6 q^{32}-8 q^{30}+10 q^{28}-9 q^{26}+5 q^{24}+2 q^{22}-10 q^{20}+16 q^{18}-16 q^{16}+14 q^{14}-4 q^{12}-5 q^{10}+15 q^8-14 q^6+12 q^4-4 q^2-4+10 q^{-2} -9 q^{-4} +3 q^{-6} +6 q^{-8} -10 q^{-10} +14 q^{-12} -9 q^{-14} -2 q^{-16} +8 q^{-18} -16 q^{-20} +18 q^{-22} -17 q^{-24} +7 q^{-26} +3 q^{-28} -13 q^{-30} +18 q^{-32} -18 q^{-34} +11 q^{-36} -3 q^{-38} -6 q^{-40} +10 q^{-42} -12 q^{-44} +8 q^{-46} -5 q^{-50} +6 q^{-52} -4 q^{-54} -2 q^{-56} +7 q^{-58} -8 q^{-60} +7 q^{-62} -4 q^{-64} - q^{-66} +6 q^{-68} -8 q^{-70} +11 q^{-72} -8 q^{-74} +6 q^{-76} - q^{-78} -2 q^{-80} +5 q^{-82} -8 q^{-84} +10 q^{-86} -7 q^{-88} +4 q^{-90} -4 q^{-94} +6 q^{-96} -6 q^{-98} +5 q^{-100} -3 q^{-102} + q^{-106} -3 q^{-108} +2 q^{-110} - q^{-112} + q^{-114} }

Further Quantum Invariants

Further quantum knot invariants for 10_9.

