10 23

10_22

10_24

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 23's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 23 at Knotilus!

[edit 10 23 Quick Notes]

[edit 10 23 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_3,12,4,13 X_13,1,14,20 X_5,15,6,14 X_7,17,8,16 X_15,7,16,6 X_19,9,20,8 X_9,19,10,18 X_17,11,18,10 X_11,2,12,3
Gauss code	-1, 10, -2, 1, -4, 6, -5, 7, -8, 9, -10, 2, -3, 4, -6, 5, -9, 8, -7, 3
Dowker-Thistlethwaite code	4 12 14 16 18 2 20 6 10 8
Conway Notation	[33112]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 4,

Braid index is 4

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

[edit Notes on presentations of 10 23]

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 23"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X_3,12,4,13 X_13,1,14,20 X_5,15,6,14 X_7,17,8,16 X_15,7,16,6 X_19,9,20,8 X_9,19,10,18 X_17,11,18,10 X_11,2,12,3

In[5]:= GaussCode[K]

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

In[6]:= DTCode[K]

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

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [33112]

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

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

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

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

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

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

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

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-2][-10]
Hyperbolic Volume	11.3932
A-Polynomial	See Data:10 23/A-polynomial

[edit Notes for 10 23's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Topological 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Concordance genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}
Rasmussen s-Invariant	2

[edit Notes for 10 23'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^3-7 t^2+13 t-15+13 t^{-1} -7 t^{-2} +2 t^{-3} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^6+5 z^4+3 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 59, 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^8+2 q^7-4 q^6+7 q^5-9 q^4+10 q^3-9 q^2+8 q-5+3 q^{-1} - q^{-2} }
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^{-4} +3 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -z^4+2 z^2 a^{-2} +6 z^2 a^{-4} -3 z^2 a^{-6} -2 z^2+3 a^{-4} -2 a^{-6} }
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-3} +z^9 a^{-5} +3 z^8 a^{-2} +5 z^8 a^{-4} +2 z^8 a^{-6} +4 z^7 a^{-1} +3 z^7 a^{-3} +z^7 a^{-5} +2 z^7 a^{-7} -5 z^6 a^{-2} -13 z^6 a^{-4} -3 z^6 a^{-6} +2 z^6 a^{-8} +3 z^6+a z^5-9 z^5 a^{-1} -9 z^5 a^{-3} -2 z^5 a^{-5} -2 z^5 a^{-7} +z^5 a^{-9} +3 z^4 a^{-2} +20 z^4 a^{-4} +5 z^4 a^{-6} -5 z^4 a^{-8} -7 z^4-2 a z^3+5 z^3 a^{-1} +9 z^3 a^{-3} +3 z^3 a^{-5} -2 z^3 a^{-7} -3 z^3 a^{-9} -z^2 a^{-2} -13 z^2 a^{-4} -6 z^2 a^{-6} +3 z^2 a^{-8} +3 z^2-z a^{-1} -2 z a^{-3} -2 z a^{-5} +z a^{-7} +2 z a^{-9} +3 a^{-4} +2 a^{-6} }
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^6+q^4+2 q^{-2} -2 q^{-4} +2 q^{-6} + q^{-10} +2 q^{-12} - q^{-14} +2 q^{-16} - q^{-18} - q^{-20} - q^{-24} }
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^{32}-2 q^{30}+4 q^{28}-7 q^{26}+6 q^{24}-5 q^{22}-2 q^{20}+14 q^{18}-24 q^{16}+33 q^{14}-32 q^{12}+17 q^{10}+7 q^8-37 q^6+63 q^4-72 q^2+58-22 q^{-2} -25 q^{-4} +64 q^{-6} -81 q^{-8} +74 q^{-10} -37 q^{-12} -9 q^{-14} +45 q^{-16} -59 q^{-18} +42 q^{-20} -2 q^{-22} -37 q^{-24} +60 q^{-26} -51 q^{-28} +17 q^{-30} +35 q^{-32} -80 q^{-34} +104 q^{-36} -91 q^{-38} +45 q^{-40} +17 q^{-42} -77 q^{-44} +113 q^{-46} -110 q^{-48} +76 q^{-50} -19 q^{-52} -35 q^{-54} +71 q^{-56} -75 q^{-58} +48 q^{-60} -4 q^{-62} -31 q^{-64} +48 q^{-66} -34 q^{-68} +2 q^{-70} +40 q^{-72} -62 q^{-74} +64 q^{-76} -42 q^{-78} -2 q^{-80} +40 q^{-82} -68 q^{-84} +72 q^{-86} -53 q^{-88} +24 q^{-90} +4 q^{-92} -31 q^{-94} +41 q^{-96} -42 q^{-98} +30 q^{-100} -17 q^{-102} + q^{-104} +9 q^{-106} -15 q^{-108} +16 q^{-110} -13 q^{-112} +9 q^{-114} -2 q^{-116} -2 q^{-118} +3 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128} }

