10 157

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10 156.gif

10_156

10 158.gif

10_158

10 157.gif
(KnotPlot image)

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[edit 10 157 Quick Notes]
[edit 10 157 Further Notes and Views]


Knot presentations

Planar diagram presentation X1627 X10,4,11,3 X16,11,17,12 X7,15,8,14 X15,9,16,8 X13,1,14,20 X19,13,20,12 X18,6,19,5 X2,10,3,9 X4,18,5,17
Gauss code -1, -9, 2, -10, 8, 1, -4, 5, 9, -2, 3, 7, -6, 4, -5, -3, 10, -8, -7, 6
Dowker-Thistlethwaite code 6 -10 -18 14 -2 -16 20 8 -4 12
Conway Notation [-3:20:20]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gif

Length is 10, width is 3,

Braid index is 3

10 157 ML.gif 10 157 AP.gif
[{4, 10}, {6, 9}, {5, 8}, {3, 6}, {11, 4}, {9, 2}, {10, 7}, {8, 3}, {7, 1}, {2, 11}, {1, 5}]

[edit Notes on presentations of 10 157]


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["10 157"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1627 X10,4,11,3 X16,11,17,12 X7,15,8,14 X15,9,16,8 X13,1,14,20 X19,13,20,12 X18,6,19,5 X2,10,3,9 X4,18,5,17
In[5]:= GaussCode[K]
Out[5]= -1, -9, 2, -10, 8, 1, -4, 5, 9, -2, 3, 7, -6, 4, -5, -3, 10, -8, -7, 6
In[6]:= DTCode[K]
Out[6]= 6 -10 -18 14 -2 -16 20 8 -4 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [-3:20:20]
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]=
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 3, 10, 3 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gif
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."
10 157 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{4, 10}, {6, 9}, {5, 8}, {3, 6}, {11, 4}, {9, 2}, {10, 7}, {8, 3}, {7, 1}, {2, 11}, {1, 5}]
In[14]:= Draw[ap]
10 157 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 2
Maximal Thurston-Bennequin number [3][-13]
Hyperbolic Volume 12.6653
A-Polynomial See Data:10 157/A-polynomial

[edit Notes for 10 157's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant 4

[edit Notes for 10 157's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6+4 z^2+1}
2nd Alexander ideal (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{7,t+1\}}
Determinant and Signature { 49, 4 }
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^{10}-4 q^9+6 q^8-8 q^7+9 q^6-8 q^5+7 q^4-4 q^3+2 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^{-6} +2 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +5 z^2 a^{-4} -2 z^2 a^{-6} +z^2 a^{-8} +2 a^{-4} - a^{-8} }
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^8 a^{-6} +z^8 a^{-8} +z^7 a^{-5} +5 z^7 a^{-7} +4 z^7 a^{-9} +2 z^6 a^{-6} +8 z^6 a^{-8} +6 z^6 a^{-10} +z^5 a^{-5} -3 z^5 a^{-7} +4 z^5 a^{-11} +3 z^4 a^{-4} -3 z^4 a^{-6} -15 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} -2 z^3 a^{-5} -6 z^3 a^{-7} -8 z^3 a^{-9} -4 z^3 a^{-11} -5 z^2 a^{-4} +7 z^2 a^{-8} +2 z^2 a^{-10} +4 z a^{-7} +4 z a^{-9} +2 a^{-4} - a^{-8} }
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 2 q^{-6} - q^{-8} +2 q^{-10} +3 q^{-16} - q^{-18} +2 q^{-20} -2 q^{-22} - q^{-24} -2 q^{-28} + q^{-30} }
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^{-28} -2 q^{-32} +12 q^{-34} -18 q^{-36} +19 q^{-38} -9 q^{-40} -10 q^{-42} +42 q^{-44} -60 q^{-46} +66 q^{-48} -44 q^{-50} -4 q^{-52} +56 q^{-54} -96 q^{-56} +101 q^{-58} -67 q^{-60} +7 q^{-62} +49 q^{-64} -79 q^{-66} +74 q^{-68} -31 q^{-70} -19 q^{-72} +61 q^{-74} -66 q^{-76} +37 q^{-78} +18 q^{-80} -70 q^{-82} +105 q^{-84} -99 q^{-86} +61 q^{-88} +4 q^{-90} -72 q^{-92} +119 q^{-94} -134 q^{-96} +99 q^{-98} -38 q^{-100} -37 q^{-102} +85 q^{-104} -102 q^{-106} +73 q^{-108} -17 q^{-110} -38 q^{-112} +64 q^{-114} -58 q^{-116} +13 q^{-118} +43 q^{-120} -81 q^{-122} +85 q^{-124} -51 q^{-126} +52 q^{-130} -83 q^{-132} +85 q^{-134} -59 q^{-136} +20 q^{-138} +14 q^{-140} -40 q^{-142} +45 q^{-144} -34 q^{-146} +22 q^{-148} -5 q^{-150} -4 q^{-152} +8 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} }
Further Quantum Invariants
Further quantum knot invariants for 10_157.

