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The Non Alternating Knot 10130Visit 10130's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10130's page at Knotilus! |
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PD Presentation: | X4251 X8493 X5,14,6,15 X15,20,16,1 X9,16,10,17 X19,10,20,11 X11,18,12,19 X17,12,18,13 X13,6,14,7 X2837 |
Gauss Code: | {1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, 4} |
DT (Dowker-Thistlethwaite) Code: | 4 8 -14 2 -16 -18 -6 -20 -12 -10 |
Minimum Braid Representative:
Length is 11, width is 4 Braid index is 4 |
A Morse Link Presentation:
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3D Invariants: |
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Alexander Polynomial: | 2t-2 - 4t-1 + 5 - 4t + 2t2 |
Conway Polynomial: | 1 + 4z2 + 2z4 |
Other knots with the same Alexander/Conway Polynomial: | {75, ...} |
Determinant and Signature: | {17, 0} |
Jones Polynomial: | - q-7 + q-6 - 2q-5 + 3q-4 - 2q-3 + 3q-2 - 2q-1 + 2 - q |
Other knots (up to mirrors) with the same Jones Polynomial: | {K11n61, ...} |
A2 (sl(3)) Invariant: | - q-22 - q-20 - q-18 - q-16 + q-14 + q-12 + 2q-10 + 2q-8 + q-6 + q-4 - q4 |
HOMFLY-PT Polynomial: | - 1 - z2 + 2a2 + 3a2z2 + a2z4 + 2a4 + 3a4z2 + a4z4 - 2a6 - a6z2 |
Kauffman Polynomial: | a-1z - 1 + 2z2 + az - 2az3 + az5 - 2a2 + 6a2z2 - 7a2z4 + 2a2z6 - 3a3z + 8a3z3 - 8a3z5 + 2a3z7 + 2a4 - 3a4z6 + a4z8 - 9a5z + 21a5z3 - 15a5z5 + 3a5z7 + 2a6 - 4a6z2 + 7a6z4 - 5a6z6 + a6z8 - 6a7z + 11a7z3 - 6a7z5 + a7z7 |
V2 and V3, the type 2 and 3 Vassiliev invariants: | {4, -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=0 is the signature of 10130. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) |
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n | Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2)) |
2 | q-21 - q-20 - q-19 + 3q-18 - q-17 - 4q-16 + 4q-15 - 5q-13 + 4q-12 + q-11 - 4q-10 + 2q-9 + 2q-8 - 2q-7 + 2q-5 + q-4 - 2q-3 + q-2 + q-1 - 1 |
3 | - q-42 + q-41 + q-40 - 3q-38 + 3q-36 + 3q-35 - 4q-34 - 3q-33 + 2q-32 + 5q-31 - 2q-30 - 3q-29 + 2q-27 - q-26 + 2q-24 - 3q-23 - 3q-22 + 3q-21 + 6q-20 - 6q-19 - 6q-18 + 4q-17 + 10q-16 - 7q-15 - 9q-14 + 5q-13 + 15q-12 - 9q-11 - 13q-10 + 6q-9 + 18q-8 - 8q-7 - 15q-6 + 4q-5 + 17q-4 - 4q-3 - 12q-2 + q-1 + 9 - q - 5q2 + q3 + 2q4 - 2q6 + q7 |
4 | q-70 - q-69 - q-68 + 4q-65 - q-64 - 2q-63 - 2q-62 - 4q-61 + 7q-60 + 2q-59 + q-58 - 2q-57 - 9q-56 + 5q-55 + 4q-53 + 3q-52 - 7q-51 + 5q-50 - 5q-49 - 2q-48 + 3q-47 - 2q-46 + 14q-45 - 3q-44 - 11q-43 - 5q-42 - 5q-41 + 22q-40 + 9q-39 - 13q-38 - 12q-37 - 15q-36 + 20q-35 + 21q-34 - 7q-33 - 11q-32 - 24q-31 + 10q-30 + 27q-29 - 5q-27 - 29q-26 + 29q-24 + 5q-23 + q-22 - 33q-21 - 8q-20 + 32q-19 + 11q-18 + 4q-17 - 38q-16 - 17q-15 + 35q-14 + 19q-13 + 11q-12 - 41q-11 - 27q-10 + 30q-9 + 22q-8 + 18q-7 - 31q-6 - 28q-5 + 16q-4 + 13q-3 + 19q-2 - 15q-1 - 17 + 7q + q2 + 10q3 - 5q4 - 5q5 + 4q6 - 2q7 + 2q8 - 2q9 + 2q11 - q12 |
