© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: |
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The Alternating Knot 1025Visit 1025's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 1025's page at Knotilus! |
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PD Presentation: | X1425 X5,14,6,15 X3,13,4,12 X13,3,14,2 X11,20,12,1 X19,6,20,7 X9,18,10,19 X7,16,8,17 X17,8,18,9 X15,10,16,11 |
Gauss Code: | {-1, 4, -3, 1, -2, 6, -8, 9, -7, 10, -5, 3, -4, 2, -10, 8, -9, 7, -6, 5} |
DT (Dowker-Thistlethwaite) Code: | 4 12 14 16 18 20 2 10 8 6 |
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-3 + 8t-2 - 14t-1 + 17 - 14t + 8t2 - 2t3 |
Conway Polynomial: | 1 - 4z4 - 2z6 |
Other knots with the same Alexander/Conway Polynomial: | {1056, K11a140, ...} |
Determinant and Signature: | {65, -4} |
Jones Polynomial: | q-10 - 3q-9 + 6q-8 - 9q-7 + 10q-6 - 11q-5 + 10q-4 - 7q-3 + 5q-2 - 2q-1 + 1 |
Other knots (up to mirrors) with the same Jones Polynomial: | {1056, ...} |
A2 (sl(3)) Invariant: | q-30 - q-28 + q-26 + q-24 - 2q-22 + q-20 - 3q-18 - q-12 + 3q-10 - q-8 + 2q-6 + q-4 + 1 |
HOMFLY-PT Polynomial: | 2a2 + 3a2z2 + a2z4 - 2a4z2 - 3a4z4 - a4z6 - 2a6 - 3a6z2 - 3a6z4 - a6z6 + a8 + 2a8z2 + a8z4 |
Kauffman Polynomial: | - 2a2 + 5a2z2 - 4a2z4 + a2z6 + a3z + 4a3z3 - 6a3z5 + 2a3z7 + 4a4z2 - 3a4z4 - 3a4z6 + 2a4z8 + 2a5z3 - 7a5z5 + 2a5z7 + a5z9 + 2a6 - 4a6z2 + 3a6z4 - 8a6z6 + 5a6z8 - 2a7z + 3a7z3 - 9a7z5 + 5a7z7 + a7z9 + a8 + a8z2 - 5a8z4 + a8z6 + 3a8z8 + 2a9z3 - 5a9z5 + 5a9z7 + 3a10z2 - 6a10z4 + 5a10z6 + a11z - 3a11z3 + 3a11z5 - a12z2 + a12z4 |
V2 and V3, the type 2 and 3 Vassiliev invariants: | {0, 2} |
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=-4 is the signature of 1025. 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-28 - 3q-27 + 2q-26 + 7q-25 - 17q-24 + 7q-23 + 26q-22 - 46q-21 + 11q-20 + 58q-19 - 78q-18 + 8q-17 + 86q-16 - 93q-15 - 4q-14 + 97q-13 - 82q-12 - 18q-11 + 85q-10 - 54q-9 - 24q-8 + 56q-7 - 23q-6 - 19q-5 + 26q-4 - 5q-3 - 9q-2 + 7q-1 - 2q + q2 |
3 | q-54 - 3q-53 + 2q-52 + 3q-51 - q-50 - 11q-49 + 5q-48 + 22q-47 - 10q-46 - 41q-45 + 17q-44 + 70q-43 - 21q-42 - 117q-41 + 27q-40 + 174q-39 - 25q-38 - 239q-37 + 9q-36 + 314q-35 + 8q-34 - 372q-33 - 48q-32 + 429q-31 + 79q-30 - 449q-29 - 128q-28 + 459q-27 + 165q-26 - 438q-25 - 203q-24 + 402q-23 + 232q-22 - 351q-21 - 246q-20 + 281q-19 + 261q-18 - 223q-17 - 243q-16 + 144q-15 + 230q-14 - 92q-13 - 186q-12 + 33q-11 + 153q-10 - 7q-9 - 101q-8 - 20q-7 + 71q-6 + 18q-5 - 34q-4 - 21q-3 + 20q-2 + 11q-1 - 6 - 8q + 4q2 + 2q3 - 2q5 + q6 |
4 | q-88 - 3q-87 + 2q-86 + 3q-85 - 5q-84 + 5q-83 - 13q-82 + 11q-81 + 16q-80 - 24q-79 + 12q-78 - 42q-77 + 42q-76 + 60q-75 - 77q-74 + q-73 - 106q-72 + 134q-71 + 180q-70 - 178q-69 - 88q-68 - 262q-67 + 329q-66 + 476q-65 - 286q-64 - 331q-63 - 613q-62 + 593q-61 + 1015q-60 - 268q-59 - 682q-58 - 1221q-57 + 758q-56 + 1697q-55 - 9q-54 - 950q-53 - 1951q-52 + 677q-51 + 2241q-50 + 421q-49 - 948q-48 - 2532q-47 + 374q-46 + 2427q-45 + 831q-44 - 676q-43 - 2775q-42 - 16q-41 + 2234q-40 + 1105q-39 - 245q-38 - 2677q-37 - 402q-36 + 1775q-35 + 1225q-34 + 233q-33 - 2296q-32 - 727q-31 + 1145q-30 + 1180q-29 + 668q-28 - 1697q-27 - 898q-26 + 473q-25 + 929q-24 + 919q-23 - 986q-22 - 811q-21 - 57q-20 + 515q-19 + 877q-18 - 369q-17 - 505q-16 - 284q-15 + 123q-14 + 591q-13 - 26q-12 - 178q-11 - 237q-10 - 73q-9 + 272q-8 + 49q-7 - 3q-6 - 102q-5 - 83q-4 + 83q-3 + 20q-2 + 27q-1 - 23 - 37q + 20q2 + 11q4 - 2q5 - 10q6 + 5q7 - q8 + 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, 25]] |
