 K11n128 |
 K11n130 |
| © | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: |
|
 Knotscape |
This page is passe. Go here
instead!
The Knot K11n129Visit K11n129's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X4251 X10,3,11,4 X5,19,6,18 X7,14,8,15 X9,16,10,17 X2,11,3,12 X13,20,14,21 X15,8,16,9 X17,1,18,22 X19,12,20,13 X21,7,22,6 |
| Gauss Code: | {1, -6, 2, -1, -3, 11, -4, 8, -5, -2, 6, 10, -7, 4, -8, 5, -9, 3, -10, 7, -11, 9} |
| DT (Dowker-Thistlethwaite) Code: | 4 10 -18 -14 -16 2 -20 -8 -22 -12 -6 |
| Alexander Polynomial: | t-3 - 4t-2 + 10t-1 - 13 + 10t - 4t2 + t3 |
| Conway Polynomial: | 1 + 3z2 + 2z4 + z6 |
| Other knots with the same Alexander/Conway Polynomial: | {10151, K11n54, ...} |
| Determinant and Signature: | {43, -2} |
| Jones Polynomial: | - q-8 + 2q-7 - 4q-6 + 6q-5 - 7q-4 + 8q-3 - 6q-2 + 5q-1 - 3 + q |
| Other knots (up to mirrors) with the same Jones Polynomial: | {921, ...} |
| A2 (sl(3)) Invariant: | - q-24 - 2q-20 - q-18 + 2q-16 + 3q-12 + q-10 + q-8 + q-6 - 2q-4 + q-2 - 1 + q4 |
| HOMFLY-PT Polynomial: | 1 + z2 - 3a2 - 5a2z2 - 2a2z4 + 6a4 + 10a4z2 + 5a4z4 + a4z6 - 3a6 - 3a6z2 - a6z4 |
| Kauffman Polynomial: | 1 - 2z2 + z4 - 5az3 + 3az5 + 3a2 - 11a2z2 + 9a2z4 - 3a2z6 + a2z8 - 2a3z - a3z3 + 6a3z5 - 3a3z7 + a3z9 + 6a4 - 21a4z2 + 30a4z4 - 16a4z6 + 4a4z8 - 6a5z + 15a5z3 - 7a5z5 + a5z9 + 3a6 - 11a6z2 + 17a6z4 - 11a6z6 + 3a6z8 - 3a7z + 8a7z3 - 9a7z5 + 3a7z7 + a8z2 - 5a8z4 + 2a8z6 + a9z - 3a9z3 + a9z5 |
| V2 and V3, the type 2 and 3 Vassiliev invariants: | {3, -6} |
|
Khovanov Homology:
(The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=-2 is the signature of 11129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) |
|
Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 30, 2005, 10:15:35)... | |
In[2]:= | Show[DrawMorseLink[Knot[11, NonAlternating, 129]]] |
| |
Out[2]= | -Graphics- |
In[3]:= | PD[Knot[11, NonAlternating, 129]] |
Out[3]= | PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 19, 6, 18], X[7, 14, 8, 15], > X[9, 16, 10, 17], X[2, 11, 3, 12], X[13, 20, 14, 21], X[15, 8, 16, 9], > X[17, 1, 18, 22], X[19, 12, 20, 13], X[21, 7, 22, 6]] |
In[4]:= | GaussCode[Knot[11, NonAlternating, 129]] |
Out[4]= | GaussCode[1, -6, 2, -1, -3, 11, -4, 8, -5, -2, 6, 10, -7, 4, -8, 5, -9, 3, -10, > 7, -11, 9] |
In[5]:= | DTCode[Knot[11, NonAlternating, 129]] |
Out[5]= | DTCode[4, 10, -18, -14, -16, 2, -20, -8, -22, -12, -6] |
In[6]:= | alex = Alexander[Knot[11, NonAlternating, 129]][t] |
Out[6]= | -3 4 10 2 3
-13 + t - -- + -- + 10 t - 4 t + t
2 t
t |
In[7]:= | Conway[Knot[11, NonAlternating, 129]][z] |
Out[7]= | 2 4 6 1 + 3 z + 2 z + z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[10, 151], Knot[11, NonAlternating, 54], Knot[11, NonAlternating, 129]} |
In[9]:= | {KnotDet[Knot[11, NonAlternating, 129]], KnotSignature[Knot[11, NonAlternating, 129]]} |
Out[9]= | {43, -2} |
In[10]:= | J=Jones[Knot[11, NonAlternating, 129]][q] |
Out[10]= | -8 2 4 6 7 8 6 5
-3 - q + -- - -- + -- - -- + -- - -- + - + q
7 6 5 4 3 2 q
q q q q q q |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[9, 21], Knot[11, NonAlternating, 129]} |
In[12]:= | A2Invariant[Knot[11, NonAlternating, 129]][q] |
Out[12]= | -24 2 -18 2 3 -10 -8 -6 2 -2 4
-1 - q - --- - q + --- + --- + q + q + q - -- + q + q
20 16 12 4
q q q q |
In[13]:= | HOMFLYPT[Knot[11, NonAlternating, 129]][a, z] |
Out[13]= | 2 4 6 2 2 2 4 2 6 2 2 4
1 - 3 a + 6 a - 3 a + z - 5 a z + 10 a z - 3 a z - 2 a z +
4 4 6 4 4 6
> 5 a z - a z + a z |
In[14]:= | Kauffman[Knot[11, NonAlternating, 129]][a, z] |
Out[14]= | 2 4 6 3 5 7 9 2 2 2
1 + 3 a + 6 a + 3 a - 2 a z - 6 a z - 3 a z + a z - 2 z - 11 a z -
4 2 6 2 8 2 3 3 3 5 3 7 3
> 21 a z - 11 a z + a z - 5 a z - a z + 15 a z + 8 a z -
9 3 4 2 4 4 4 6 4 8 4 5 3 5
> 3 a z + z + 9 a z + 30 a z + 17 a z - 5 a z + 3 a z + 6 a z -
5 5 7 5 9 5 2 6 4 6 6 6 8 6
> 7 a z - 9 a z + a z - 3 a z - 16 a z - 11 a z + 2 a z -
3 7 7 7 2 8 4 8 6 8 3 9 5 9
> 3 a z + 3 a z + a z + 4 a z + 3 a z + a z + a z |
In[15]:= | {Vassiliev[2][Knot[11, NonAlternating, 129]], Vassiliev[3][Knot[11, NonAlternating, 129]]} |
Out[15]= | {3, -6} |
In[16]:= | Kh[Knot[11, NonAlternating, 129]][q, t] |
Out[16]= | 3 3 1 1 1 3 1 3 3 4
-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- +
3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3
q q t q t q t q t q t q t q t q t
3 4 4 2 4 t 3 2
> ----- + ----- + ----- + ---- + ---- + - + 2 q t + q t
7 3 7 2 5 2 5 3 q
q t q t q t q t q t |
| Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n129 |
|