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^7-q^5+q^3-q+2 q^{-1} + q^{-7} -2 q^{-9} + q^{-11} - q^{-13} + q^{-15} }
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^{22}-q^{20}-q^{18}+3 q^{16}-q^{14}-3 q^{12}+4 q^{10}+q^8-5 q^6+2 q^4+3 q^2-4+ q^{-2} +4 q^{-4} -2 q^{-6} - q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} - q^{-16} +4 q^{-18} -3 q^{-20} -3 q^{-22} +5 q^{-24} -2 q^{-26} -2 q^{-28} +3 q^{-30} - q^{-32} - q^{-34} +2 q^{-36} - q^{-38} - q^{-40} + q^{-42} }
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}-q^{41}+q^{39}+2 q^{37}-q^{35}-4 q^{33}+q^{31}+6 q^{29}-7 q^{25}-3 q^{23}+8 q^{21}+7 q^{19}-6 q^{17}-9 q^{15}+2 q^{13}+9 q^{11}+2 q^9-9 q^7-6 q^5+5 q^3+10 q- q^{-1} -9 q^{-3} - q^{-5} +11 q^{-7} +4 q^{-9} -9 q^{-11} -3 q^{-13} +8 q^{-15} +5 q^{-17} -6 q^{-19} -5 q^{-21} +2 q^{-23} +5 q^{-25} -6 q^{-29} -6 q^{-31} +7 q^{-33} +6 q^{-35} -4 q^{-37} -9 q^{-39} +2 q^{-41} +9 q^{-43} + q^{-45} -4 q^{-47} -3 q^{-49} +2 q^{-51} +3 q^{-53} +2 q^{-55} -3 q^{-57} -3 q^{-59} +3 q^{-63} + q^{-65} -2 q^{-67} -2 q^{-69} + q^{-71} +2 q^{-73} - q^{-77} - q^{-79} + q^{-81} }
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}-q^{72}+q^{70}+2 q^{66}-3 q^{64}-2 q^{62}+3 q^{60}+q^{58}+6 q^{56}-6 q^{54}-9 q^{52}+2 q^{50}+5 q^{48}+16 q^{46}-3 q^{44}-17 q^{42}-10 q^{40}-3 q^{38}+25 q^{36}+14 q^{34}-8 q^{32}-17 q^{30}-21 q^{28}+13 q^{26}+19 q^{24}+13 q^{22}+q^{20}-24 q^{18}-11 q^{16}-5 q^{14}+14 q^{12}+27 q^{10}-16 q^6-32 q^4-6 q^2+36+28 q^{-2} -39 q^{-6} -27 q^{-8} +25 q^{-10} +36 q^{-12} +11 q^{-14} -30 q^{-16} -29 q^{-18} +11 q^{-20} +26 q^{-22} +11 q^{-24} -17 q^{-26} -20 q^{-28} +2 q^{-30} +18 q^{-32} +9 q^{-34} -7 q^{-36} -15 q^{-38} -10 q^{-40} +9 q^{-42} +16 q^{-44} +14 q^{-46} -11 q^{-48} -32 q^{-50} -7 q^{-52} +20 q^{-54} +41 q^{-56} +9 q^{-58} -44 q^{-60} -34 q^{-62} - q^{-64} +50 q^{-66} +36 q^{-68} -25 q^{-70} -39 q^{-72} -28 q^{-74} +27 q^{-76} +43 q^{-78} +4 q^{-80} -20 q^{-82} -35 q^{-84} +26 q^{-88} +15 q^{-90} +2 q^{-92} -24 q^{-94} -10 q^{-96} +9 q^{-98} +11 q^{-100} +10 q^{-102} -11 q^{-104} -8 q^{-106} - q^{-108} +3 q^{-110} +9 q^{-112} -3 q^{-114} -3 q^{-116} -2 q^{-118} - q^{-120} +4 q^{-122} - q^{-128} - q^{-130} + q^{-132} }
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}-q^{111}+q^{109}-q^{101}-q^{99}+3 q^{97}+4 q^{95}-q^{93}-4 q^{91}-6 q^{89}-4 q^{87}+4 q^{85}+14 q^{83}+11 q^{81}-4 q^{79}-17 q^{77}-22 q^{75}-8 q^{73}+18 q^{71}+35 q^{69}+24 q^{67}-9 q^{65}-37 q^{63}-42 q^{61}-14 q^{59}+29 q^{57}+54 q^{55}+37 q^{53}-9 q^{51}-46 q^{49}-52 q^{47}-21 q^{45}+24 q^{43}+48 q^{41}+38 q^{39}+10 q^{37}-20 q^{35}-37 q^{33}-36 q^{31}-22 q^{29}+12 q^{27}+47 q^{25}+60 q^{23}+35 q^{21}-28 q^{19}-87 q^{17}-87 q^{15}-9 q^{13}+87 q^{11}+122 q^9+61 q^7-63 q^5-146 q^3-107 q+29 q^{-1} +142 q^{-3} +140 q^{-5} +18 q^{-7} -124 q^{-9} -155 q^{-11} -47 q^{-13} +97 q^{-15} +148 q^{-17} +71 q^{-19} -63 q^{-21} -133 q^{-23} -75 q^{-25} +40 q^{-27} +105 q^{-29} +69 q^{-31} -20 q^{-33} -78 q^{-35} -59 q^{-37} +12 q^{-39} +58 q^{-41} +40 q^{-43} -8 q^{-45} -42 q^{-47} -35 q^{-49} +5 q^{-51} +34 q^{-53} +38 q^{-55} +8 q^{-57} -33 q^{-59} -52 q^{-61} -34 q^{-63} +18 q^{-65} +74 q^{-67} +79 q^{-69} +5 q^{-71} -91 q^{-73} -119 q^{-75} -52 q^{-77} +86 q^{-79} +170 q^{-81} +104 q^{-83} -65 q^{-85} -189 q^{-87} -156 q^{-89} +16 q^{-91} +184 q^{-93} +191 q^{-95} +34 q^{-97} -150 q^{-99} -194 q^{-101} -73 q^{-103} +95 q^{-105} +175 q^{-107} +99 q^{-109} -50 q^{-111} -134 q^{-113} -99 q^{-115} +9 q^{-117} +91 q^{-119} +85 q^{-121} +11 q^{-123} -55 q^{-125} -65 q^{-127} -21 q^{-129} +32 q^{-131} +47 q^{-133} +21 q^{-135} -14 q^{-137} -33 q^{-139} -23 q^{-141} +8 q^{-143} +24 q^{-145} +18 q^{-147} -16 q^{-151} -17 q^{-153} -5 q^{-155} +11 q^{-157} +14 q^{-159} +5 q^{-161} -4 q^{-163} -9 q^{-165} -7 q^{-167} + q^{-169} +7 q^{-171} +4 q^{-173} -2 q^{-177} -4 q^{-179} - q^{-181} +2 q^{-183} +2 q^{-185} - q^{-191} - q^{-193} + q^{-195} }
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^{162}-q^{160}-q^{158}+q^{156}-2 q^{150}+2 q^{148}-q^{144}+5 q^{142}+2 q^{140}-2 q^{138}-9 q^{136}-2 q^{134}-3 q^{132}+15 q^{128}+14 q^{126}+6 q^{124}-17 q^{122}-15 q^{120}-22 q^{118}-16 q^{116}+21 q^{114}+39 q^{112}+42 q^{110}+4 q^{108}-17 q^{106}-55 q^{104}-69 q^{102}-22 q^{100}+34 q^{98}+85 q^{96}+70 q^{94}+49 q^{92}-31 q^{90}-104 q^{88}-107 q^{86}-58 q^{84}+40 q^{82}+89 q^{80}+134 q^{78}+78 q^{76}-22 q^{74}-99 q^{72}-125 q^{70}-71 q^{68}-23 q^{66}+74 q^{64}+99 q^{62}+79 q^{60}+32 q^{58}-9 q^{56}-25 q^{54}-81 q^{52}-72 q^{50}-76 q^{48}-38 q^{46}+33 q^{44}+131 q^{42}+197 q^{40}+117 q^{38}-7 q^{36}-189 q^{34}-289 q^{32}-238 q^{30}-7 q^{28}+269 q^{26}+379 q^{24}+313 q^{22}+22 q^{20}-311 q^{18}-497 q^{16}-368 q^{14}+q^{12}+358 q^{10}+553 q^8+407 q^6-5 q^4-445 q^2-598-379 q^{-2} +52 q^{-4} +484 q^{-6} +620 q^{-8} +353 q^{-10} -157 q^{-12} -534 q^{-14} -558 q^{-16} -252 q^{-18} +225 q^{-20} +554 q^{-22} +495 q^{-24} +98 q^{-26} -314 q^{-28} -485 q^{-30} -349 q^{-32} +11 q^{-34} +350 q^{-36} +410 q^{-38} +176 q^{-40} -125 q^{-42} -302 q^{-44} -267 q^{-46} -62 q^{-48} +167 q^{-50} +236 q^{-52} +121 q^{-54} -40 q^{-56} -142 q^{-58} -130 q^{-60} -36 q^{-62} +77 q^{-64} +113 q^{-66} +53 q^{-68} -34 q^{-70} -93 q^{-72} -81 q^{-74} -20 q^{-76} +71 q^{-78} +127 q^{-80} +106 q^{-82} +5 q^{-84} -124 q^{-86} -191 q^{-88} -163 q^{-90} +207 q^{-94} +327 q^{-96} +231 q^{-98} -48 q^{-100} -325 q^{-102} -461 q^{-104} -290 q^{-106} +115 q^{-108} +502 q^{-110} +577 q^{-112} +264 q^{-114} -220 q^{-116} -629 q^{-118} -639 q^{-120} -216 q^{-122} +367 q^{-124} +697 q^{-126} +573 q^{-128} +125 q^{-130} -429 q^{-132} -678 q^{-134} -479 q^{-136} +8 q^{-138} +431 q^{-140} +535 q^{-142} +349 q^{-144} -63 q^{-146} -371 q^{-148} -399 q^{-150} -187 q^{-152} +84 q^{-154} +234 q^{-156} +262 q^{-158} +102 q^{-160} -70 q^{-162} -146 q^{-164} -118 q^{-166} -42 q^{-168} +12 q^{-170} +77 q^{-172} +48 q^{-174} +11 q^{-176} -3 q^{-178} -2 q^{-180} -4 q^{-182} -24 q^{-184} +2 q^{-186} -17 q^{-188} -17 q^{-190} + q^{-192} +22 q^{-194} +24 q^{-196} +4 q^{-198} +15 q^{-200} -16 q^{-202} -23 q^{-204} -19 q^{-206} - q^{-208} +9 q^{-210} +7 q^{-212} +26 q^{-214} +5 q^{-216} -4 q^{-218} -14 q^{-220} -10 q^{-222} -8 q^{-224} -4 q^{-226} +15 q^{-228} +7 q^{-230} +7 q^{-232} -3 q^{-234} -3 q^{-236} -7 q^{-238} -7 q^{-240} +5 q^{-242} +2 q^{-244} +5 q^{-246} -3 q^{-252} -3 q^{-254} +2 q^{-256} +2 q^{-260} - q^{-266} - q^{-268} + q^{-270} }