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 2 q^{-3} -2 q^{-5} +3 q^{-7} - q^{-9} + q^{-11} + q^{-13} -2 q^{-15} +2 q^{-17} -3 q^{-19} + q^{-21} }
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-4} +3 q^{-6} -5 q^{-8} -3 q^{-10} +14 q^{-12} -4 q^{-14} -14 q^{-16} +18 q^{-18} +3 q^{-20} -18 q^{-22} +10 q^{-24} +8 q^{-26} -11 q^{-28} -2 q^{-30} +8 q^{-32} +2 q^{-34} -14 q^{-36} +4 q^{-38} +14 q^{-40} -18 q^{-42} -3 q^{-44} +19 q^{-46} -9 q^{-48} -7 q^{-50} +10 q^{-52} - q^{-54} -3 q^{-56} + q^{-58} }
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 2 q^{-5} +2 q^{-7} -12 q^{-11} -4 q^{-13} +20 q^{-15} +25 q^{-17} -19 q^{-19} -51 q^{-21} +9 q^{-23} +75 q^{-25} +19 q^{-27} -86 q^{-29} -54 q^{-31} +86 q^{-33} +82 q^{-35} -68 q^{-37} -100 q^{-39} +41 q^{-41} +105 q^{-43} -16 q^{-45} -96 q^{-47} -5 q^{-49} +79 q^{-51} +25 q^{-53} -59 q^{-55} -46 q^{-57} +34 q^{-59} +59 q^{-61} -9 q^{-63} -79 q^{-65} -19 q^{-67} +88 q^{-69} +56 q^{-71} -89 q^{-73} -81 q^{-75} +73 q^{-77} +101 q^{-79} -45 q^{-81} -105 q^{-83} +15 q^{-85} +91 q^{-87} +10 q^{-89} -62 q^{-91} -22 q^{-93} +34 q^{-95} +21 q^{-97} -19 q^{-99} -11 q^{-101} +5 q^{-103} +7 q^{-105} - q^{-107} -3 q^{-109} + q^{-111} }
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 q^{-3} +2 q^{-5} +4 q^{-7} -12 q^{-11} -24 q^{-13} -14 q^{-15} +6 q^{-17} +54 q^{-19} +98 q^{-21} +54 q^{-23} -66 q^{-25} -197 q^{-27} -237 q^{-29} -79 q^{-31} +273 q^{-33} +535 q^{-35} +423 q^{-37} -103 q^{-39} -775 q^{-41} -1014 q^{-43} -438 q^{-45} +748 q^{-47} +1636 q^{-49} +1350 q^{-51} -186 q^{-53} -1966 q^{-55} -2462 q^{-57} -942 q^{-59} +1726 q^{-61} +3388 q^{-63} +2420 q^{-65} -764 q^{-67} -3756 q^{-69} -3916 q^{-71} -748 q^{-73} +3407 q^{-75} +4987 q^{-77} +2429 q^{-79} -2355 q^{-81} -5384 q^{-83} -3920 q^{-85} +933 q^{-87} +5085 q^{-89} +4872 q^{-91} +496 q^{-93} -4226 q^{-95} -5202 q^{-97} -1663 q^{-99} +3129 q^{-101} +4966 q^{-103} +2386 q^{-105} -2053 q^{-107} -4345 q^{-109} -2691 q^{-111} +1141 q^{-113} +3596 q^{-115} +2688 q^{-117} -467 q^{-119} -2879 q^{-121} -2543 q^{-123} -35 q^{-125} +2286 q^{-127} +2442 q^{-129} +473 q^{-131} -1827 q^{-133} -2490 q^{-135} -974 q^{-137} +1424 q^{-139} +2684 q^{-141} +1676 q^{-143} -928 q^{-145} -2999 q^{-147} -2587 q^{-149} +235 q^{-151} +3207 q^{-153} +3657 q^{-155} +825 q^{-157} -3155 q^{-159} -4695 q^{-161} -2126 q^{-163} +2590 q^{-165} +5395 q^{-167} +3579 q^{-169} -1510 q^{-171} -5506 q^{-173} -4812 q^{-175} +32 q^{-177} +4847 q^{-179} +5499 q^{-181} +1509 q^{-183} -3539 q^{-185} -5406 q^{-187} -2730 q^{-189} +1925 q^{-191} +4553 q^{-193} +3288 q^{-195} -396 q^{-197} -3228 q^{-199} -3148 q^{-201} -663 q^{-203} +1861 q^{-205} +2472 q^{-207} +1112 q^{-209} -761 q^{-211} -1605 q^{-213} -1085 q^{-215} +103 q^{-217} +886 q^{-219} +774 q^{-221} +145 q^{-223} -371 q^{-225} -453 q^{-227} -193 q^{-229} +135 q^{-231} +231 q^{-233} +109 q^{-235} -27 q^{-237} -92 q^{-239} -64 q^{-241} -2 q^{-243} +42 q^{-245} +30 q^{-247} -2 q^{-249} -13 q^{-251} -9 q^{-253} -2 q^{-255} +2 q^{-257} +7 q^{-259} - q^{-261} -3 q^{-263} + q^{-265} }
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+3 q^{-2} +3 q^{-4} +3 q^{-6} -3 q^{-8} -11 q^{-10} -27 q^{-12} -33 q^{-14} -9 q^{-16} +43 q^{-18} +99 q^{-20} +129 q^{-22} +113 q^{-24} -45 q^{-26} -267 q^{-28} -445 q^{-30} -381 q^{-32} -53 q^{-34} +477 q^{-36} +1073 q^{-38} +1130 q^{-40} +478 q^{-42} -828 q^{-44} -2046 q^{-46} -2515 q^{-48} -1544 q^{-50} +1000 q^{-52} +3654 q^{-54} +4885 q^{-56} +3239 q^{-58} -749 q^{-60} -5699 q^{-62} -8412 q^{-64} -6194 q^{-66} +354 q^{-68} +8364 q^{-70} +12685 q^{-72} +10360 q^{-74} +548 q^{-76} -11562 q^{-78} -18333 q^{-80} -14889 q^{-82} -1383 q^{-84} +14847 q^{-86} +24778 q^{-88} +19812 q^{-90} +1786 q^{-92} -19340 q^{-94} -30753 q^{-96} -23892 q^{-98} -1614 q^{-100} +24502 q^{-102} +36410 q^{-104} +26299 q^{-106} -1001 q^{-108} -29247 q^{-110} -40339 q^{-112} -26570 q^{-114} +5472 q^{-116} +34040 q^{-118} +41714 q^{-120} +22840 q^{-122} -10634 q^{-124} -37368 q^{-126} -40159 q^{-128} -16268 q^{-130} +16700 q^{-132} +38262 q^{-134} +34261 q^{-136} +8499 q^{-138} -21677 q^{-140} -36303 q^{-142} -25761 q^{-144} +101 q^{-146} +24307 q^{-148} +30561 q^{-150} +16556 q^{-152} -7303 q^{-154} -24199 q^{-156} -23017 q^{-158} -7265 q^{-160} +12008 q^{-162} +21086 q^{-164} +15416 q^{-166} -312 q^{-168} -14278 q^{-170} -17026 q^{-172} -8260 q^{-174} +5758 q^{-176} +14712 q^{-178} +13350 q^{-180} +2510 q^{-182} -9742 q^{-184} -15244 q^{-186} -10312 q^{-188} +2144 q^{-190} +13543 q^{-192} +16584 q^{-194} +8064 q^{-196} -6636 q^{-198} -18576 q^{-200} -18720 q^{-202} -5820 q^{-204} +12496 q^{-206} +24642 q^{-208} +20997 q^{-210} +2705 q^{-212} -20211 q^{-214} -31317 q^{-216} -22258 q^{-218} +2772 q^{-220} +28476 q^{-222} +37046 q^{-224} +21702 q^{-226} -10072 q^{-228} -36442 q^{-230} -40322 q^{-232} -17887 q^{-234} +17147 q^{-236} +42003 q^{-238} +40479 q^{-240} +12046 q^{-242} -23262 q^{-244} -43806 q^{-246} -36344 q^{-248} -6373 q^{-250} +26589 q^{-252} +41873 q^{-254} +29729 q^{-256} +1035 q^{-258} -26503 q^{-260} -35741 q^{-262} -22871 q^{-264} +2567 q^{-266} +23853 q^{-268} +27849 q^{-270} +15756 q^{-272} -4144 q^{-274} -18656 q^{-276} -20412 q^{-278} -9881 q^{-280} +4539 q^{-282} +13189 q^{-284} +13297 q^{-286} +5691 q^{-288} -3444 q^{-290} -8886 q^{-292} -7864 q^{-294} -2695 q^{-296} +2329 q^{-298} +5135 q^{-300} +4255 q^{-302} +1325 q^{-304} -1704 q^{-306} -2686 q^{-308} -1904 q^{-310} -528 q^{-312} +898 q^{-314} +1300 q^{-316} +892 q^{-318} -2 q^{-320} -452 q^{-322} -478 q^{-324} -334 q^{-326} +26 q^{-328} +217 q^{-330} +227 q^{-332} +39 q^{-334} -39 q^{-336} -64 q^{-338} -74 q^{-340} -12 q^{-342} +26 q^{-344} +44 q^{-346} -2 q^{-348} -4 q^{-350} - q^{-352} -12 q^{-354} -2 q^{-356} +2 q^{-358} +7 q^{-360} - q^{-362} -3 q^{-364} + q^{-366} }

