10 102

10_101

10_103

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 102's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 102 at Knotilus!

[edit 10 102 Quick Notes]

[edit 10 102 Further Notes and Views]

Knot presentations

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

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 4,

Braid index is 4

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

[edit Notes on presentations of 10 102]

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["10 102"];

In[4]:= PD[K]

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

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

In[5]:= GaussCode[K]

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

In[6]:= DTCode[K]

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

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [3:2:20]

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

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

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(4,\{-1,-1,2,-1,-3,2,-1,2,2,3,3\})}

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

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-5][-7]
Hyperbolic Volume	13.7273
A-Polynomial	See Data:10 102/A-polynomial

[edit Notes for 10 102'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	0

[edit Notes for 10 102's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^3+8 t^2-16 t+21-16 t^{-1} +8 t^{-2} -2 t^{-3} }
Conway polynomial	$-2z^{6}-4z^{4}-2z^{2}+1$
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	{ 73, 0 }
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+6 q^4-9 q^3+11 q^2-12 q+12-9 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }
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} -z^6+a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -3 z^2+a^2- a^{-2} + a^{-4} }
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+6 a z^7+4 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} +5 a^2 z^6-24 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -7 z^6+3 a^3 z^5-9 a z^5-17 z^5 a^{-1} -14 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-6 a^2 z^4+21 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} +3 z^4-3 a^3 z^3+7 a z^3+16 z^3 a^{-1} +13 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+3 a^2 z^2-8 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -2 z a^{-5} -a^2+ a^{-2} + a^{-4} }
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^{12}-q^{10}+q^8+q^6-2 q^4+3 q^2-1+ q^{-2} -2 q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} }
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^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+11 q^{52}-18 q^{50}+26 q^{48}-28 q^{46}+19 q^{44}-4 q^{42}-19 q^{40}+42 q^{38}-59 q^{36}+68 q^{34}-61 q^{32}+33 q^{30}+15 q^{28}-64 q^{26}+110 q^{24}-123 q^{22}+100 q^{20}-42 q^{18}-40 q^{16}+108 q^{14}-134 q^{12}+108 q^{10}-29 q^8-57 q^6+110 q^4-105 q^2+39+57 q^{-2} -141 q^{-4} +164 q^{-6} -116 q^{-8} +14 q^{-10} +107 q^{-12} -193 q^{-14} +216 q^{-16} -165 q^{-18} +60 q^{-20} +55 q^{-22} -151 q^{-24} +192 q^{-26} -166 q^{-28} +88 q^{-30} +14 q^{-32} -100 q^{-34} +135 q^{-36} -108 q^{-38} +29 q^{-40} +61 q^{-42} -125 q^{-44} +126 q^{-46} -65 q^{-48} -34 q^{-50} +131 q^{-52} -171 q^{-54} +148 q^{-56} -68 q^{-58} -33 q^{-60} +111 q^{-62} -142 q^{-64} +125 q^{-66} -69 q^{-68} +6 q^{-70} +42 q^{-72} -63 q^{-74} +57 q^{-76} -36 q^{-78} +16 q^{-80} + q^{-82} -10 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94} }

Further Quantum Invariants

Further quantum knot invariants for 10_102.