Further Quantum Invariants

Further quantum knot invariants for 10_23.

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^5+2 q^3-2 q+3 q^{-1} - q^{-3} + q^{-5} + q^{-7} -2 q^{-9} +3 q^{-11} -2 q^{-13} + q^{-15} - q^{-17} }
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^{16}-2 q^{14}-q^{12}+6 q^{10}-5 q^8-7 q^6+13 q^4-q^2-14+15 q^{-2} +5 q^{-4} -16 q^{-6} +8 q^{-8} +8 q^{-10} -8 q^{-12} -2 q^{-14} +7 q^{-16} +4 q^{-18} -13 q^{-20} +2 q^{-22} +14 q^{-24} -15 q^{-26} -4 q^{-28} +16 q^{-30} -8 q^{-32} -5 q^{-34} +9 q^{-36} -3 q^{-38} -3 q^{-40} +3 q^{-42} - q^{-44} - q^{-46} + q^{-48} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{33}+2 q^{31}+q^{29}-3 q^{27}-3 q^{25}+5 q^{23}+9 q^{21}-9 q^{19}-17 q^{17}+7 q^{15}+28 q^{13}-40 q^9-15 q^7+49 q^5+32 q^3-46 q-54 q^{-1} +43 q^{-3} +70 q^{-5} -27 q^{-7} -77 q^{-9} +12 q^{-11} +78 q^{-13} +5 q^{-15} -67 q^{-17} -18 q^{-19} +53 q^{-21} +28 q^{-23} -33 q^{-25} -40 q^{-27} +11 q^{-29} +47 q^{-31} +11 q^{-33} -53 q^{-35} -32 q^{-37} +52 q^{-39} +54 q^{-41} -46 q^{-43} -69 q^{-45} +33 q^{-47} +72 q^{-49} -16 q^{-51} -69 q^{-53} -2 q^{-55} +60 q^{-57} +11 q^{-59} -40 q^{-61} -16 q^{-63} +26 q^{-65} +14 q^{-67} -14 q^{-69} -10 q^{-71} +7 q^{-73} +5 q^{-75} -4 q^{-77} -2 q^{-79} +2 q^{-81} + q^{-83} -2 q^{-85} + q^{-89} + q^{-91} - q^{-93} }
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^{56}-2 q^{54}-q^{52}+3 q^{50}+3 q^{46}-8 q^{44}-4 q^{42}+11 q^{40}+4 q^{38}+12 q^{36}-25 q^{34}-26 q^{32}+18 q^{30}+25 q^{28}+52 q^{26}-36 q^{24}-81 q^{22}-26 q^{20}+32 q^{18}+150 q^{16}+30 q^{14}-121 q^{12}-152 q^{10}-62 q^8+232 q^6+200 q^4-33 q^2-265-271 q^{-2} +177 q^{-4} +352 q^{-6} +172 q^{-8} -236 q^{-10} -442 q^{-12} -6 q^{-14} +348 q^{-16} +341 q^{-18} -86 q^{-20} -444 q^{-22} -167 q^{-24} +212 q^{-26} +356 q^{-28} +60 q^{-30} -300 q^{-32} -229 q^{-34} +47 q^{-36} +267 q^{-38} +156 q^{-40} -114 q^{-42} -235 q^{-44} -104 q^{-46} +149 q^{-48} +233 q^{-50} +81 q^{-52} -232 q^{-54} -257 q^{-56} +13 q^{-58} +292 q^{-60} +286 q^{-62} -170 q^{-64} -372 q^{-66} -167 q^{-68} +257 q^{-70} +442 q^{-72} -16 q^{-74} -351 q^{-76} -317 q^{-78} +87 q^{-80} +428 q^{-82} +148 q^{-84} -174 q^{-86} -322 q^{-88} -94 q^{-90} +255 q^{-92} +184 q^{-94} +11 q^{-96} -182 q^{-98} -143 q^{-100} +73 q^{-102} +99 q^{-104} +73 q^{-106} -48 q^{-108} -84 q^{-110} +16 q^{-114} +44 q^{-116} +2 q^{-118} -25 q^{-120} -2 q^{-122} -10 q^{-124} +12 q^{-126} +4 q^{-128} -3 q^{-130} +5 q^{-132} -7 q^{-134} + q^{-136} - q^{-140} +4 q^{-142} - q^{-144} - q^{-148} - q^{-150} + q^{-152} }