A2 Invariants.

Weight	Invariant
1,0	$q^{8}+q^{4}+q^{-2}-q^{-4}+2q^{-6}-q^{-8}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
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}+4 q^{24}-8 q^{22}+15 q^{20}-20 q^{18}+30 q^{16}-38 q^{14}+45 q^{12}-50 q^{10}+50 q^8-46 q^6+31 q^4-14 q^2-2+30 q^{-2} -50 q^{-4} +68 q^{-6} -76 q^{-8} +82 q^{-10} -84 q^{-12} +74 q^{-14} -62 q^{-16} +46 q^{-18} -32 q^{-20} +18 q^{-22} -2 q^{-24} -4 q^{-26} +17 q^{-28} -18 q^{-30} +18 q^{-32} -22 q^{-34} +22 q^{-36} -26 q^{-38} +20 q^{-40} -22 q^{-42} +25 q^{-44} -20 q^{-46} +18 q^{-48} -14 q^{-50} +11 q^{-52} -8 q^{-54} +4 q^{-56} -2 q^{-58} + q^{-60} }
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^{24}+q^{18}+q^{16}-q^{14}-q^{12}+2 q^{10}+q^8-2 q^6+2 q^2-1-3 q^{-2} + q^{-4} - q^{-8} + q^{-10} +3 q^{-12} + q^{-14} + q^{-16} +4 q^{-18} + q^{-20} -3 q^{-22} - q^{-24} + q^{-26} -2 q^{-28} - q^{-30} + q^{-32} + q^{-34} - q^{-36} - q^{-38} + q^{-52} }