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^9-2 q^7+3 q^5-3 q^3+3 q- q^{-3} +2 q^{-5} -3 q^{-7} +3 q^{-9} -2 q^{-11} + q^{-13} }
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^{26}-2 q^{24}+5 q^{20}-7 q^{18}+q^{16}+13 q^{14}-17 q^{12}-3 q^{10}+25 q^8-17 q^6-13 q^4+24 q^2-2-15 q^{-2} +6 q^{-4} +13 q^{-6} -9 q^{-8} -12 q^{-10} +20 q^{-12} + q^{-14} -24 q^{-16} +16 q^{-18} +13 q^{-20} -24 q^{-22} +4 q^{-24} +17 q^{-26} -12 q^{-28} -5 q^{-30} +8 q^{-32} - q^{-34} -2 q^{-36} + q^{-38} }
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^{51}-2 q^{49}+2 q^{45}+q^{43}-3 q^{41}-q^{39}+6 q^{37}-5 q^{35}-10 q^{33}+14 q^{31}+24 q^{29}-23 q^{27}-53 q^{25}+28 q^{23}+91 q^{21}-13 q^{19}-129 q^{17}-24 q^{15}+148 q^{13}+70 q^{11}-140 q^9-107 q^7+99 q^5+134 q^3-46 q-131 q^{-1} -6 q^{-3} +111 q^{-5} +57 q^{-7} -87 q^{-9} -90 q^{-11} +58 q^{-13} +118 q^{-15} -35 q^{-17} -136 q^{-19} +5 q^{-21} +148 q^{-23} +29 q^{-25} -150 q^{-27} -67 q^{-29} +135 q^{-31} +105 q^{-33} -98 q^{-35} -133 q^{-37} +47 q^{-39} +140 q^{-41} +4 q^{-43} -116 q^{-45} -47 q^{-47} +77 q^{-49} +65 q^{-51} -36 q^{-53} -56 q^{-55} +3 q^{-57} +36 q^{-59} +10 q^{-61} -17 q^{-63} -8 q^{-65} +5 q^{-67} +4 q^{-69} - q^{-71} -2 q^{-73} + q^{-75} }
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^{84}-2 q^{82}+2 q^{78}-2 q^{76}+5 q^{74}-5 q^{72}-q^{70}+q^{68}-11 q^{66}+21 q^{64}+5 q^{62}+5 q^{60}-18 q^{58}-67 q^{56}+25 q^{54}+70 q^{52}+104 q^{50}-17 q^{48}-250 q^{46}-139 q^{44}+118 q^{42}+426 q^{40}+261 q^{38}-418 q^{36}-636 q^{34}-223 q^{32}+708 q^{30}+945 q^{28}-70 q^{26}-1017 q^{24}-1010 q^{22}+343 q^{20}+1386 q^{18}+739 q^{16}-626 q^{14}-1428 q^{12}-481 q^{10}+947 q^8+1150 q^6+222 q^4-987 q^2-928+87 q^{-2} +845 q^{-4} +724 q^{-6} -234 q^{-8} -813 q^{-10} -493 q^{-12} +369 q^{-14} +818 q^{-16} +262 q^{-18} -620 q^{-20} -779 q^{-22} +102 q^{-24} +874 q^{-26} +588 q^{-28} -530 q^{-30} -1064 q^{-32} -171 q^{-34} +923 q^{-36} +1002 q^{-38} -243 q^{-40} -1256 q^{-42} -685 q^{-44} +591 q^{-46} +1318 q^{-48} +423 q^{-50} -934 q^{-52} -1115 q^{-54} -211 q^{-56} +1022 q^{-58} +978 q^{-60} -69 q^{-62} -868 q^{-64} -845 q^{-66} +174 q^{-68} +789 q^{-70} +566 q^{-72} -105 q^{-74} -684 q^{-76} -390 q^{-78} +129 q^{-80} +441 q^{-82} +324 q^{-84} -144 q^{-86} -277 q^{-88} -183 q^{-90} +58 q^{-92} +199 q^{-94} +72 q^{-96} -25 q^{-98} -93 q^{-100} -47 q^{-102} +32 q^{-104} +26 q^{-106} +19 q^{-108} -11 q^{-110} -14 q^{-112} +2 q^{-114} + q^{-116} +4 q^{-118} - q^{-120} -2 q^{-122} + q^{-124} }