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^{85}+2 q^{83}+q^{81}-3 q^{79}+3 q^{71}+3 q^{69}-7 q^{67}-8 q^{65}+4 q^{63}+10 q^{61}+12 q^{59}+4 q^{57}-17 q^{55}-37 q^{53}-17 q^{51}+34 q^{49}+61 q^{47}+51 q^{45}-16 q^{43}-101 q^{41}-128 q^{39}-32 q^{37}+127 q^{35}+216 q^{33}+156 q^{31}-76 q^{29}-323 q^{27}-357 q^{25}-74 q^{23}+358 q^{21}+600 q^{19}+372 q^{17}-260 q^{15}-818 q^{13}-794 q^{11}-40 q^9+932 q^7+1247 q^5+520 q^3-807 q-1644 q^{-1} -1142 q^{-3} +480 q^{-5} +1864 q^{-7} +1736 q^{-9} +70 q^{-11} -1820 q^{-13} -2233 q^{-15} -682 q^{-17} +1552 q^{-19} +2486 q^{-21} +1251 q^{-23} -1093 q^{-25} -2482 q^{-27} -1674 q^{-29} +579 q^{-31} +2250 q^{-33} +1879 q^{-35} -102 q^{-37} -1852 q^{-39} -1886 q^{-41} -291 q^{-43} +1409 q^{-45} +1735 q^{-47} +562 q^{-49} -962 q^{-51} -1515 q^{-53} -745 q^{-55} +561 q^{-57} +1282 q^{-59} +888 q^{-61} -209 q^{-63} -1089 q^{-65} -1041 q^{-67} -121 q^{-69} +922 q^{-71} +1237 q^{-73} +485 q^{-75} -786 q^{-77} -1464 q^{-79} -883 q^{-81} +589 q^{-83} +1692 q^{-85} +1354 q^{-87} -308 q^{-89} -1859 q^{-91} -1820 q^{-93} -95 q^{-95} +1858 q^{-97} +2237 q^{-99} +607 q^{-101} -1657 q^{-103} -2506 q^{-105} -1132 q^{-107} +1234 q^{-109} +2517 q^{-111} +1602 q^{-113} -652 q^{-115} -2265 q^{-117} -1898 q^{-119} +45 q^{-121} +1786 q^{-123} +1919 q^{-125} +495 q^{-127} -1169 q^{-129} -1723 q^{-131} -838 q^{-133} +587 q^{-135} +1323 q^{-137} +940 q^{-139} -92 q^{-141} -882 q^{-143} -859 q^{-145} -195 q^{-147} +475 q^{-149} +652 q^{-151} +315 q^{-153} -178 q^{-155} -423 q^{-157} -302 q^{-159} +5 q^{-161} +233 q^{-163} +228 q^{-165} +64 q^{-167} -99 q^{-169} -145 q^{-171} -76 q^{-173} +25 q^{-175} +79 q^{-177} +60 q^{-179} +5 q^{-181} -35 q^{-183} -36 q^{-185} -17 q^{-187} +9 q^{-189} +24 q^{-191} +14 q^{-193} -2 q^{-195} -6 q^{-197} -10 q^{-199} -7 q^{-201} +5 q^{-203} +6 q^{-205} + q^{-207} +2 q^{-209} - q^{-211} -4 q^{-213} - q^{-215} + q^{-217} + q^{-221} + q^{-223} - q^{-225} }