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}+q^{14}-2 q^{12}+q^{10}+2 q^8-q^6+4 q^4+3 q^2-2+3 q^{-2} +2 q^{-4} -5 q^{-6} + q^{-10} -3 q^{-12} -2 q^{-14} -2 q^{-20} +5 q^{-24} - q^{-26} - q^{-28} +6 q^{-30} - q^{-32} -3 q^{-34} +3 q^{-36} -3 q^{-40} + q^{-42} - q^{-46} + q^{-48} }
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^9+2 q^5+2 q- q^{-1} + q^{-3} - q^{-5} - q^{-11} -2 q^{-15} + q^{-17} - q^{-19} +2 q^{-21} + q^{-25} }
1,0,1	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^{34}-2 q^{32}+3 q^{30}-3 q^{28}+q^{26}+4 q^{24}-7 q^{22}+13 q^{20}-6 q^{18}+2 q^{16}+9 q^{14}-17 q^{12}+16 q^{10}-14 q^8+5 q^6+4 q^4-11 q^2+22-17 q^{-2} +19 q^{-4} -13 q^{-6} +3 q^{-8} -2 q^{-10} -7 q^{-12} +8 q^{-14} -12 q^{-16} +16 q^{-18} -18 q^{-20} +28 q^{-22} -22 q^{-24} +16 q^{-26} +2 q^{-28} -17 q^{-30} +26 q^{-32} -28 q^{-34} +22 q^{-36} -6 q^{-38} -4 q^{-40} +10 q^{-42} -6 q^{-44} -5 q^{-46} +6 q^{-48} -7 q^{-50} -7 q^{-52} +12 q^{-54} -12 q^{-56} +12 q^{-58} -2 q^{-60} -4 q^{-62} +9 q^{-64} -9 q^{-66} +6 q^{-68} - q^{-70} -3 q^{-72} +3 q^{-74} -2 q^{-76} + q^{-78} }

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^{22}+q^{16}-q^{12}+q^8+q^6+4 q^4+3 q^2+3+4 q^{-2} +3 q^{-4} - q^{-6} -4 q^{-8} -3 q^{-12} -6 q^{-14} -4 q^{-16} + q^{-18} -5 q^{-20} -2 q^{-22} +3 q^{-24} +5 q^{-30} +5 q^{-32} +3 q^{-36} +4 q^{-38} -4 q^{-42} -3 q^{-48} - q^{-50} + q^{-52} + q^{-58} }

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^{10}+2 q^6+q^4+2 q^2+1+ q^{-4} -2 q^{-6} -2 q^{-10} -2 q^{-14} - q^{-18} + q^{-22} +2 q^{-26} + q^{-30} }

B2 Invariants.

Weight

Invariant

0,1

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

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}+q^{26}+2 q^{24}-q^{22}-3 q^{20}+4 q^{16}+3 q^{14}-3 q^{12}-4 q^{10}+2 q^8+7 q^6+2 q^4-5 q^2-4+3 q^{-2} +5 q^{-4} + q^{-6} -5 q^{-8} -2 q^{-10} +3 q^{-12} +2 q^{-14} -2 q^{-16} -2 q^{-18} +2 q^{-20} +2 q^{-22} -2 q^{-24} -4 q^{-26} +3 q^{-30} -4 q^{-34} -2 q^{-36} +4 q^{-38} +4 q^{-40} -2 q^{-42} -4 q^{-44} + q^{-46} +6 q^{-48} +3 q^{-50} -3 q^{-52} -4 q^{-54} +4 q^{-58} +2 q^{-60} -2 q^{-62} -3 q^{-64} - q^{-66} +2 q^{-68} + q^{-70} - q^{-72} - q^{-74} + q^{-78} }

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}+q^{22}-2 q^{20}+3 q^{18}-4 q^{16}+3 q^{14}-4 q^{12}+6 q^{10}-3 q^8+6 q^6-q^4+6 q^2+2+2 q^{-2} +2 q^{-4} -2 q^{-6} +2 q^{-8} -8 q^{-10} +4 q^{-12} -10 q^{-14} +5 q^{-16} -11 q^{-18} +6 q^{-20} -8 q^{-22} +7 q^{-24} -5 q^{-26} +5 q^{-28} -2 q^{-30} +4 q^{-32} +2 q^{-34} +2 q^{-38} -2 q^{-40} +5 q^{-42} -3 q^{-44} +3 q^{-46} -4 q^{-48} +4 q^{-50} -3 q^{-52} +2 q^{-54} -3 q^{-56} +2 q^{-58} -2 q^{-60} + q^{-62} - q^{-64} + q^{-66} }