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 2 q^{-6} - q^{-8} +2 q^{-10} +3 q^{-16} - q^{-18} +2 q^{-20} -2 q^{-22} - q^{-24} -2 q^{-28} + q^{-30} }
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 2 q^{-10} -4 q^{-14} +28 q^{-16} -60 q^{-18} +118 q^{-20} -190 q^{-22} +272 q^{-24} -340 q^{-26} +385 q^{-28} -374 q^{-30} +316 q^{-32} -196 q^{-34} +37 q^{-36} +150 q^{-38} -346 q^{-40} +502 q^{-42} -635 q^{-44} +696 q^{-46} -698 q^{-48} +630 q^{-50} -493 q^{-52} +324 q^{-54} -130 q^{-56} -56 q^{-58} +206 q^{-60} -314 q^{-62} +358 q^{-64} -354 q^{-66} +314 q^{-68} -248 q^{-70} +184 q^{-72} -120 q^{-74} +69 q^{-76} -38 q^{-78} +18 q^{-80} -6 q^{-82} + q^{-84} }
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^{-10} +4 q^{-12} -2 q^{-14} -4 q^{-16} +5 q^{-18} +4 q^{-20} -4 q^{-22} -2 q^{-24} +7 q^{-26} +6 q^{-28} -6 q^{-30} +2 q^{-32} +8 q^{-34} -4 q^{-36} -3 q^{-38} +4 q^{-40} - q^{-42} -6 q^{-44} +2 q^{-48} -7 q^{-50} -5 q^{-52} +5 q^{-54} +2 q^{-56} -9 q^{-58} +2 q^{-60} +7 q^{-62} - q^{-66} +4 q^{-70} -2 q^{-72} -2 q^{-74} + q^{-76} }