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^6+q^4+2 q^{-2} -2 q^{-4} +2 q^{-6} + q^{-10} +2 q^{-12} - q^{-14} +2 q^{-16} - q^{-18} - q^{-20} - q^{-24} }
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^{20}-4 q^{18}+10 q^{16}-22 q^{14}+44 q^{12}-76 q^{10}+116 q^8-168 q^6+224 q^4-276 q^2+308-318 q^{-2} +297 q^{-4} -230 q^{-6} +130 q^{-8} +10 q^{-10} -153 q^{-12} +310 q^{-14} -442 q^{-16} +548 q^{-18} -605 q^{-20} +612 q^{-22} -568 q^{-24} +476 q^{-26} -355 q^{-28} +210 q^{-30} -64 q^{-32} -68 q^{-34} +168 q^{-36} -238 q^{-38} +272 q^{-40} -278 q^{-42} +253 q^{-44} -218 q^{-46} +182 q^{-48} -140 q^{-50} +106 q^{-52} -76 q^{-54} +52 q^{-56} -36 q^{-58} +21 q^{-60} -12 q^{-62} +6 q^{-64} -2 q^{-66} + q^{-68} }
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^{18}-q^{16}-2 q^{14}+2 q^{12}+3 q^{10}-3 q^8-6 q^6+3 q^4+6 q^2-5-4 q^{-2} +8 q^{-4} +3 q^{-6} -5 q^{-8} + q^{-10} +7 q^{-12} - q^{-14} - q^{-16} +6 q^{-18} +2 q^{-20} -6 q^{-22} +3 q^{-24} +5 q^{-26} -8 q^{-28} -4 q^{-30} +6 q^{-32} +2 q^{-34} -7 q^{-36} -2 q^{-38} +6 q^{-40} -4 q^{-44} +2 q^{-48} - q^{-50} - q^{-52} + q^{-62} }

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^{14}-2 q^{12}+3 q^8-6 q^6+3 q^4+7 q^2-11+4 q^{-2} +10 q^{-4} -14 q^{-6} +2 q^{-8} +11 q^{-10} -7 q^{-12} - q^{-14} +8 q^{-16} +2 q^{-18} -2 q^{-20} +10 q^{-24} -3 q^{-26} -10 q^{-28} +12 q^{-30} -3 q^{-32} -14 q^{-34} +10 q^{-36} -9 q^{-40} +5 q^{-42} + q^{-44} -4 q^{-46} +2 q^{-48} + q^{-50} - q^{-52} + q^{-54} }

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^7+q^5-q^3+2 q- q^{-1} +2 q^{-3} -2 q^{-5} + q^{-7} + q^{-11} +2 q^{-13} + q^{-15} +3 q^{-17} - q^{-19} +2 q^{-21} -2 q^{-23} -2 q^{-27} - q^{-31} }

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^{16}-q^{14}-q^{12}+q^{10}-q^8-2 q^6+3 q^4+3 q^2-4- q^{-2} +6 q^{-4} + q^{-6} -10 q^{-8} +2 q^{-10} +10 q^{-12} -5 q^{-14} -7 q^{-16} +10 q^{-18} +5 q^{-20} -7 q^{-22} +4 q^{-24} +11 q^{-26} -2 q^{-30} +11 q^{-32} +5 q^{-34} -9 q^{-36} +2 q^{-38} +7 q^{-40} -9 q^{-42} -11 q^{-44} +3 q^{-46} -9 q^{-50} -4 q^{-52} +4 q^{-54} + q^{-56} -3 q^{-58} +2 q^{-60} +3 q^{-62} + q^{-68} }

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^8+q^6-q^4+q^2+1- q^{-2} +2 q^{-4} -2 q^{-6} + q^{-8} - q^{-10} + q^{-12} + q^{-14} +2 q^{-16} +2 q^{-18} +2 q^{-20} +3 q^{-22} - q^{-24} +2 q^{-26} -2 q^{-28} - q^{-30} - q^{-32} -2 q^{-34} - q^{-38} }