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}+2 q^{42}-3 q^{40}+2 q^{38}-2 q^{36}-q^{34}+6 q^{32}-8 q^{30}+10 q^{28}-9 q^{26}+5 q^{24}+2 q^{22}-10 q^{20}+16 q^{18}-16 q^{16}+14 q^{14}-4 q^{12}-5 q^{10}+15 q^8-14 q^6+12 q^4-4 q^2-4+10 q^{-2} -9 q^{-4} +3 q^{-6} +6 q^{-8} -10 q^{-10} +14 q^{-12} -9 q^{-14} -2 q^{-16} +8 q^{-18} -16 q^{-20} +18 q^{-22} -17 q^{-24} +7 q^{-26} +3 q^{-28} -13 q^{-30} +18 q^{-32} -18 q^{-34} +11 q^{-36} -3 q^{-38} -6 q^{-40} +10 q^{-42} -12 q^{-44} +8 q^{-46} -5 q^{-50} +6 q^{-52} -4 q^{-54} -2 q^{-56} +7 q^{-58} -8 q^{-60} +7 q^{-62} -4 q^{-64} - q^{-66} +6 q^{-68} -8 q^{-70} +11 q^{-72} -8 q^{-74} +6 q^{-76} - q^{-78} -2 q^{-80} +5 q^{-82} -8 q^{-84} +10 q^{-86} -7 q^{-88} +4 q^{-90} -4 q^{-94} +6 q^{-96} -6 q^{-98} +5 q^{-100} -3 q^{-102} + q^{-106} -3 q^{-108} +2 q^{-110} - q^{-112} + q^{-114} }

.

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 9"];

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

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^8-5 z^6-7 z^4-2 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]= { 39, 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^7-2 q^6+3 q^5-5 q^4+6 q^3-6 q^2+6 q-4+3 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^8 a^{-2} -7 z^6 a^{-2} +z^6 a^{-4} +z^6-17 z^4 a^{-2} +5 z^4 a^{-4} +5 z^4-16 z^2 a^{-2} +7 z^2 a^{-4} +7 z^2-4 a^{-2} +2 a^{-4} +3}

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

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

Out[10]= $z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+2z^{8}a^{-4}+2z^{8}+2az^{7}-2z^{7}a^{-1}-2z^{7}a^{-3}+2z^{7}a^{-5}+a^{2}z^{6}-18z^{6}a^{-2}-7z^{6}a^{-4}+2z^{6}a^{-6}-8z^{6}-8az^{5}-2z^{5}a^{-1}-4z^{5}a^{-5}+2z^{5}a^{-7}-4a^{2}z^{4}+31z^{4}a^{-2}+13z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+10z^{4}+7az^{3}+4z^{3}a^{-1}+5z^{3}a^{-3}+4z^{3}a^{-5}-4z^{3}a^{-7}+3a^{2}z^{2}-22z^{2}a^{-2}-8z^{2}a^{-4}+z^{2}a^{-6}-2z^{2}a^{-8}-8z^{2}-az-2za^{-1}-2za^{-3}+za^{-7}+4a^{-2}+2a^{-4}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk

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

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

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["10 9"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

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

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

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

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:

(-2, -2)

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 -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 -16}

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{260}{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{220}{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{1088}{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{320}{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 144}

Failed to parse (SVG (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{256}{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}

-{\frac {2080}{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{1760}{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{1231}{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{3588}{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{54844}{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{2911}{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 -\frac{6271}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 2 is the signature of 10 9. 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