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^{12}-q^{10}+q^8+q^6-2 q^4+3 q^2-1+ q^{-2} -2 q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} + 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^{36}-4 q^{34}+10 q^{32}-20 q^{30}+36 q^{28}-58 q^{26}+90 q^{24}-128 q^{22}+171 q^{20}-218 q^{18}+278 q^{16}-348 q^{14}+407 q^{12}-458 q^{10}+496 q^8-480 q^6+399 q^4-242 q^2+24+242 q^{-2} -534 q^{-4} +798 q^{-6} -1008 q^{-8} +1140 q^{-10} -1165 q^{-12} +1098 q^{-14} -934 q^{-16} +698 q^{-18} -408 q^{-20} +96 q^{-22} +188 q^{-24} -420 q^{-26} +583 q^{-28} -656 q^{-30} +638 q^{-32} -556 q^{-34} +445 q^{-36} -324 q^{-38} +208 q^{-40} -122 q^{-42} +66 q^{-44} -30 q^{-46} +12 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^{32}-q^{30}-q^{28}+3 q^{26}-4 q^{22}+2 q^{20}+7 q^{18}-2 q^{16}-10 q^{14}+4 q^{12}+12 q^{10}-9 q^8-9 q^6+9 q^4+4 q^2-5-3 q^{-2} +8 q^{-4} -2 q^{-6} -4 q^{-8} +7 q^{-10} + q^{-12} -7 q^{-14} +4 q^{-16} +9 q^{-18} -8 q^{-20} -5 q^{-22} +6 q^{-24} +6 q^{-26} -7 q^{-28} -7 q^{-30} +8 q^{-32} +3 q^{-34} -5 q^{-36} -3 q^{-38} +2 q^{-40} +3 q^{-42} - q^{-44} - q^{-46} + q^{-48} }

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

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^{15}-q^{13}+2 q^{11}-q^9+2 q^7-2 q^5+3 q^3-q+ q^{-1} - q^{-5} -2 q^{-9} +2 q^{-11} -2 q^{-13} +2 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^{34}-q^{32}-q^{30}+3 q^{28}-4 q^{24}+q^{22}+7 q^{20}-9 q^{16}+3 q^{14}+13 q^{12}-7 q^{10}-12 q^8+14 q^6+7 q^4-12 q^2+1+15 q^{-2} -3 q^{-4} -12 q^{-6} +8 q^{-8} +5 q^{-10} -19 q^{-12} -3 q^{-14} +17 q^{-16} -11 q^{-18} -12 q^{-20} +17 q^{-22} +9 q^{-24} -12 q^{-26} - q^{-28} +14 q^{-30} -10 q^{-34} +3 q^{-36} +6 q^{-38} -6 q^{-40} -2 q^{-42} +4 q^{-44} - q^{-46} - q^{-48} + q^{-50} }

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^{18}-q^{16}+2 q^{14}+2 q^8-2 q^6+3 q^4-q^2+1- q^{-6} - q^{-8} -2 q^{-12} +2 q^{-14} -2 q^{-16} +2 q^{-18} +2 q^{-24} - q^{-26} + q^{-28} }

B2 Invariants.

Weight

Invariant

0,1

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}-2 q^{26}+4 q^{24}-7 q^{22}+10 q^{20}-14 q^{18}+19 q^{16}-21 q^{14}+24 q^{12}-22 q^{10}+18 q^8-10 q^6+12 q^2-24+34 q^{-2} -42 q^{-4} +46 q^{-6} -46 q^{-8} +41 q^{-10} -32 q^{-12} +22 q^{-14} -9 q^{-16} -3 q^{-18} +13 q^{-20} -20 q^{-22} +23 q^{-24} -24 q^{-26} +23 q^{-28} -19 q^{-30} +14 q^{-32} -9 q^{-34} +5 q^{-36} -2 q^{-38} + q^{-40} }

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}-2 q^{42}-2 q^{40}+2 q^{38}+6 q^{36}+q^{34}-8 q^{32}-8 q^{30}+5 q^{28}+16 q^{26}+4 q^{24}-16 q^{22}-16 q^{20}+9 q^{18}+24 q^{16}+3 q^{14}-23 q^{12}-13 q^{10}+17 q^8+20 q^6-9 q^4-20 q^2+3+20 q^{-2} +3 q^{-4} -17 q^{-6} -7 q^{-8} +13 q^{-10} +9 q^{-12} -12 q^{-14} -12 q^{-16} +10 q^{-18} +15 q^{-20} -7 q^{-22} -21 q^{-24} +24 q^{-28} +12 q^{-30} -20 q^{-32} -22 q^{-34} +11 q^{-36} +26 q^{-38} +3 q^{-40} -21 q^{-42} -12 q^{-44} +13 q^{-46} +14 q^{-48} -4 q^{-50} -11 q^{-52} -2 q^{-54} +6 q^{-56} +3 q^{-58} -2 q^{-60} -2 q^{-62} + q^{-66} }

