|
|
Line 1: |
Line 1: |
|
{{Rolfsen Knot Page|n=7|k=6}} |
|
{{Knot Navigation Links|name=7_6}} |
|
|
|
|
|
{{Knot Site Links|n=7|k=6}} |
|
|
|
|
|
{{Knot Presentations|name=7_6}} |
|
|
{{3D Invariants|name=7_6}} |
|
|
{{Polynomial Invariants|name=7_6}} |
|
|
{{Vassiliev Invariants|name=7_6}} |
|
|
{{Khovanov Invariants|name=7_6}} |
|
|
{{Quantum Invariants|name=7_6}} |
Revision as of 17:10, 25 August 2005
[[Image:7_5.{{{ext}}}|80px|link=7_5]]
7_5
|
[[Image:7_7.{{{ext}}}|80px|link=7_7]]
7_7
|
Visit 7 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit [{{{KnotilusURL}}} 7 6's page] at Knotilus!
Visit 7 6's page at the original Knot Atlas!
Knot presentations
Planar diagram presentation
|
X1425 X3849 X5,12,6,13 X9,1,10,14 X13,11,14,10 X11,6,12,7 X7283
|
Gauss code
|
-1, 7, -2, 1, -3, 6, -7, 2, -4, 5, -6, 3, -5, 4
|
Dowker-Thistlethwaite code
|
4 8 12 2 14 6 10
|
Conway Notation
|
[2212]
|
Polynomial invariants
Alexander polynomial |
![{\displaystyle -t^{2}-t^{-2}+5t+5t^{-1}-7}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c66a96a2342212a446259b59b416a611ab917d41) |
Conway polynomial |
![{\displaystyle -z^{4}+z^{2}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5081747f0ca27cf83546090a48dd0ef7f075e4) |
2nd Alexander ideal (db, data sources) |
![{\displaystyle \{1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124) |
Determinant and Signature |
{ 19, -2 } |
Jones polynomial |
![{\displaystyle q-2+3q^{-1}-3q^{-2}+4q^{-3}-3q^{-4}+2q^{-5}-q^{-6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77492910f48613034ac0af20336a5705ad23adbf) |
HOMFLY-PT polynomial (db, data sources) |
![{\displaystyle -a^{6}+2a^{4}z^{2}+2a^{4}-a^{2}z^{4}-2a^{2}z^{2}-a^{2}+z^{2}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce753e61949908851cb1e32e20736535f7de52c4) |
Kauffman polynomial (db, data sources) |
![{\displaystyle a^{7}z^{3}-a^{7}z+2a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{5}-a^{5}z^{3}+a^{4}z^{6}+2a^{4}z^{4}-4a^{4}z^{2}+2a^{4}+4a^{3}z^{5}-6a^{3}z^{3}+2a^{3}z+a^{2}z^{6}+a^{2}z^{4}-4a^{2}z^{2}+a^{2}+2az^{5}-4az^{3}+az+z^{4}-2z^{2}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5040e975a177140f70bbd4bf3237b6dd22add11) |
The A2 invariant |
![{\displaystyle -q^{20}-q^{18}+q^{16}+q^{12}+q^{10}+q^{6}-q^{4}+q^{2}+q^{-4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9592aef1bc44d965a12f952e6afcf47cfdea41b2) |
The G2 invariant |
![{\displaystyle q^{100}-q^{98}+2q^{96}-2q^{94}-3q^{88}+5q^{86}-6q^{84}+4q^{82}-4q^{80}-q^{78}+5q^{76}-8q^{74}+8q^{72}-5q^{70}+q^{68}+2q^{66}-5q^{64}+4q^{62}-2q^{58}+6q^{56}-5q^{54}+2q^{52}+6q^{50}-9q^{48}+11q^{46}-9q^{44}+5q^{42}+2q^{40}-6q^{38}+10q^{36}-9q^{34}+8q^{32}-3q^{30}-3q^{28}+5q^{26}-5q^{24}+3q^{22}-3q^{18}+5q^{16}-3q^{14}-q^{12}+5q^{10}-8q^{8}+8q^{6}-5q^{4}-q^{2}+5-6q^{-2}+8q^{-4}-4q^{-6}+2q^{-8}+q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b69d71f93a5faaf3bf9cd7eab47212e08a91299b) |
Further Quantum Invariants
Further quantum knot invariants for 7_6.
A1 Invariants.
Weight
|
Invariant
|
1
|
|
2
|
|
3
|
|
4
|
|
5
|
|
6
|
|
A2 Invariants.
Weight
|
Invariant
|
1,0
|
|
1,1
|
|
2,0
|
|
3,0
|
|
A3 Invariants.
Weight
|
Invariant
|
0,1,0
|
|
1,0,0
|
|
A4 Invariants.
Weight
|
Invariant
|
0,1,0,0
|
|
1,0,0,0
|
|
B2 Invariants.
Weight
|
Invariant
|
0,1
|
|
1,0
|
|
D4 Invariants.
Weight
|
Invariant
|
1,0,0,0
|
|
G2 Invariants.
Weight
|
Invariant
|
1,0
|
|
.
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`
|
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
|
Out[5]=
|
|
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]=
|
|
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
|
In[9]:=
|
HOMFLYPT[K][a, z]
|
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
|
In[10]:=
|
Kauffman[K][a, z]
|
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
|
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
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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