Visit 7 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit [{{{KnotilusURL}}} 7 5's page] at Knotilus!
Visit 7 5's page at the original Knot Atlas!
Knot presentations
Planar diagram presentation
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X1425 X3,10,4,11 X5,12,6,13 X7,14,8,1 X13,6,14,7 X11,8,12,9 X9,2,10,3
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Gauss code
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-1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4
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Dowker-Thistlethwaite code
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4 10 12 14 2 8 6
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Conway Notation
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[322]
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Symmetry type
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Reversible
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Unknotting number
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2
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3-genus
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2
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Bridge index (super bridge index)
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2 (4)
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Nakanishi index
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1
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Polynomial invariants
Alexander polynomial |
![{\displaystyle 2t^{2}-4t+5-4t^{-1}+2t^{-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20931fda7029f71deb45081150956b4b6c71a49c) |
Conway polynomial |
![{\displaystyle 2z^{4}+4z^{2}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9da0db1ae422913f62b32da6db19dc8e90b470a4) |
2nd Alexander ideal (db, data sources) |
![{\displaystyle \{1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124) |
Determinant and Signature |
{ 17, -4 } |
Jones polynomial |
![{\displaystyle -q^{-9}+2q^{-8}-3q^{-7}+3q^{-6}-3q^{-5}+3q^{-4}-q^{-3}+q^{-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3cd50c09af052e7199103523487105aa00935b) |
HOMFLY-PT polynomial (db, data sources) |
![{\displaystyle a^{8}\left(-z^{2}\right)-a^{8}+a^{6}z^{4}+2a^{6}z^{2}+a^{4}z^{4}+3a^{4}z^{2}+2a^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a51c940a689fb625ef08f81dbdf77ee39383083c) |
Kauffman polynomial (db, data sources) |
![{\displaystyle a^{11}z^{3}-a^{11}z+2a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+a^{8}z^{6}+a^{8}z^{2}-a^{8}+3a^{7}z^{5}-4a^{7}z^{3}+a^{7}z+a^{6}z^{6}-a^{6}z^{4}+a^{5}z^{5}-a^{5}z^{3}-a^{5}z+a^{4}z^{4}-3a^{4}z^{2}+2a^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09ae50d319073ccdd673e96675a71b675c227c1d) |
The A2 invariant |
![{\displaystyle -q^{28}-q^{22}-q^{18}+q^{16}+q^{14}+q^{12}+2q^{10}+q^{6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/283b71965d40095f3c747c5ee7c5a9440913749c) |
The G2 invariant |
![{\displaystyle q^{148}-q^{146}+2q^{144}-2q^{142}+q^{138}-2q^{136}+5q^{134}-5q^{132}+4q^{130}-2q^{128}-3q^{126}+4q^{124}-6q^{122}+5q^{120}-3q^{118}-q^{116}+3q^{114}-3q^{112}+q^{110}+2q^{108}-5q^{106}+4q^{104}-3q^{102}-2q^{100}+5q^{98}-7q^{96}+8q^{94}-7q^{92}+2q^{90}+2q^{88}-6q^{86}+6q^{84}-7q^{82}+4q^{80}-2q^{76}+3q^{74}-3q^{72}+2q^{70}+3q^{68}-5q^{66}+3q^{64}-2q^{60}+7q^{58}-5q^{56}+5q^{54}-q^{52}+4q^{48}-4q^{46}+5q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45f051627bd5a8f3573f6a5c2a832ec960265a1d) |
Further Quantum Invariants
Further quantum knot invariants for 7_5.
A1 Invariants.
Weight
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Invariant
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1
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2
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3
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4
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5
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6
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A2 Invariants.
Weight
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Invariant
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1,0
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1,1
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2,0
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3,0
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A3 Invariants.
Weight
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Invariant
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0,1,0
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1,0,0
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1,0,1
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A4 Invariants.
Weight
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Invariant
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0,1,0,0
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1,0,0,0
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B2 Invariants.
Weight
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Invariant
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0,1
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1,0
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D4 Invariants.
Weight
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Invariant
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1,0,0,0
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G2 Invariants.
Weight
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Invariant
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1,0
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.
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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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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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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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V2,1 through V6,9:
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V2,1
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V3,1
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V4,1
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V4,2
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V4,3
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V5,1
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V5,2
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V5,3
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V5,4
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V6,1
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V6,2
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V6,3
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V6,4
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V6,5
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V6,6
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V6,7
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V6,8
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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.
Template:Khovanov Invariants
Template:Quantum Invariants