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^{38}-2 q^{36}+2 q^{34}-3 q^{32}+6 q^{30}-8 q^{28}+8 q^{26}-10 q^{24}+16 q^{22}-16 q^{20}+15 q^{18}-17 q^{16}+21 q^{14}-15 q^{12}+11 q^{10}-9 q^8+5 q^6+7 q^4-10 q^2+17-24 q^{-2} +31 q^{-4} -32 q^{-6} +33 q^{-8} -39 q^{-10} +33 q^{-12} -32 q^{-14} +26 q^{-16} -24 q^{-18} +14 q^{-20} -7 q^{-22} +3 q^{-24} +5 q^{-26} -9 q^{-28} +18 q^{-30} -16 q^{-32} +19 q^{-34} -19 q^{-36} +20 q^{-38} -17 q^{-40} +13 q^{-42} -12 q^{-44} +8 q^{-46} -5 q^{-48} +3 q^{-50} -2 q^{-52} + q^{-54} }

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^{66}-2 q^{64}+4 q^{62}-6 q^{60}+5 q^{58}-3 q^{56}-2 q^{54}+11 q^{52}-18 q^{50}+26 q^{48}-28 q^{46}+19 q^{44}-4 q^{42}-19 q^{40}+42 q^{38}-59 q^{36}+68 q^{34}-61 q^{32}+33 q^{30}+15 q^{28}-64 q^{26}+110 q^{24}-123 q^{22}+100 q^{20}-42 q^{18}-40 q^{16}+108 q^{14}-134 q^{12}+108 q^{10}-29 q^8-57 q^6+110 q^4-105 q^2+39+57 q^{-2} -141 q^{-4} +164 q^{-6} -116 q^{-8} +14 q^{-10} +107 q^{-12} -193 q^{-14} +216 q^{-16} -165 q^{-18} +60 q^{-20} +55 q^{-22} -151 q^{-24} +192 q^{-26} -166 q^{-28} +88 q^{-30} +14 q^{-32} -100 q^{-34} +135 q^{-36} -108 q^{-38} +29 q^{-40} +61 q^{-42} -125 q^{-44} +126 q^{-46} -65 q^{-48} -34 q^{-50} +131 q^{-52} -171 q^{-54} +148 q^{-56} -68 q^{-58} -33 q^{-60} +111 q^{-62} -142 q^{-64} +125 q^{-66} -69 q^{-68} +6 q^{-70} +42 q^{-72} -63 q^{-74} +57 q^{-76} -36 q^{-78} +16 q^{-80} + q^{-82} -10 q^{-84} +10 q^{-86} -9 q^{-88} +5 q^{-90} -2 q^{-92} + q^{-94} }

.

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["10 102"];

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

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

Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^3+8 t^2-16 t+21-16 t^{-1} +8 t^{-2} -2 t^{-3} }

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

Out[5]= $-2z^{6}-4z^{4}-2z^{2}+1$

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]= { 73, 0 }

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+6 q^4-9 q^3+11 q^2-12 q+12-9 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} }

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} -z^6+a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-4} -3 z^2+a^2- a^{-2} + a^{-4} }

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

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

Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+6 a z^7+4 z^7 a^{-1} +z^7 a^{-3} +3 z^7 a^{-5} +5 a^2 z^6-24 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -7 z^6+3 a^3 z^5-9 a z^5-17 z^5 a^{-1} -14 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-6 a^2 z^4+21 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} +3 z^4-3 a^3 z^3+7 a z^3+16 z^3 a^{-1} +13 z^3 a^{-3} +7 z^3 a^{-5} -a^4 z^2+3 a^2 z^2-8 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -2 z a^{-5} -a^2+ a^{-2} + a^{-4} }

"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["10 102"];

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

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

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

Out[4]= { Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t^3+8 t^2-16 t+21-16 t^{-1} +8 t^{-2} -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+6 q^4-9 q^3+11 q^2-12 q+12-9 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4} } }

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, -1)

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 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{212}{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{148}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 64}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{496}{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{64}{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 88}

Failed to parse (SVG (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 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{1696}{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{1184}{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{5431}{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{5044}{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{37924}{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{1831}{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{3511}{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=} 0 is the signature of 10 102. 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