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 3 q^{-12} -3 q^{-14} +2 q^{-16} +9 q^{-18} -9 q^{-20} +4 q^{-22} +13 q^{-24} -13 q^{-26} +4 q^{-28} +11 q^{-30} -9 q^{-32} -2 q^{-34} +5 q^{-36} -3 q^{-38} -5 q^{-40} -4 q^{-42} +6 q^{-44} -2 q^{-46} -11 q^{-48} +13 q^{-50} -13 q^{-54} +12 q^{-56} + q^{-58} -8 q^{-60} +7 q^{-62} -3 q^{-66} + q^{-68} }
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 2 q^{-9} - q^{-11} +3 q^{-13} - q^{-15} +2 q^{-17} +2 q^{-21} + q^{-23} + q^{-27} -2 q^{-29} -3 q^{-33} + q^{-35} -2 q^{-37} + q^{-39} }
1,0,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 q^{-16} + q^{-18} -5 q^{-20} +20 q^{-22} -10 q^{-24} -12 q^{-26} +80 q^{-28} -122 q^{-30} +123 q^{-32} -23 q^{-34} -142 q^{-36} +301 q^{-38} -363 q^{-40} +274 q^{-42} -35 q^{-44} -229 q^{-46} +427 q^{-48} -447 q^{-50} +281 q^{-52} -60 q^{-54} -153 q^{-56} +180 q^{-58} -114 q^{-60} -4 q^{-62} +38 q^{-64} +73 q^{-66} -226 q^{-68} +349 q^{-70} -294 q^{-72} +85 q^{-74} +185 q^{-76} -396 q^{-78} +423 q^{-80} -286 q^{-82} +48 q^{-84} +170 q^{-86} -267 q^{-88} +235 q^{-90} -108 q^{-92} -23 q^{-94} +94 q^{-96} -91 q^{-98} +46 q^{-100} -3 q^{-102} -17 q^{-104} +15 q^{-106} -6 q^{-108} + q^{-110} }