6

χ

15

1

13

1

-1

11

2

1

9

3

1

-2

7

3

2

1

5

3

0

3

0

1

2

4

2

-1

1

2

-1

-3

1

2

1

-5

1

-1

-7

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}}	$i=1$	$i=3$
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}^{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=-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}^{4}\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}^{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}^{3}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{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}\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}	Failed to parse (SVG (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^{20}-2 q^{19}+4 q^{17}-5 q^{16}+8 q^{14}-10 q^{13}+15 q^{11}-18 q^{10}+22 q^8-23 q^7-q^6+25 q^5-21 q^4-5 q^3+24 q^2-15 q-8+19 q^{-1} -8 q^{-2} -9 q^{-3} +12 q^{-4} -2 q^{-5} -6 q^{-6} +5 q^{-7} -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^{39}-2 q^{38}+q^{36}+3 q^{35}-3 q^{34}-3 q^{33}+q^{32}+6 q^{31}-q^{30}-6 q^{29}-2 q^{28}+6 q^{27}+4 q^{26}-5 q^{25}-3 q^{24}+q^{23}+3 q^{22}+5 q^{20}-6 q^{19}-8 q^{18}+5 q^{17}+15 q^{16}-5 q^{15}-21 q^{14}+5 q^{13}+21 q^{12}-24 q^{10}-2 q^9+20 q^8+11 q^7-21 q^6-13 q^5+14 q^4+24 q^3-14 q^2-25 q+6+32 q^{-1} -3 q^{-2} -30 q^{-3} -5 q^{-4} +29 q^{-5} +8 q^{-6} -23 q^{-7} -12 q^{-8} +18 q^{-9} +11 q^{-10} -10 q^{-11} -11 q^{-12} +7 q^{-13} +7 q^{-14} -3 q^{-15} -5 q^{-16} +2 q^{-17} +2 q^{-18} -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^{64}-2 q^{63}+q^{61}+5 q^{59}-7 q^{58}-q^{57}+17 q^{54}-13 q^{53}-5 q^{52}-7 q^{51}-3 q^{50}+38 q^{49}-12 q^{48}-7 q^{47}-26 q^{46}-17 q^{45}+64 q^{44}+q^{43}+4 q^{42}-52 q^{41}-52 q^{40}+79 q^{39}+25 q^{38}+43 q^{37}-68 q^{36}-107 q^{35}+68 q^{34}+39 q^{33}+104 q^{32}-54 q^{31}-158 q^{30}+35 q^{29}+29 q^{28}+157 q^{27}-22 q^{26}-179 q^{25}+8 q^{24}+4 q^{23}+178 q^{22}+3 q^{21}-177 q^{20}+q^{19}-15 q^{18}+173 q^{17}+11 q^{16}-161 q^{15}+10 q^{14}-31 q^{13}+151 q^{12}+14 q^{11}-133 q^{10}+25 q^9-46 q^8+111 q^7+13 q^6-92 q^5+50 q^4-57 q^3+59 q^2+q-53+78 q^{-1} -49 q^{-2} +17 q^{-3} -25 q^{-4} -37 q^{-5} +94 q^{-6} -22 q^{-7} +4 q^{-8} -44 q^{-9} -43 q^{-10} +81 q^{-11} +3 q^{-12} +16 q^{-13} -38 q^{-14} -49 q^{-15} +47 q^{-16} +7 q^{-17} +25 q^{-18} -16 q^{-19} -38 q^{-20} +19 q^{-21} +18 q^{-23} -2 q^{-24} -19 q^{-25} +8 q^{-26} -3 q^{-27} +7 q^{-28} + q^{-29} -7 q^{-30} +3 q^{-31} - q^{-32} +2 q^{-33} -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^{95}-2 q^{94}+q^{92}+2 q^{90}+q^{89}-5 q^{88}-3 q^{87}+3 q^{86}+2 q^{85}+6 q^{84}+4 q^{83}-11 q^{82}-11 q^{81}+q^{80}+7 q^{79}+15 q^{78}+13 q^{77}-14 q^{76}-27 q^{75}-11 q^{74}+8 q^{73}+31 q^{72}+31 q^{71}-8 q^{70}-43 q^{69}-42 q^{68}-2 q^{67}+50 q^{66}+66 q^{65}+18 q^{64}-58 q^{63}-95 q^{62}-46 q^{61}+60 q^{60}+132 q^{59}+92 q^{58}-52 q^{57}-177 q^{56}-154 q^{55}+25 q^{54}+216 q^{53}+241 q^{52}+24 q^{51}-257 q^{50}-322 q^{49}-96 q^{48}+260 q^{47}+425 q^{46}+181 q^{45}-264 q^{44}-490 q^{43}-268 q^{42}+227 q^{41}+549 q^{40}+350 q^{39}-198 q^{38}-574 q^{37}-406 q^{36}+160 q^{35}+577 q^{34}+446 q^{33}-124 q^{32}-579 q^{31}-462 q^{30}+108 q^{29}+559 q^{28}+465 q^{27}-83 q^{26}-549 q^{25}-466 q^{24}+79 q^{23}+519 q^{22}+458 