13

1

11

2

-2

9

4

1

3

7

5

2

-3

5

6

4

2

3

6

5

-1

1

6

0

-1

4

7

3

-3

2

5

-3

-5

1

4

3

-7

2

-2

-9

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=1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-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}^{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=-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}^{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}^{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=-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}^{5}\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=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}^{7}\oplus{\mathbb Z}_2^{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}^{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 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}^{6}\oplus{\mathbb Z}_2^{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}^{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 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}^{5}\oplus{\mathbb Z}_2^{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}^{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 r=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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}^{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 r=4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{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=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^{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=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^{18}-3 q^{17}+q^{16}+10 q^{15}-16 q^{14}-6 q^{13}+39 q^{12}-29 q^{11}-34 q^{10}+76 q^9-26 q^8-74 q^7+101 q^6-7 q^5-106 q^4+104 q^3+15 q^2-113 q+83+28 q^{-1} -87 q^{-2} +46 q^{-3} +24 q^{-4} -45 q^{-5} +18 q^{-6} +10 q^{-7} -15 q^{-8} +6 q^{-9} +2 q^{-10} -3 q^{-11} + q^{-12} }
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^{36}-3 q^{35}+q^{34}+5 q^{33}+2 q^{32}-16 q^{31}-8 q^{30}+32 q^{29}+28 q^{28}-49 q^{27}-67 q^{26}+52 q^{25}+129 q^{24}-37 q^{23}-191 q^{22}-17 q^{21}+249 q^{20}+99 q^{19}-284 q^{18}-197 q^{17}+284 q^{16}+302 q^{15}-254 q^{14}-399 q^{13}+201 q^{12}+481 q^{11}-135 q^{10}-542 q^9+60 q^8+582 q^7+18 q^6-602 q^5-88 q^4+585 q^3+162 q^2-548 q-205+460 q^{-1} +247 q^{-2} -368 q^{-3} -240 q^{-4} +254 q^{-5} +214 q^{-6} -158 q^{-7} -162 q^{-8} +82 q^{-9} +109 q^{-10} -42 q^{-11} -58 q^{-12} +19 q^{-13} +28 q^{-14} -12 q^{-15} -11 q^{-16} +9 q^{-17} +4 q^{-18} -7 q^{-19} +2 q^{-21} +2 q^{-22} -3 q^{-23} + q^{-24} }