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 3 q^{-18} -2 q^{-20} + q^{-22} +7 q^{-24} -3 q^{-28} +11 q^{-30} +6 q^{-32} -7 q^{-34} +3 q^{-36} +12 q^{-38} - q^{-40} -11 q^{-42} +7 q^{-44} +8 q^{-46} -14 q^{-48} -5 q^{-50} +11 q^{-52} -11 q^{-54} -13 q^{-56} +8 q^{-58} - q^{-60} -12 q^{-62} +2 q^{-64} +11 q^{-66} -3 q^{-68} -6 q^{-70} +9 q^{-72} +6 q^{-74} -8 q^{-76} - q^{-78} +6 q^{-80} -2 q^{-82} -2 q^{-84} + q^{-86} }
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 2 q^{-12} - q^{-14} +3 q^{-16} + q^{-20} +2 q^{-22} +2 q^{-26} +2 q^{-30} - q^{-32} + q^{-34} -2 q^{-36} -2 q^{-40} -2 q^{-42} + q^{-44} -2 q^{-46} + q^{-48} }

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 3 q^{-12} -5 q^{-14} +10 q^{-16} -11 q^{-18} +15 q^{-20} -14 q^{-22} +13 q^{-24} -9 q^{-26} +4 q^{-28} +3 q^{-30} -11 q^{-32} +18 q^{-34} -23 q^{-36} +27 q^{-38} -27 q^{-40} +24 q^{-42} -18 q^{-44} +12 q^{-46} -5 q^{-48} -3 q^{-50} +8 q^{-52} -13 q^{-54} +14 q^{-56} -15 q^{-58} +12 q^{-60} -9 q^{-62} +6 q^{-64} -3 q^{-66} + q^{-68} }
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 3 q^{-18} + q^{-20} -4 q^{-22} -4 q^{-24} +6 q^{-26} +10 q^{-28} -13 q^{-32} -5 q^{-34} +14 q^{-36} +13 q^{-38} -7 q^{-40} -14 q^{-42} + q^{-44} +15 q^{-46} +5 q^{-48} -11 q^{-50} -7 q^{-52} +8 q^{-54} +7 q^{-56} -6 q^{-58} -10 q^{-60} +2 q^{-62} +10 q^{-64} - q^{-66} -12 q^{-68} -3 q^{-70} +10 q^{-72} +5 q^{-74} -11 q^{-76} -10 q^{-78} +9 q^{-80} +14 q^{-82} -4 q^{-84} -16 q^{-86} -3 q^{-88} +13 q^{-90} +10 q^{-92} -7 q^{-94} -10 q^{-96} + q^{-98} +8 q^{-100} +3 q^{-102} -3 q^{-104} -3 q^{-106} + q^{-110} }

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 3 q^{-18} -4 q^{-20} +7 q^{-22} -5 q^{-24} +12 q^{-26} -10 q^{-28} +13 q^{-30} -9 q^{-32} +13 q^{-34} -7 q^{-36} +4 q^{-38} - q^{-40} + q^{-42} +8 q^{-44} -13 q^{-46} +13 q^{-48} -17 q^{-50} +19 q^{-52} -23 q^{-54} +16 q^{-56} -22 q^{-58} +17 q^{-60} -13 q^{-62} +8 q^{-64} -8 q^{-66} +3 q^{-68} +5 q^{-70} -6 q^{-72} +7 q^{-74} -11 q^{-76} +13 q^{-78} -10 q^{-80} +10 q^{-82} -10 q^{-84} +8 q^{-86} -4 q^{-88} +3 q^{-90} -3 q^{-92} + q^{-94} }