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

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^{24}-2 q^{20}-2 q^{18}+2 q^{16}+5 q^{14}-q^{12}-8 q^{10}-4 q^8+9 q^6+11 q^4-5 q^2-15-3 q^{-2} +15 q^{-4} +12 q^{-6} -10 q^{-8} -15 q^{-10} +2 q^{-12} +15 q^{-14} +4 q^{-16} -10 q^{-18} -6 q^{-20} +9 q^{-22} +8 q^{-24} -5 q^{-26} -8 q^{-28} +5 q^{-30} +10 q^{-32} - q^{-34} -10 q^{-36} +12 q^{-40} +4 q^{-42} -12 q^{-44} -9 q^{-46} +10 q^{-48} +13 q^{-50} -5 q^{-52} -17 q^{-54} -5 q^{-56} +13 q^{-58} +10 q^{-60} -6 q^{-62} -12 q^{-64} -2 q^{-66} +8 q^{-68} +5 q^{-70} -3 q^{-72} -5 q^{-74} - q^{-76} +3 q^{-78} +2 q^{-80} - q^{-82} - q^{-84} + q^{-88} }

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^{18}-2 q^{16}+2 q^{14}-4 q^{12}+6 q^{10}-8 q^8+9 q^6-10 q^4+13 q^2-12+11 q^{-2} -10 q^{-4} +10 q^{-6} -7 q^{-8} +2 q^{-12} -4 q^{-14} +11 q^{-16} -15 q^{-18} +18 q^{-20} -17 q^{-22} +26 q^{-24} -20 q^{-26} +23 q^{-28} -18 q^{-30} +22 q^{-32} -12 q^{-34} +10 q^{-36} -9 q^{-38} + q^{-40} +2 q^{-42} -7 q^{-44} +4 q^{-46} -12 q^{-48} +11 q^{-50} -11 q^{-52} +9 q^{-54} -11 q^{-56} +9 q^{-58} -7 q^{-60} +5 q^{-62} -5 q^{-64} +4 q^{-66} -2 q^{-68} +2 q^{-70} - q^{-72} + q^{-74} }

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^{32}-2 q^{30}+4 q^{28}-7 q^{26}+6 q^{24}-5 q^{22}-2 q^{20}+14 q^{18}-24 q^{16}+33 q^{14}-32 q^{12}+17 q^{10}+7 q^8-37 q^6+63 q^4-72 q^2+58-22 q^{-2} -25 q^{-4} +64 q^{-6} -81 q^{-8} +74 q^{-10} -37 q^{-12} -9 q^{-14} +45 q^{-16} -59 q^{-18} +42 q^{-20} -2 q^{-22} -37 q^{-24} +60 q^{-26} -51 q^{-28} +17 q^{-30} +35 q^{-32} -80 q^{-34} +104 q^{-36} -91 q^{-38} +45 q^{-40} +17 q^{-42} -77 q^{-44} +113 q^{-46} -110 q^{-48} +76 q^{-50} -19 q^{-52} -35 q^{-54} +71 q^{-56} -75 q^{-58} +48 q^{-60} -4 q^{-62} -31 q^{-64} +48 q^{-66} -34 q^{-68} +2 q^{-70} +40 q^{-72} -62 q^{-74} +64 q^{-76} -42 q^{-78} -2 q^{-80} +40 q^{-82} -68 q^{-84} +72 q^{-86} -53 q^{-88} +24 q^{-90} +4 q^{-92} -31 q^{-94} +41 q^{-96} -42 q^{-98} +30 q^{-100} -17 q^{-102} + q^{-104} +9 q^{-106} -15 q^{-108} +16 q^{-110} -13 q^{-112} +9 q^{-114} -2 q^{-116} -2 q^{-118} +3 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128} }

.

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 23"];

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^3-7 t^2+13 t-15+13 t^{-1} -7 t^{-2} +2 t^{-3} }

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

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

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

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

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 59, 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^8+2 q^7-4 q^6+7 q^5-9 q^4+10 q^3-9 q^2+8 q-5+3 q^{-1} - q^{-2} }

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

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

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

Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-3} +z^9 a^{-5} +3 z^8 a^{-2} +5 z^8 a^{-4} +2 z^8 a^{-6} +4 z^7 a^{-1} +3 z^7 a^{-3} +z^7 a^{-5} +2 z^7 a^{-7} -5 z^6 a^{-2} -13 z^6 a^{-4} -3 z^6 a^{-6} +2 z^6 a^{-8} +3 z^6+a z^5-9 z^5 a^{-1} -9 z^5 a^{-3} -2 z^5 a^{-5} -2 z^5 a^{-7} +z^5 a^{-9} +3 z^4 a^{-2} +20 z^4 a^{-4} +5 z^4 a^{-6} -5 z^4 a^{-8} -7 z^4-2 a z^3+5 z^3 a^{-1} +9 z^3 a^{-3} +3 z^3 a^{-5} -2 z^3 a^{-7} -3 z^3 a^{-9} -z^2 a^{-2} -13 z^2 a^{-4} -6 z^2 a^{-6} +3 z^2 a^{-8} +3 z^2-z a^{-1} -2 z a^{-3} -2 z a^{-5} +z a^{-7} +2 z a^{-9} +3 a^{-4} +2 a^{-6} }