q^{21}-49 q^{20}-501 q^{19}-448 q^{18}+33 q^{17}+448 q^{16}+439 q^{15}+9 q^{14}-412 q^{13}-412 q^{12}-32 q^{11}+333 q^{10}+381 q^9+79 q^8-278 q^7-335 q^6-83 q^5+189 q^4+273 q^3+110 q^2-136 q-213-81 q^{-1} +76 q^{-2} +137 q^{-3} +71 q^{-4} -53 q^{-5} -89 q^{-6} -20 q^{-7} +41 q^{-8} +41 q^{-9} -7 q^{-10} -53 q^{-11} -30 q^{-12} +43 q^{-13} +66 q^{-14} +28 q^{-15} -42 q^{-16} -87 q^{-17} -44 q^{-18} +42 q^{-19} +83 q^{-20} +58 q^{-21} -14 q^{-22} -77 q^{-23} -68 q^{-24} -3 q^{-25} +52 q^{-26} +64 q^{-27} +23 q^{-28} -31 q^{-29} -51 q^{-30} -28 q^{-31} +9 q^{-32} +36 q^{-33} +28 q^{-34} -3 q^{-35} -18 q^{-36} -17 q^{-37} -8 q^{-38} +10 q^{-39} +14 q^{-40} +2 q^{-41} -5 q^{-42} -2 q^{-43} -5 q^{-44} +6 q^{-46} -3 q^{-48} + q^{-49} - q^{-51} +2 q^{-52} -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^{132}-2 q^{131}+q^{129}+2 q^{127}-2 q^{126}+3 q^{125}-7 q^{124}+4 q^{122}+7 q^{120}-5 q^{119}+6 q^{118}-19 q^{117}+8 q^{115}+17 q^{113}-5 q^{112}+14 q^{111}-38 q^{110}-4 q^{109}+6 q^{108}-4 q^{107}+27 q^{106}+4 q^{105}+35 q^{104}-57 q^{103}-2 q^{102}-4 q^{101}-22 q^{100}+23 q^{99}+11 q^{98}+66 q^{97}-68 q^{96}+18 q^{95}-6 q^{94}-43 q^{93}+5 q^{92}+11 q^{91}+85 q^{90}-94 q^{89}+38 q^{88}-4 q^{87}-44 q^{86}+19 q^{85}+48 q^{84}+114 q^{83}-159 q^{82}-16 q^{81}-80 q^{80}-72 q^{79}+95 q^{78}+220 q^{77}+274 q^{76}-187 q^{75}-166 q^{74}-351 q^{73}-284 q^{72}+123 q^{71}+528 q^{70}+686 q^{69}-q^{68}-270 q^{67}-774 q^{66}-771 q^{65}-76 q^{64}+777 q^{63}+1240 q^{62}+447 q^{61}-146 q^{60}-1104 q^{59}-1354 q^{58}-499 q^{57}+787 q^{56}+1649 q^{55}+931 q^{54}+167 q^{53}-1179 q^{52}-1741 q^{51}-904 q^{50}+616 q^{49}+1785 q^{48}+1208 q^{47}+446 q^{46}-1083 q^{45}-1861 q^{44}-1111 q^{43}+453 q^{42}+1757 q^{41}+1275 q^{40}+575 q^{39}-982 q^{38}-1840 q^{37}-1167 q^{36}+362 q^{35}+1696 q^{34}+1263 q^{33}+634 q^{32}-895 q^{31}-1780 q^{30}-1203 q^{29}+249 q^{28}+1602 q^{27}+1251 q^{26}+736 q^{25}-734 q^{24}-1665 q^{23}-1272 q^{22}+20 q^{21}+1397 q^{20}+1216 q^{19}+913 q^{18}-433 q^{17}-1431 q^{16}-1332 q^{15}-319 q^{14}+1037 q^{13}+1080 q^{12}+1084 q^{11}-21 q^{10}-1034 q^9-1273 q^8-648 q^7+560 q^6+774 q^5+1108 q^4+356 q^3-524 q^2-1006 q-784+128 q^{-1} +343 q^{-2} +889 q^{-3} +509 q^{-4} -84 q^{-5} -594 q^{-6} -638 q^{-7} -67 q^{-8} -14 q^{-9} +520 q^{-10} +380 q^{-11} +102 q^{-12} -261 q^{-13} -347 q^{-14} - q^{-15} -124 q^{-16} +244 q^{-17} +149 q^{-18} +51 q^{-19} -161 q^{-20} -165 q^{-21} +123 q^{-22} -44 q^{-23} +178 q^{-24} +51 q^{-25} -20 q^{-26} -199 q^{-27} -161 q^{-28} +114 q^{-29} +12 q^{-30} +194 q^{-31} +92 q^{-32} +27 q^{-33} -179 q^{-34} -186 q^{-35} +17 q^{-36} -36 q^{-37} +140 q^{-38} +118 q^{-39} +104 q^{-40} -79 q^{-41} -130 q^{-42} -28 q^{-43} -85 q^{-44} +42 q^{-45} +69 q^{-46} +107 q^{-47} -6 q^{-48} -50 q^{-49} -7 q^{-50} -70 q^{-51} -9 q^{-52} +13 q^{-53} +60 q^{-54} +8 q^{-55} -12 q^{-56} +14 q^{-57} -32 q^{-58} -12 q^{-59} -5 q^{-60} +23 q^{-61} +2 q^{-62} -5 q^{-63} +12 q^{-64} -9 q^{-65} -4 q^{-66} -4 q^{-67} +8 q^{-68} - q^{-69} -4 q^{-70} +5 q^{-71} -2 q^{-72} - q^{-74} +2 q^{-75} -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^{175}-2 q^{174}+q^{172}+2 q^{170}-2 q^{169}+q^{167}-4 q^{166}+q^{165}+2 q^{164}+7 q^{162}-4 q^{161}-5 q^{160}+2 