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]:= | PD[Knot[10, 130]] |
Out[2]= | PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[15, 20, 16, 1], > X[9, 16, 10, 17], X[19, 10, 20, 11], X[11, 18, 12, 19], X[17, 12, 18, 13], > X[13, 6, 14, 7], X[2, 8, 3, 7]] |
In[3]:= | GaussCode[Knot[10, 130]] |
Out[3]= | GaussCode[1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, > 4] |
In[4]:= | DTCode[Knot[10, 130]] |
Out[4]= | DTCode[4, 8, -14, 2, -16, -18, -6, -20, -12, -10] |
In[5]:= | br = BR[Knot[10, 130]] |
Out[5]= | BR[4, {1, 1, 1, -2, -1, -1, -2, -2, -3, 2, -3}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {4, 11} |
In[7]:= | BraidIndex[Knot[10, 130]] |
Out[7]= | 4 |
In[8]:= | Show[DrawMorseLink[Knot[10, 130]]] |
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Out[8]= | -Graphics- |
In[9]:= | #[Knot[10, 130]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Reversible, 2, 2, 3, NotAvailable, 1} |
In[10]:= | alex = Alexander[Knot[10, 130]][t] |
Out[10]= | 2 4 2 5 + -- - - - 4 t + 2 t 2 t t |
In[11]:= | Conway[Knot[10, 130]][z] |
Out[11]= | 2 4 1 + 4 z + 2 z |
In[12]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[12]= | {Knot[7, 5], Knot[10, 130]} |
In[13]:= | {KnotDet[Knot[10, 130]], KnotSignature[Knot[10, 130]]} |
Out[13]= | {17, 0} |
In[14]:= | Jones[Knot[10, 130]][q] |
Out[14]= | -7 -6 2 3 2 3 2 2 - q + q - -- + -- - -- + -- - - - q 5 4 3 2 q q q q q |
In[15]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[15]= | {Knot[10, 130], Knot[11, NonAlternating, 61]} |
In[16]:= | A2Invariant[Knot[10, 130]][q] |
Out[16]= | -22 -20 -18 -16 -14 -12 2 2 -6 -4 4 -q - q - q - q + q + q + --- + -- + q + q - q 10 8 q q |
In[17]:= | HOMFLYPT[Knot[10, 130]][a, z] |
Out[17]= | 2 4 6 2 2 2 4 2 6 2 2 4 4 4 -1 + 2 a + 2 a - 2 a - z + 3 a z + 3 a z - a z + a z + a z |
In[18]:= | Kauffman[Knot[10, 130]][a, z] |
Out[18]= | 2 4 6 z 3 5 7 2 2 2 -1 - 2 a + 2 a + 2 a + - + a z - 3 a z - 9 a z - 6 a z + 2 z + 6 a z - a 6 2 3 3 3 5 3 7 3 2 4 6 4 > 4 a z - 2 a z + 8 a z + 21 a z + 11 a z - 7 a z + 7 a z + 5 3 5 5 5 7 5 2 6 4 6 6 6 > a z - 8 a z - 15 a z - 6 a z + 2 a z - 3 a z - 5 a z + 3 7 5 7 7 7 4 8 6 8 > 2 a z + 3 a z + a z + a z + a z |
In[19]:= | {Vassiliev[2][Knot[10, 130]], Vassiliev[3][Knot[10, 130]]} |
Out[19]= | {4, -6} |
In[20]:= | Kh[Knot[10, 130]][q, t] |
Out[20]= | 2 1 1 2 1 2 1 1 2 - + q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 15 7 11 6 11 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 1 2 3 > ----- + --- + q t 3 2 q t q t |
In[21]:= | ColouredJones[Knot[10, 130], 2][q] |
Out[21]= | -21 -20 -19 3 -17 4 4 5 4 -11 4 -1 + q - q - q + --- - q - --- + --- - --- + --- + q - --- + 18 16 15 13 12 10 q q q q q q 2 2 2 2 -4 2 -2 1 > -- + -- - -- + -- + q - -- + q + - 9 8 7 5 3 q q q q q q |
Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10130 |
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