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_52,}

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 23"];

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^3-7 t^2+13 t-15+13 t^{-1} -7 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^8+2 q^7-4 q^6+7 q^5-9 q^4+10 q^3-9 q^2+8 q-5+3 q^{-1} - q^{-2} } }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {10_52,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

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

Out[6]= {}

Vassiliev invariants

V₂ and V₃:

(3, 5)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 12}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 40}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 72}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 142}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -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 480}

Failed to parse (SVG (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{2032}{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{352}{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 -24}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 800}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1704}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -72}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{32991}{10}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{12494}{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{982}{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{1}{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 -\frac{1249}{10}}

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

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 2 is the signature of 10 23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

1

13

3

1

-2

11

4

1

3

9

5

3

-2

7

5

4

1

5

4

5

1

3

4

5

-1

1

2

5

3

-1

1

3

-2

-3

2

-5

1

-1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-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}}
Failed to parse (SVG (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}^{3}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=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}^{5}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{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=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^{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=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}^{3}\oplus{\mathbb Z}_2^{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=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^{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=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}\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=7}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_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^{23}-2 q^{22}+5 q^{20}-8 q^{19}+17 q^{17}-22 q^{16}-3 q^{15}+41 q^{14}-42 q^{13}-14 q^{12}+70 q^{11}-54 q^{10}-29 q^9+87 q^8-51 q^7-38 q^6+81 q^5-35 q^4-38 q^3+57 q^2-14 q-28+28 q^{-1} - q^{-2} -14 q^{-3} +8 q^{-4} + q^{-5} -3 q^{-6} + q^{-7} }