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^{60}-3 q^{59}+q^{58}+5 q^{57}-3 q^{56}+2 q^{55}-19 q^{54}+4 q^{53}+35 q^{52}+4 q^{51}+8 q^{50}-98 q^{49}-42 q^{48}+103 q^{47}+101 q^{46}+135 q^{45}-239 q^{44}-283 q^{43}+9 q^{42}+234 q^{41}+603 q^{40}-122 q^{39}-595 q^{38}-510 q^{37}-60 q^{36}+1182 q^{35}+549 q^{34}-372 q^{33}-1125 q^{32}-1079 q^{31}+1159 q^{30}+1348 q^{29}+675 q^{28}-1081 q^{27}-2312 q^{26}+255 q^{25}+1529 q^{24}+2032 q^{23}-186 q^{22}-3039 q^{21}-1021 q^{20}+958 q^{19}+3045 q^{18}+1059 q^{17}-3118 q^{16}-2115 q^{15}+65 q^{14}+3579 q^{13}+2177 q^{12}-2832 q^{11}-2887 q^{10}-816 q^9+3738 q^8+3059 q^7-2276 q^6-3336 q^5-1678 q^4+3418 q^3+3638 q^2-1318 q-3215-2436 q^{-1} +2403 q^{-2} +3579 q^{-3} -109 q^{-4} -2287 q^{-5} -2639 q^{-6} +975 q^{-7} +2632 q^{-8} +693 q^{-9} -922 q^{-10} -1992 q^{-11} -68 q^{-12} +1279 q^{-13} +686 q^{-14} +25 q^{-15} -977 q^{-16} -305 q^{-17} +348 q^{-18} +273 q^{-19} +243 q^{-20} -298 q^{-21} -140 q^{-22} +40 q^{-23} +16 q^{-24} +132 q^{-25} -65 q^{-26} -19 q^{-27} +6 q^{-28} -29 q^{-29} +40 q^{-30} -16 q^{-31} +4 q^{-32} +6 q^{-33} -13 q^{-34} +8 q^{-35} -4 q^{-36} +2 q^{-37} +2 q^{-38} -3 q^{-39} + q^{-40} }
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^{90}-3 q^{89}+q^{88}+5 q^{87}-3 q^{86}-3 q^{85}-q^{84}-7 q^{83}+6 q^{82}+30 q^{81}+8 q^{80}-30 q^{79}-45 q^{78}-50 q^{77}+16 q^{76}+128 q^{75}+157 q^{74}+17 q^{73}-192 q^{72}-335 q^{71}-235 q^{70}+179 q^{69}+605 q^{68}+637 q^{67}+82 q^{66}-758 q^{65}-1232 q^{64}-757 q^{63}+550 q^{62}+1791 q^{61}+1853 q^{60}+283 q^{59}-1920 q^{58}-3045 q^{57}-1897 q^{56}+1146 q^{55}+3922 q^{54}+3977 q^{53}+714 q^{52}-3730 q^{51}-5956 q^{50}-3646 q^{49}+2126 q^{48}+7099 q^{47}+6977 q^{46}+1015 q^{45}-6689 q^{44}-9976 q^{43}-5332 q^{42}+4510 q^{41}+11851 q^{40}+10028 q^{39}-659 q^{38}-12077 q^{37}-14368 q^{36}-4326 q^{35}+10632 q^{34}+17699 q^{33}+9687 q^{32}-7753 q^{31}-19725 q^{30}-14815 q^{29}+4003 q^{28}+20514 q^{27}+19230 q^{26}+44 q^{25}-20330 q^{24}-22779 q^{23}-3923 q^{22}+19550 q^{21}+25525 q^{20}+7395 q^{19}-18533 q^{18}-27651 q^{17}-10427 q^{16}+17472 q^{15}+29344 q^{14}+13141 q^{13}-16275 q^{12}-30781 q^{11}-15790 q^{10}+14891 q^9+31811 q^8+18420 q^7-12795 q^6-32257 q^5-21220 q^4+10011 q^3+31695 q^2+23621 q-6124-29716 q^{-1} -25525 q^{-2} +1714 q^{-3} +26177 q^{-4} +25991 q^{-5} +2937 q^{-6} -21127 q^{-7} -24932 q^{-8} -6866 q^{-9} +15224 q^{-10} +22028 q^{-11} +9535 q^{-12} -9293 q^{-13} -17801 q^{-14} -10441 q^{-15} +4184 q^{-16} +12919 q^{-17} +9782 q^{-18} -549 q^{-19} -8329 q^{-20} -7902 q^{-21} -1488 q^{-22} +4561 q^{-23} +5657 q^{-24} +2170 q^{-25} -2047 q^{-26} -3534 q^{-27} -1961 q^{-28} +586 q^{-29} +1934 q^{-30} +1431 q^{-31} +42 q^{-32} -923 q^{-33} -875 q^{-34} -207 q^{-35} +370 q^{-36} +459 q^{-37} +187 q^{-38} -109 q^{-39} -225 q^{-40} -124 q^{-41} +41 q^{-42} +84 q^{-43} +50 q^{-44} +13 q^{-45} -32 q^{-46} -40 q^{-47} +8 q^{-48} +13 q^{-49} -3 q^{-50} +10 q^{-51} + q^{-52} -11 q^{-53} +2 q^{-54} +4 q^{-55} -4 q^{-56} +2 q^{-57} +2 q^{-58} -3 q^{-59} + q^{-60} }