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^{-28} -2 q^{-32} +12 q^{-34} -18 q^{-36} +19 q^{-38} -9 q^{-40} -10 q^{-42} +42 q^{-44} -60 q^{-46} +66 q^{-48} -44 q^{-50} -4 q^{-52} +56 q^{-54} -96 q^{-56} +101 q^{-58} -67 q^{-60} +7 q^{-62} +49 q^{-64} -79 q^{-66} +74 q^{-68} -31 q^{-70} -19 q^{-72} +61 q^{-74} -66 q^{-76} +37 q^{-78} +18 q^{-80} -70 q^{-82} +105 q^{-84} -99 q^{-86} +61 q^{-88} +4 q^{-90} -72 q^{-92} +119 q^{-94} -134 q^{-96} +99 q^{-98} -38 q^{-100} -37 q^{-102} +85 q^{-104} -102 q^{-106} +73 q^{-108} -17 q^{-110} -38 q^{-112} +64 q^{-114} -58 q^{-116} +13 q^{-118} +43 q^{-120} -81 q^{-122} +85 q^{-124} -51 q^{-126} +52 q^{-130} -83 q^{-132} +85 q^{-134} -59 q^{-136} +20 q^{-138} +14 q^{-140} -40 q^{-142} +45 q^{-144} -34 q^{-146} +22 q^{-148} -5 q^{-150} -4 q^{-152} +8 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} }

.

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

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

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 157"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]=
In[5]:= Conway[K][z]
Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6+4 z^2+1}
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{7,t+1\}}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 49, 4 }
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^{10}-4 q^9+6 q^8-8 q^7+9 q^6-8 q^5+7 q^4-4 q^3+2 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^{-6} +2 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +5 z^2 a^{-4} -2 z^2 a^{-6} +z^2 a^{-8} +2 a^{-4} - a^{-8} }
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^8 a^{-6} +z^8 a^{-8} +z^7 a^{-5} +5 z^7 a^{-7} +4 z^7 a^{-9} +2 z^6 a^{-6} +8 z^6 a^{-8} +6 z^6 a^{-10} +z^5 a^{-5} -3 z^5 a^{-7} +4 z^5 a^{-11} +3 z^4 a^{-4} -3 z^4 a^{-6} -15 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} -2 z^3 a^{-5} -6 z^3 a^{-7} -8 z^3 a^{-9} -4 z^3 a^{-11} -5 z^2 a^{-4} +7 z^2 a^{-8} +2 z^2 a^{-10} +4 z a^{-7} +4 z a^{-9} +2 a^{-4} - a^{-8} }

"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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 157"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^3+6 t^2-11 t+13-11 t^{-1} +6 t^{-2} - t^{-3} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-4 q^9+6 q^8-8 q^7+9 q^6-8 q^5+7 q^4-4 q^3+2 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]= {}
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: (4, 8)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 128} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1016}{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{184}{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 1024} Failed to parse (SVG (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{5824}{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{1120}{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 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 \frac{2048}{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 2048} Failed to parse (SVG (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{16256}{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{2944}{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{168062}{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{584}{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{218888}{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{610}{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{10142}{15}}