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}+2 q^{44}-q^{42}-3 q^{41}+5 q^{40}+q^{39}-5 q^{38}-5 q^{37}+14 q^{36}+3 q^{35}-22 q^{34}-9 q^{33}+42 q^{32}+15 q^{31}-64 q^{30}-33 q^{29}+93 q^{28}+64 q^{27}-126 q^{26}-100 q^{25}+146 q^{24}+152 q^{23}-165 q^{22}-202 q^{21}+169 q^{20}+252 q^{19}-167 q^{18}-286 q^{17}+148 q^{16}+316 q^{15}-131 q^{14}-322 q^{13}+97 q^{12}+323 q^{11}-70 q^{10}-297 q^9+26 q^8+274 q^7+2 q^6-224 q^5-40 q^4+185 q^3+52 q^2-127 q-67+88 q^{-1} +60 q^{-2} -49 q^{-3} -50 q^{-4} +24 q^{-5} +35 q^{-6} -9 q^{-7} -22 q^{-8} +3 q^{-9} +11 q^{-10} - q^{-11} -4 q^{-12} - q^{-13} +3 q^{-14} - q^{-15} }
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^{74}-2 q^{73}+q^{71}-q^{70}+6 q^{69}-7 q^{68}+q^{67}+2 q^{66}-9 q^{65}+18 q^{64}-15 q^{63}+8 q^{62}+10 q^{61}-31 q^{60}+26 q^{59}-38 q^{58}+35 q^{57}+52 q^{56}-59 q^{55}+10 q^{54}-122 q^{53}+71 q^{52}+173 q^{51}-33 q^{50}-16 q^{49}-338 q^{48}+32 q^{47}+366 q^{46}+140 q^{45}+55 q^{44}-687 q^{43}-196 q^{42}+514 q^{41}+462 q^{40}+335 q^{39}-1028 q^{38}-600 q^{37}+480 q^{36}+797 q^{35}+793 q^{34}-1213 q^{33}-1024 q^{32}+275 q^{31}+999 q^{30}+1249 q^{29}-1207 q^{28}-1303 q^{27}+5 q^{26}+1024 q^{25}+1562 q^{24}-1055 q^{23}-1387 q^{22}-248 q^{21}+893 q^{20}+1683 q^{19}-785 q^{18}-1276 q^{17}-468 q^{16}+617 q^{15}+1612 q^{14}-425 q^{13}-980 q^{12}-612 q^{11}+238 q^{10}+1335 q^9-67 q^8-553 q^7-605 q^6-116 q^5+899 q^4+139 q^3-145 q^2-425 q-291+451 q^{-1} +145 q^{-2} +87 q^{-3} -192 q^{-4} -259 q^{-5} +157 q^{-6} +55 q^{-7} +118 q^{-8} -41 q^{-9} -139 q^{-10} +39 q^{-11} -3 q^{-12} +63 q^{-13} +4 q^{-14} -51 q^{-15} +12 q^{-16} -10 q^{-17} +19 q^{-18} +5 q^{-19} -14 q^{-20} +4 q^{-21} -3 q^{-22} +4 q^{-23} + q^{-24} -3 q^{-25} + q^{-26} }
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^{110}+2 q^{109}-q^{107}+q^{106}-2 q^{105}-4 q^{104}+5 q^{103}+3 q^{102}-2 q^{101}+6 q^{100}-3 q^{99}-16 q^{98}+2 q^{97}+7 q^{96}+2 q^{95}+22 q^{94}+7 q^{93}-31 q^{92}-24 q^{91}-12 q^{90}+3 q^{89}+62 q^{88}+62 q^{87}-12 q^{86}-78 q^{85}-113 q^{84}-66 q^{83}+108 q^{82}+225 q^{81}+152 q^{80}-73 q^{79}-341 q^{78}-373 q^{77}-13 q^{76}+470 q^{75}+645 q^{74}+264 q^{73}-518 q^{72}-1043 q^{71}-677 q^{70}+447 q^{69}+1435 q^{68}+1296 q^{67}-135 q^{66}-1779 q^{65}-2102 q^{64}-438 q^{63}+1989 q^{62}+2960 q^{61}+1289 q^{60}-1912 q^{59}-3843 q^{58}-2381 q^{57}+1622 q^{56}+4573 q^{55}+3543 q^{54}-997 q^{53}-5126 q^{52}-4747 q^{51}+248 q^{50}+5422 q^{49}+5807 q^{48}+633 q^{47}-5505 q^{46}-6700 q^{45}-1477 q^{44}+5383 q^{43}+7358 q^{42}+2295 q^{41}-5167 q^{40}-7803 q^{39}-2949 q^{38}+4802 q^{37}+8036 q^{36}+3566 q^{35}-4415 q^{34}-8118 q^{33}-3992 q^{32}+3882 q^{31}+7988 q^{30}+4446 q^{29}-3318 q^{28}-7724 q^{27}-4713 q^{26}+2576 q^{25}+7218 q^{24}+4999 q^{23}-1794 q^{22}-6551 q^{21}-5039 q^{20}+876 q^{19}+5623 q^{18}+5033 q^{17}-44 q^{16}-4570 q^{15}-4668 q^{14}-795 q^{13}+3370 q^{12}+4225 q^{11}+1345 q^{10}-2226 q^9-3433 q^8-1729 q^7+1136 q^6+2674 q^5+1758 q^4-336 q^3-1767 q^2-1601 q-248+1052 q^{-1} +1256 q^{-2} +501 q^{-3} -440 q^{-4} -874 q^{-5} -563 q^{-6} +80 q^{-7} +502 q^{-8} +477 q^{-9} +118 q^{-10} -242 q^{-11} -335 q^{-12} -162 q^{-13} +70 q^{-14} +194 q^{-15} +152 q^{-16} +5 q^{-17} -103 q^{-18} -102 q^{-19} -19 q^{-20} +35 q^{-21} +56 q^{-22} +32 q^{-23} -18 q^{-24} -35 q^{-25} -9 q^{-26} +8 q^{-27} +5 q^{-28} +12 q^{-29} +2 q^{-30} -14 q^{-31} - q^{-32} +6 q^{-33} - q^{-34} +3 q^{-36} -4 q^{-37} - q^{-38} +3 q^{-39} - q^{-40} }