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^{126}-3 q^{125}+q^{124}+5 q^{123}-3 q^{122}-3 q^{121}-6 q^{120}+11 q^{119}-5 q^{118}+q^{117}+33 q^{116}-11 q^{115}-30 q^{114}-59 q^{113}+12 q^{112}+7 q^{111}+50 q^{110}+184 q^{109}+62 q^{108}-79 q^{107}-318 q^{106}-221 q^{105}-224 q^{104}+66 q^{103}+714 q^{102}+764 q^{101}+506 q^{100}-475 q^{99}-950 q^{98}-1696 q^{97}-1349 q^{96}+541 q^{95}+2130 q^{94}+3217 q^{93}+2014 q^{92}+358 q^{91}-3476 q^{90}-5774 q^{89}-4408 q^{88}-654 q^{87}+4951 q^{86}+7925 q^{85}+9048 q^{84}+2364 q^{83}-6422 q^{82}-12597 q^{81}-13210 q^{80}-5607 q^{79}+5394 q^{78}+19052 q^{77}+20077 q^{76}+10828 q^{75}-6186 q^{74}-22897 q^{73}-28992 q^{72}-20834 q^{71}+6473 q^{70}+29024 q^{69}+40296 q^{68}+28989 q^{67}-1052 q^{66}-36130 q^{65}-56952 q^{64}-39274 q^{63}-1927 q^{62}+45337 q^{61}+70330 q^{60}+56932 q^{59}+3884 q^{58}-60852 q^{57}-87335 q^{56}-70063 q^{55}-2028 q^{54}+73356 q^{53}+113490 q^{52}+80747 q^{51}-9705 q^{50}-93783 q^{49}-133650 q^{48}-84650 q^{47}+21804 q^{46}+127695 q^{45}+151189 q^{44}+73937 q^{43}-48988 q^{42}-156602 q^{41}-160004 q^{40}-58160 q^{39}+95729 q^{38}+184571 q^{37}+150009 q^{36}+19514 q^{35}-139593 q^{34}-202709 q^{33}-130123 q^{32}+44450 q^{31}+184290 q^{30}+197880 q^{29}+80015 q^{28}-107233 q^{27}-217422 q^{26}-177844 q^{25}+117 q^{24}+171609 q^{23}+222886 q^{22}+120894 q^{21}-80040 q^{20}-221925 q^{19}-207773 q^{18}-30735 q^{17}+161460 q^{16}+240301 q^{15}+150885 q^{14}-59247 q^{13}-226308 q^{12}-234155 q^{11}-60229 q^{10}+149756 q^9+256463 q^8+184503 q^7-28946 q^6-221404 q^5-260122 q^4-103337 q^3+116818 q^2+257165 q+221314+25412 q^{-1} -184231 q^{-2} -265980 q^{-3} -154497 q^{-4} +49941 q^{-5} +216685 q^{-6} +235317 q^{-7} +91357 q^{-8} -105680 q^{-9} -224132 q^{-10} -180644 q^{-11} -30045 q^{-12} +131162 q^{-13} +198022 q^{-14} +129097 q^{-15} -15560 q^{-16} -137253 q^{-17} -153672 q^{-18} -78103 q^{-19} +38491 q^{-20} +118814 q^{-21} +113070 q^{-22} +38645 q^{-23} -49053 q^{-24} -89136 q^{-25} -73297 q^{-26} -15256 q^{-27} +43494 q^{-28} +64186 q^{-29} +42586 q^{-30} -315 q^{-31} -31353 q^{-32} -40384 q^{-33} -23473 q^{-34} +4683 q^{-35} +22655 q^{-36} +22982 q^{-37} +9688 q^{-38} -3983 q^{-39} -13618 q^{-40} -12778 q^{-41} -3840 q^{-42} +4221 q^{-43} +7427 q^{-44} +5249 q^{-45} +1824 q^{-46} -2636 q^{-47} -4221 q^{-48} -2246 q^{-49} +28 q^{-50} +1504 q^{-51} +1457 q^{-52} +1212 q^{-53} -185 q^{-54} -1038 q^{-55} -619 q^{-56} -191 q^{-57} +202 q^{-58} +215 q^{-59} +411 q^{-60} +51 q^{-61} -239 q^{-62} -102 q^{-63} -52 q^{-64} +27 q^{-65} -11 q^{-66} +110 q^{-67} +22 q^{-68} -60 q^{-69} -4 q^{-70} -8 q^{-71} +11 q^{-72} -19 q^{-73} +23 q^{-74} +7 q^{-75} -16 q^{-76} +4 q^{-77} -2 q^{-78} +4 q^{-79} -4 q^{-80} +2 q^{-81} +2 q^{-82} -3 q^{-83} + q^{-84} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

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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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