V2,1 through V6,9 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=} 4 is the signature of 10 157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 012345678χ
21        11
19       3 -3
17      31 2
15     53  -2
13    43   1
11   45    1
9  34     -1
7 14      3
513       -2
32        2
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=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=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=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (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} Failed to parse (SVG (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^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\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=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}^{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=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}^{3}\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=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}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=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}\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=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 {\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^{28}-4 q^{27}+2 q^{26}+12 q^{25}-21 q^{24}+40 q^{22}-43 q^{21}-15 q^{20}+72 q^{19}-53 q^{18}-33 q^{17}+88 q^{16}-47 q^{15}-43 q^{14}+79 q^{13}-28 q^{12}-41 q^{11}+51 q^{10}-7 q^9-26 q^8+19 q^7+3 q^6-8 q^5+2 q^4+q^3}
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^{54}-4 q^{53}+2 q^{52}+8 q^{51}-q^{50}-20 q^{49}-6 q^{48}+48 q^{47}+12 q^{46}-76 q^{45}-46 q^{44}+120 q^{43}+93 q^{42}-152 q^{41}-166 q^{40}+180 q^{39}+239 q^{38}-180 q^{37}-320 q^{36}+172 q^{35}+384 q^{34}-148 q^{33}-427 q^{32}+112 q^{31}+454 q^{30}-80 q^{29}-452 q^{28}+32 q^{27}+441 q^{26}+4 q^{25}-398 q^{24}-52 q^{23}+350 q^{22}+84 q^{21}-277 q^{20}-116 q^{19}+209 q^{18}+116 q^{17}-127 q^{16}-112 q^{15}+69 q^{14}+84 q^{13}-22 q^{12}-56 q^{11}+3 q^{10}+24 q^9+10 q^8-12 q^7-2 q^6+2 q^4}
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^{88}-4 q^{87}+2 q^{86}+8 q^{85}-5 q^{84}-26 q^{82}+12 q^{81}+49 q^{80}-7 q^{79}-8 q^{78}-128 q^{77}+10 q^{76}+190 q^{75}+77 q^{74}-6 q^{73}-427 q^{72}-144 q^{71}+418 q^{70}+425 q^{69}+212 q^{68}-910 q^{67}-676 q^{66}+470 q^{65}+1014 q^{64}+867 q^{63}-1256 q^{62}-1492 q^{61}+94 q^{60}+1491 q^{59}+1803 q^{58}-1204 q^{57}-2184 q^{56}-555 q^{55}+1597 q^{54}+2587 q^{53}-857 q^{52}-2482 q^{51}-1143 q^{50}+1399 q^{49}+2981 q^{48}-434 q^{47}-2416 q^{46}-1523 q^{45}+1034 q^{44}+3000 q^{43}+8 q^{42}-2064 q^{41}-1729 q^{40}+527 q^{39}+2696 q^{38}+468 q^{37}-1448 q^{36}-1728 q^{35}-79 q^{34}+2050 q^{33}+815 q^{32}-651 q^{31}-1408 q^{30}-569 q^{29}+1159 q^{28}+822 q^{27}+35 q^{26}-798 q^{25}-666 q^{24}+357 q^{23}+482 q^{22}+301 q^{21}-224 q^{20}-397 q^{19}-25 q^{18}+118 q^{17}+193 q^{16}+24 q^{15}-107 q^{14}-51 q^{13}-16 q^{12}+42 q^{11}+26 q^{10}-5 q^9-6 q^8-8 q^7+2 q^5+q^4}