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^{153}-2 q^{152}+q^{150}-q^{149}+2 q^{148}+6 q^{146}-9 q^{145}-3 q^{144}+4 q^{143}-7 q^{142}+5 q^{141}+4 q^{140}+26 q^{139}-21 q^{138}-14 q^{137}+7 q^{136}-27 q^{135}+14 q^{133}+80 q^{132}-26 q^{131}-30 q^{130}+6 q^{129}-81 q^{128}-42 q^{127}+18 q^{126}+201 q^{125}+17 q^{124}-17 q^{123}+9 q^{122}-224 q^{121}-208 q^{120}-55 q^{119}+409 q^{118}+220 q^{117}+184 q^{116}+134 q^{115}-498 q^{114}-715 q^{113}-507 q^{112}+527 q^{111}+653 q^{110}+952 q^{109}+899 q^{108}-586 q^{107}-1684 q^{106}-1970 q^{105}-260 q^{104}+775 q^{103}+2458 q^{102}+3250 q^{101}+771 q^{100}-2277 q^{99}-4655 q^{98}-3262 q^{97}-1227 q^{96}+3484 q^{95}+7381 q^{94}+5300 q^{93}-102 q^{92}-6893 q^{91}-8586 q^{90}-7429 q^{89}+1084 q^{88}+11113 q^{87}+12954 q^{86}+7100 q^{85}-5282 q^{84}-13580 q^{83}-17441 q^{82}-6965 q^{81}+10716 q^{80}+20597 q^{79}+18465 q^{78}+2210 q^{77}-14418 q^{76}-27630 q^{75}-19214 q^{74}+4499 q^{73}+24329 q^{72}+29944 q^{71}+13646 q^{70}-9826 q^{69}-34056 q^{68}-31338 q^{67}-5273 q^{66}+23073 q^{65}+37723 q^{64}+24760 q^{63}-2185 q^{62}-35722 q^{61}-39791 q^{60}-14694 q^{59}+18893 q^{58}+40962 q^{57}+32583 q^{56}+5219 q^{55}-34283 q^{54}-43984 q^{53}-21536 q^{52}+14213 q^{51}+41035 q^{50}+36931 q^{49}+10954 q^{48}-31401 q^{47}-45125 q^{46}-26050 q^{45}+9668 q^{44}+39080 q^{43}+38960 q^{42}+15662 q^{41}-27138 q^{40}-43968 q^{39}-29319 q^{38}+4319 q^{37}+34768 q^{36}+39083 q^{35}+20320 q^{34}-20434 q^{33}-39874 q^{32}-31363 q^{31}-2562 q^{30}+27008 q^{29}+36274 q^{28}+24452 q^{27}-10960 q^{26}-31701 q^{25}-30622 q^{24}-9790 q^{23}+15895 q^{22}+29167 q^{21}+25815 q^{20}-711 q^{19}-19866 q^{18}-25357 q^{17}-14301 q^{16}+4128 q^{15}+18301 q^{14}+22281 q^{13}+6499 q^{12}-7574 q^{11}-16143 q^{10}-13598 q^9-4053 q^8+7118 q^7+14566 q^6+8058 q^5+824 q^4-6573 q^3-8568 q^2-6308 q-208+6421 q^{-1} +5171 q^{-2} +3461 q^{-3} -523 q^{-4} -2972 q^{-5} -4304 q^{-6} -2388 q^{-7} +1378 q^{-8} +1551 q^{-9} +2352 q^{-10} +1220 q^{-11} +157 q^{-12} -1614 q^{-13} -1599 q^{-14} -176 q^{-15} -284 q^{-16} +703 q^{-17} +761 q^{-18} +781 q^{-19} -247 q^{-20} -509 q^{-21} -118 q^{-22} -487 q^{-23} -18 q^{-24} +153 q^{-25} +457 q^{-26} +23 q^{-27} -71 q^{-28} +73 q^{-29} -224 q^{-30} -87 q^{-31} -38 q^{-32} +170 q^{-33} - q^{-34} -15 q^{-35} +77 q^{-36} -59 q^{-37} -31 q^{-38} -32 q^{-39} +58 q^{-40} -12 q^{-41} -15 q^{-42} +31 q^{-43} -13 q^{-44} -5 q^{-45} -12 q^{-46} +21 q^{-47} -4 q^{-48} -10 q^{-49} +9 q^{-50} -3 q^{-51} -3 q^{-53} +4 q^{-54} + q^{-55} -3 q^{-56} + q^{-57} }

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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