10 75
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Visit 10 75's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 10 75's page at Knotilus! Visit 10 75's page at the original Knot Atlas! |
Knot presentations
Planar diagram presentation | X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,16,14,17 X7,15,8,14 X15,7,16,6 X17,20,18,1 X9,19,10,18 X19,9,20,8 |
Gauss code | -1, 4, -3, 1, -2, 7, -6, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, 8 |
Dowker-Thistlethwaite code | 4 10 12 14 18 2 16 6 20 8 |
Conway Notation | [21,21,21+] |
Length is 12, width is 5. Braid index is 5. |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
Alexander polynomial | |
Conway polynomial | |
2nd Alexander ideal (db, data sources) | |
Determinant and Signature | { 81, 0 } |
Jones polynomial | |
HOMFLY-PT polynomial (db, data sources) | |
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^{-1} +z^9 a^{-3} +7 z^8 a^{-2} +3 z^8 a^{-4} +4 z^8+7 a z^7+13 z^7 a^{-1} +9 z^7 a^{-3} +3 z^7 a^{-5} +7 a^2 z^6-4 z^6 a^{-2} -3 z^6 a^{-4} +z^6 a^{-6} +7 z^6+4 a^3 z^5-5 a z^5-29 z^5 a^{-1} -29 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-8 a^2 z^4-21 z^4 a^{-2} -9 z^4 a^{-4} -3 z^4 a^{-6} -24 z^4-3 a^3 z^3-a z^3+17 z^3 a^{-1} +24 z^3 a^{-3} +9 z^3 a^{-5} +4 a^2 z^2+20 z^2 a^{-2} +12 z^2 a^{-4} +3 z^2 a^{-6} +15 z^2-a z-5 z a^{-1} -7 z a^{-3} -3 z a^{-5} -3 a^{-2} -3 a^{-4} - a^{-6} } |
The A2 invariant | |
The G2 invariant |
A1 Invariants.
Weight | Invariant |
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1 | |
2 | |
3 | |
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^{125}-3 q^{123}+6 q^{119}-2 q^{117}-2 q^{115}-2 q^{113}-4 q^{111}+5 q^{109}+14 q^{107}-6 q^{105}-27 q^{103}-11 q^{101}+30 q^{99}+60 q^{97}+21 q^{95}-65 q^{93}-149 q^{91}-80 q^{89}+171 q^{87}+321 q^{85}+147 q^{83}-276 q^{81}-612 q^{79}-391 q^{77}+452 q^{75}+1136 q^{73}+772 q^{71}-598 q^{69}-1843 q^{67}-1513 q^{65}+611 q^{63}+2804 q^{61}+2676 q^{59}-373 q^{57}-3862 q^{55}-4272 q^{53}-408 q^{51}+4782 q^{49}+6318 q^{47}+1824 q^{45}-5292 q^{43}-8403 q^{41}-3932 q^{39}+4942 q^{37}+10239 q^{35}+6485 q^{33}-3625 q^{31}-11185 q^{29}-9089 q^{27}+1333 q^{25}+10980 q^{23}+11083 q^{21}+1516 q^{19}-9338 q^{17}-12117 q^{15}-4443 q^{13}+6725 q^{11}+11765 q^9+6815 q^7-3402 q^5-10244 q^3-8354 q+140 q^{-1} +7907 q^{-3} +8861 q^{-5} +2721 q^{-7} -5196 q^{-9} -8601 q^{-11} -4950 q^{-13} +2662 q^{-15} +7894 q^{-17} +6446 q^{-19} -426 q^{-21} -7047 q^{-23} -7609 q^{-25} -1330 q^{-27} +6335 q^{-29} +8415 q^{-31} +2851 q^{-33} -5657 q^{-35} -9279 q^{-37} -4256 q^{-39} +5029 q^{-41} +9984 q^{-43} +5791 q^{-45} -4091 q^{-47} -10585 q^{-49} -7436 q^{-51} +2722 q^{-53} +10720 q^{-55} +9145 q^{-57} -773 q^{-59} -10198 q^{-61} -10594 q^{-63} -1627 q^{-65} +8759 q^{-67} +11476 q^{-69} +4246 q^{-71} -6478 q^{-73} -11431 q^{-75} -6592 q^{-77} +3532 q^{-79} +10242 q^{-81} +8235 q^{-83} -365 q^{-85} -8093 q^{-87} -8765 q^{-89} -2409 q^{-91} +5236 q^{-93} +8065 q^{-95} +4393 q^{-97} -2297 q^{-99} -6419 q^{-101} -5148 q^{-103} -187 q^{-105} +4199 q^{-107} +4831 q^{-109} +1816 q^{-111} -2045 q^{-113} -3733 q^{-115} -2428 q^{-117} +365 q^{-119} +2334 q^{-121} +2244 q^{-123} +615 q^{-125} -1094 q^{-127} -1633 q^{-129} -913 q^{-131} +249 q^{-133} +926 q^{-135} +794 q^{-137} +169 q^{-139} -397 q^{-141} -517 q^{-143} -250 q^{-145} +96 q^{-147} +252 q^{-149} +192 q^{-151} +29 q^{-153} -100 q^{-155} -108 q^{-157} -37 q^{-159} +26 q^{-161} +40 q^{-163} +28 q^{-165} + q^{-167} -20 q^{-169} -11 q^{-171} + q^{-173} +3 q^{-175} +3 q^{-177} +3 q^{-179} -2 q^{-181} -2 q^{-183} + q^{-185} } |
A2 Invariants.
Weight | Invariant |
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1,0 | |
2,0 |
A3 Invariants.
Weight | Invariant |
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0,1,0 | |
1,0,0 |
B2 Invariants.
Weight | Invariant |
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0,1 | |
1,0 |
G2 Invariants.
Weight | Invariant |
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1,0 |
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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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 75"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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In[5]:=
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Conway[K][z]
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Out[5]=
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In[6]:=
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Alexander[K, 2][t]
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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.
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Out[6]=
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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 81, 0 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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Failed to parse (SVG (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^{-1} +z^9 a^{-3} +7 z^8 a^{-2} +3 z^8 a^{-4} +4 z^8+7 a z^7+13 z^7 a^{-1} +9 z^7 a^{-3} +3 z^7 a^{-5} +7 a^2 z^6-4 z^6 a^{-2} -3 z^6 a^{-4} +z^6 a^{-6} +7 z^6+4 a^3 z^5-5 a z^5-29 z^5 a^{-1} -29 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-8 a^2 z^4-21 z^4 a^{-2} -9 z^4 a^{-4} -3 z^4 a^{-6} -24 z^4-3 a^3 z^3-a z^3+17 z^3 a^{-1} +24 z^3 a^{-3} +9 z^3 a^{-5} +4 a^2 z^2+20 z^2 a^{-2} +12 z^2 a^{-4} +3 z^2 a^{-6} +15 z^2-a z-5 z a^{-1} -7 z a^{-3} -3 z a^{-5} -3 a^{-2} -3 a^{-4} - a^{-6} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_42, ...}
Same Jones Polynomial (up to mirroring, ): {...}
Vassiliev invariants
V2 and V3: | (0, -1) |
V2,1 through V6,9: |
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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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 0 is the signature of 10 75. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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Integral Khovanov Homology
(db, data source) |
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The Coloured Jones Polynomials
2 | |
3 | |
4 | |
5 | Not Available |
6 | Not Available |
7 | Not Available |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session.
See/edit the Rolfsen_Splice_Template.