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^{130}-4 q^{129}+2 q^{128}+8 q^{127}-5 q^{126}-4 q^{125}-6 q^{124}-8 q^{123}+13 q^{122}+40 q^{121}+7 q^{120}-48 q^{119}-68 q^{118}-36 q^{117}+78 q^{116}+176 q^{115}+129 q^{114}-144 q^{113}-396 q^{112}-296 q^{111}+160 q^{110}+692 q^{109}+758 q^{108}-32 q^{107}-1179 q^{106}-1484 q^{105}-360 q^{104}+1536 q^{103}+2631 q^{102}+1328 q^{101}-1790 q^{100}-4008 q^{99}-2845 q^{98}+1456 q^{97}+5463 q^{96}+5012 q^{95}-525 q^{94}-6636 q^{93}-7500 q^{92}-1220 q^{91}+7330 q^{90}+10060 q^{89}+3465 q^{88}-7288 q^{87}-12315 q^{86}-6064 q^{85}+6636 q^{84}+14056 q^{83}+8554 q^{82}-5472 q^{81}-15120 q^{80}-10780 q^{79}+4067 q^{78}+15596 q^{77}+12534 q^{76}-2640 q^{75}-15570 q^{74}-13752 q^{73}+1245 q^{72}+15184 q^{71}+14605 q^{70}-36 q^{69}-14562 q^{68}-15012 q^{67}-1153 q^{66}+13668 q^{65}+15268 q^{64}+2264 q^{63}-12593 q^{62}-15168 q^{61}-3474 q^{60}+11160 q^{59}+14932 q^{58}+4676 q^{57}-9438 q^{56}-14260 q^{55}-5929 q^{54}+7328 q^{53}+13278 q^{52}+6968 q^{51}-4999 q^{50}-11680 q^{49}-7766 q^{48}+2536 q^{47}+9739 q^{46}+7944 q^{45}-277 q^{44}-7304 q^{43}-7553 q^{42}-1616 q^{41}+4886 q^{40}+6480 q^{39}+2752 q^{38}-2520 q^{37}-4995 q^{36}-3196 q^{35}+731 q^{34}+3312 q^{33}+2912 q^{32}+472 q^{31}-1811 q^{30}-2228 q^{29}-931 q^{28}+644 q^{27}+1392 q^{26}+968 q^{25}-31 q^{24}-692 q^{23}-645 q^{22}-244 q^{21}+206 q^{20}+392 q^{19}+208 q^{18}-20 q^{17}-119 q^{16}-132 q^{15}-56 q^{14}+40 q^{13}+50 q^{12}+20 q^{11}+12 q^{10}-12 q^9-12 q^8-4 q^7+2 q^6+2 q^4}
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^{180}-4 q^{179}+2 q^{178}+8 q^{177}-5 q^{176}-4 q^{175}-10 q^{174}+12 q^{173}-7 q^{172}+4 q^{171}+54 q^{170}-23 q^{169}-42 q^{168}-72 q^{167}+22 q^{166}+18 q^{165}+82 q^{164}+242 q^{163}-33 q^{162}-233 q^{161}-432 q^{160}-122 q^{159}+44 q^{158}+532 q^{157}+1136 q^{156}+375 q^{155}-635 q^{154}-1858 q^{153}-1498 q^{152}-738 q^{151}+1514 q^{150}+4165 q^{149}+3305 q^{148}+245 q^{147}-4664 q^{146}-6522 q^{145}-5907 q^{144}+492 q^{143}+9607 q^{142}+12440 q^{141}+7851 q^{140}-4772 q^{139}-15172 q^{138}-20327 q^{137}-10039 q^{136}+11363 q^{135}+26952 q^{134}+27751 q^{133}+7321 q^{132}-19168 q^{131}-41613 q^{130}-35477 q^{129}-1507 q^{128}+36190 q^{127}+55289 q^{126}+36015 q^{125}-7024 q^{124}-56897 q^{123}-68439 q^{122}-31478 q^{121}+28728 q^{120}+75833 q^{119}+71323 q^{118}+21409 q^{117}-55373 q^{116}-93295 q^{115}-66512 q^{114}+6293 q^{113}+79713 q^{112}+97693 q^{111}+53183 q^{110}-40029 q^{109}-101865 q^{108}-92216 q^{107}-18737 q^{106}+70654 q^{105}+108799 q^{104}+76099 q^{103}-21737 q^{102}-98220 q^{101}-104362 q^{100}-37053 q^{99}+57754 q^{98}+108943 q^{97}+88039 q^{96}-7037 q^{95}-89700 q^{94}-107403 q^{93}-48452 q^{92}+45298 q^{91}+104011 q^{90}+93541 q^{89}+5215 q^{88}-78860 q^{87}-106041 q^{86}-57406 q^{85}+31280 q^{84}+95245 q^{83}+96289 q^{82}+19181 q^{81}-63132 q^{80}-100371 q^{79}-66484 q^{78}+12007 q^{77}+79493 q^{76}+95107 q^{75}+36077 q^{74}-39273 q^{73}-86366 q^{72}-72738 q^{71}-12199 q^{70}+53631 q^{69}+84565 q^{68}+50703 q^{67}-9097 q^{66}-60604 q^{65}-68737 q^{64}-33761 q^{63}+20663 q^{62}+60674 q^{61}+53494 q^{60}+17637 q^{59}-27130 q^{58}-49863 q^{57}-41435 q^{56}-7905 q^{55}+28632 q^{54}+39725 q^{53}+28729 q^{52}+1116 q^{51}-22563 q^{50}-31324 q^{49}-19813 q^{48}+2516 q^{47}+17447 q^{46}+21868 q^{45}+12529 q^{44}-1437 q^{43}-13298 q^{42}-14847 q^{41}-7415 q^{40}+1217 q^{39}+8362 q^{38}+9085 q^{37}+5334 q^{36}-1188 q^{35}-5035 q^{34}-5090 q^{33}-3104 q^{32}+352 q^{31}+2537 q^{30}+3116 q^{29}+1525 q^{28}-85 q^{27}-1102 q^{26}-1458 q^{25}-879 q^{24}-117 q^{23}+572 q^{22}+554 q^{21}+384 q^{20}+116 q^{19}-152 q^{18}-227 q^{17}-174 q^{16}-24 q^{15}+24 q^{14}+56 q^{13}+52 q^{12}+26 q^{11}-5 q^{10}-16 q^9-8 q^8-6 q^7+2 q^4+q^3}

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages

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

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

10 156.gif

10_156

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10_158

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