[[Image:T(13,2).{{{ext}}}|80px|link=T(13,2)]]
T(13,2)
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[[Image:T(5,4).{{{ext}}}|80px|link=T(5,4)]]
T(5,4)
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Visit T(7,3)'s page at Knotilus!
Visit T(7,3)'s page at the original Knot Atlas!
Knot presentations
Planar diagram presentation
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X1,11,2,10 X20,12,21,11 X21,3,22,2 X12,4,13,3 X13,23,14,22 X4,24,5,23 X5,15,6,14 X24,16,25,15 X25,7,26,6 X16,8,17,7 X17,27,18,26 X8,28,9,27 X9,19,10,18 X28,20,1,19
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Gauss code
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-1, 3, 4, -6, -7, 9, 10, -12, -13, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -2, -3, 5, 6, -8, -9, 11, 12, -14
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Dowker-Thistlethwaite code
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10 -12 14 -16 18 -20 22 -24 26 -28 2 -4 6 -8
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Conway Notation
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Data:T(7,3)/Conway Notation
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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 t^{6}-t^{5}+t^{3}-t^{2}+1-t^{-2}+t^{-3}-t^{-5}+t^{-6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cff4cca38ff74e1421caab9251a68ad4d221376) |
Conway polynomial |
![{\displaystyle z^{12}+11z^{10}+44z^{8}+78z^{6}+60z^{4}+16z^{2}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa77cebd97658e3de905eaff090a426984961e62) |
2nd Alexander ideal (db, data sources) |
![{\displaystyle \{1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124) |
Determinant and Signature |
{ 1, 8 } |
Jones polynomial |
![{\displaystyle -q^{14}+q^{8}+q^{6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02747b84a094ea281b9f03041a698b002bbf5205) |
HOMFLY-PT polynomial (db, data sources) |
![{\displaystyle z^{12}a^{-12}+12z^{10}a^{-12}-z^{10}a^{-14}+55z^{8}a^{-12}-11z^{8}a^{-14}+121z^{6}a^{-12}-44z^{6}a^{-14}+z^{6}a^{-16}+132z^{4}a^{-12}-78z^{4}a^{-14}+6z^{4}a^{-16}+66z^{2}a^{-12}-60z^{2}a^{-14}+10z^{2}a^{-16}+12a^{-12}-16a^{-14}+5a^{-16}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70193a94581eb691e02f7ce39c66671055a28f2b) |
Kauffman polynomial (db, data sources) |
![{\displaystyle z^{12}a^{-12}+z^{12}a^{-14}+z^{11}a^{-13}+z^{11}a^{-15}-12z^{10}a^{-12}-12z^{10}a^{-14}-11z^{9}a^{-13}-11z^{9}a^{-15}+55z^{8}a^{-12}+55z^{8}a^{-14}+44z^{7}a^{-13}+44z^{7}a^{-15}-121z^{6}a^{-12}-122z^{6}a^{-14}-z^{6}a^{-16}-78z^{5}a^{-13}-78z^{5}a^{-15}+132z^{4}a^{-12}+138z^{4}a^{-14}+6z^{4}a^{-16}+60z^{3}a^{-13}+60z^{3}a^{-15}-66z^{2}a^{-12}-76z^{2}a^{-14}-10z^{2}a^{-16}-16za^{-13}-16za^{-15}+12a^{-12}+16a^{-14}+5a^{-16}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c65602dd0d5e96e8bffbd2943e0d75c39c3eafb3) |
The A2 invariant |
Data:T(7,3)/QuantumInvariant/A2/1,0 |
The G2 invariant |
Data:T(7,3)/QuantumInvariant/G2/1,0 |
Further Quantum Invariants
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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In[3]:=
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K = Knot["T(7,3)"];
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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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Data:T(7,3)/V 2,1
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Data:T(7,3)/V 3,1
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Data:T(7,3)/V 4,1
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Data:T(7,3)/V 4,2
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Data:T(7,3)/V 4,3
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Data:T(7,3)/V 5,1
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Data:T(7,3)/V 5,2
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Data:T(7,3)/V 5,3
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Data:T(7,3)/V 5,4
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Data:T(7,3)/V 6,1
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Data:T(7,3)/V 6,2
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Data:T(7,3)/V 6,3
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Data:T(7,3)/V 6,4
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Data:T(7,3)/V 6,5
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Data:T(7,3)/V 6,6
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Data:T(7,3)/V 6,7
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Data:T(7,3)/V 6,8
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Data:T(7,3)/V 6,9
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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