q^{159}-10 q^{158}+4 q^{157}+5 q^{156}+17 q^{154}-3 q^{153}-7 q^{152}-q^{151}-26 q^{150}+7 q^{148}-4 q^{147}+34 q^{146}+10 q^{145}+7 q^{144}+11 q^{143}-49 q^{142}-21 q^{141}-16 q^{140}-28 q^{139}+36 q^{138}+28 q^{137}+52 q^{136}+69 q^{135}-31 q^{134}-37 q^{133}-65 q^{132}-110 q^{131}-17 q^{130}+4 q^{129}+88 q^{128}+178 q^{127}+80 q^{126}+38 q^{125}-79 q^{124}-218 q^{123}-157 q^{122}-140 q^{121}+14 q^{120}+245 q^{119}+251 q^{118}+259 q^{117}+89 q^{116}-200 q^{115}-283 q^{114}-398 q^{113}-272 q^{112}+68 q^{111}+253 q^{110}+500 q^{109}+477 q^{108}+159 q^{107}-89 q^{106}-492 q^{105}-657 q^{104}-470 q^{103}-246 q^{102}+322 q^{101}+741 q^{100}+799 q^{99}+709 q^{98}+73 q^{97}-617 q^{96}-1044 q^{95}-1278 q^{94}-698 q^{93}+240 q^{92}+1129 q^{91}+1837 q^{90}+1462 q^{89}+390 q^{88}-939 q^{87}-2250 q^{86}-2297 q^{85}-1256 q^{84}+516 q^{83}+2479 q^{82}+3033 q^{81}+2154 q^{80}+137 q^{79}-2416 q^{78}-3588 q^{77}-3041 q^{76}-887 q^{75}+2183 q^{74}+3905 q^{73}+3729 q^{72}+1594 q^{71}-1792 q^{70}-3979 q^{69}-4205 q^{68}-2202 q^{67}+1379 q^{66}+3918 q^{65}+4468 q^{64}+2605 q^{63}-1036 q^{62}-3750 q^{61}-4540 q^{60}-2865 q^{59}+768 q^{58}+3591 q^{57}+4546 q^{56}+2971 q^{55}-619 q^{54}-3459 q^{53}-4482 q^{52}-3010 q^{51}+525 q^{50}+3352 q^{49}+4430 q^{48}+3034 q^{47}-463 q^{46}-3278 q^{45}-4374 q^{44}-3049 q^{43}+379 q^{42}+3163 q^{41}+4322 q^{40}+3109 q^{39}-246 q^{38}-3007 q^{37}-4240 q^{36}-3197 q^{35}+22 q^{34}+2773 q^{33}+4124 q^{32}+3300 q^{31}+274 q^{30}-2402 q^{29}-3922 q^{28}-3444 q^{27}-685 q^{26}+1949 q^{25}+3645 q^{24}+3528 q^{23}+1132 q^{22}-1318 q^{21}-3221 q^{20}-3588 q^{19}-1652 q^{18}+631 q^{17}+2687 q^{16}+3507 q^{15}+2095 q^{14}+149 q^{13}-1976 q^{12}-3279 q^{11}-2476 q^{10}-923 q^9+1198 q^8+2888 q^7+2633 q^6+1580 q^5-344 q^4-2271 q^3-2604 q^2-2100 q-444+1600 q^{-1} +2314 q^{-2} +2307 q^{-3} +1094 q^{-4} -829 q^{-5} -1825 q^{-6} -2306 q^{-7} -1525 q^{-8} +196 q^{-9} +1250 q^{-10} +2003 q^{-11} +1657 q^{-12} +318 q^{-13} -652 q^{-14} -1582 q^{-15} -1579 q^{-16} -563 q^{-17} +197 q^{-18} +1087 q^{-19} +1299 q^{-20} +615 q^{-21} +102 q^{-22} -688 q^{-23} -965 q^{-24} -488 q^{-25} -205 q^{-26} +396 q^{-27} +675 q^{-28} +323 q^{-29} +168 q^{-30} -280 q^{-31} -478 q^{-32} -153 q^{-33} -92 q^{-34} +242 q^{-35} +402 q^{-36} +94 q^{-37} +26 q^{-38} -269 q^{-39} -395 q^{-40} -94 q^{-41} -18 q^{-42} +256 q^{-43} +399 q^{-44} +147 q^{-45} +78 q^{-46} -196 q^{-47} -389 q^{-48} -205 q^{-49} -144 q^{-50} +117 q^{-51} +316 q^{-52} +203 q^{-53} +206 q^{-54} + q^{-55} -224 q^{-56} -189 q^{-57} -225 q^{-58} -59 q^{-59} +130 q^{-60} +116 q^{-61} +193 q^{-62} +107 q^{-63} -41 q^{-64} -62 q^{-65} -159 q^{-66} -106 q^{-67} +12 q^{-68} +15 q^{-69} +91 q^{-70} +81 q^{-71} +21 q^{-72} +22 q^{-73} -63 q^{-74} -66 q^{-75} -12 q^{-76} -17 q^{-77} +27 q^{-78} +28 q^{-79} +11 q^{-80} +31 q^{-81} -12 q^{-82} -28 q^{-83} -6 q^{-84} -11 q^{-85} +8 q^{-86} +5 q^{-87} -3 q^{-88} +16 q^{-89} + q^{-90} -8 q^{-91} -2 q^{-92} -4 q^{-93} +4 q^{-94} + q^{-95} -5 q^{-96} +4 q^{-97} +2 q^{-98} -2 q^{-99} - q^{-101} +2 q^{-102} -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.

10_8

10_10

10 9

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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