Out[2]= | PD[X[1, 4, 2, 5], X[5, 14, 6, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], > X[11, 20, 12, 1], X[19, 6, 20, 7], X[9, 18, 10, 19], X[7, 16, 8, 17], > X[17, 8, 18, 9], X[15, 10, 16, 11]] |
In[3]:= | GaussCode[Knot[10, 25]] |
Out[3]= | GaussCode[-1, 4, -3, 1, -2, 6, -8, 9, -7, 10, -5, 3, -4, 2, -10, 8, -9, 7, -6, > 5] |
In[4]:= | DTCode[Knot[10, 25]] |
Out[4]= | DTCode[4, 12, 14, 16, 18, 20, 2, 10, 8, 6] |
In[5]:= | br = BR[Knot[10, 25]] |
Out[5]= | BR[4, {-1, -1, -1, -1, -2, 1, -2, -2, 3, -2, 3}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {4, 11} |
In[7]:= | BraidIndex[Knot[10, 25]] |
Out[7]= | 4 |
In[8]:= | Show[DrawMorseLink[Knot[10, 25]]] |
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Out[8]= | -Graphics- |
In[9]:= | #[Knot[10, 25]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Reversible, 2, 3, 2, NotAvailable, 1} |
In[10]:= | alex = Alexander[Knot[10, 25]][t] |
Out[10]= | 2 8 14 2 3 17 - -- + -- - -- - 14 t + 8 t - 2 t 3 2 t t t |
In[11]:= | Conway[Knot[10, 25]][z] |
Out[11]= | 4 6 1 - 4 z - 2 z |
In[12]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[12]= | {Knot[10, 25], Knot[10, 56], Knot[11, Alternating, 140]} |
In[13]:= | {KnotDet[Knot[10, 25]], KnotSignature[Knot[10, 25]]} |
Out[13]= | {65, -4} |
In[14]:= | Jones[Knot[10, 25]][q] |
Out[14]= | -10 3 6 9 10 11 10 7 5 2 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q |
In[15]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[15]= | {Knot[10, 25], Knot[10, 56]} |
In[16]:= | A2Invariant[Knot[10, 25]][q] |
Out[16]= | -30 -28 -26 -24 2 -20 3 -12 3 -8 2 -4 1 + q - q + q + q - --- + q - --- - q + --- - q + -- + q 22 18 10 6 q q q q |
In[17]:= | HOMFLYPT[Knot[10, 25]][a, z] |
Out[17]= | 2 6 8 2 2 4 2 6 2 8 2 2 4 4 4 2 a - 2 a + a + 3 a z - 2 a z - 3 a z + 2 a z + a z - 3 a z - 6 4 8 4 4 6 6 6 > 3 a z + a z - a z - a z |
In[18]:= | Kauffman[Knot[10, 25]][a, z] |
Out[18]= | 2 6 8 3 7 11 2 2 4 2 6 2 -2 a + 2 a + a + a z - 2 a z + a z + 5 a z + 4 a z - 4 a z + 8 2 10 2 12 2 3 3 5 3 7 3 9 3 > a z + 3 a z - a z + 4 a z + 2 a z + 3 a z + 2 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 3 a z - 4 a z - 3 a z + 3 a z - 5 a z - 6 a z + a z - 3 5 5 5 7 5 9 5 11 5 2 6 4 6 > 6 a z - 7 a z - 9 a z - 5 a z + 3 a z + a z - 3 a z - 6 6 8 6 10 6 3 7 5 7 7 7 9 7 > 8 a z + a z + 5 a z + 2 a z + 2 a z + 5 a z + 5 a z + 4 8 6 8 8 8 5 9 7 9 > 2 a z + 5 a z + 3 a z + a z + a z |
In[19]:= | {Vassiliev[2][Knot[10, 25]], Vassiliev[3][Knot[10, 25]]} |
Out[19]= | {0, 2} |
In[20]:= | Kh[Knot[10, 25]][q, t] |
Out[20]= | 2 4 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 5 5 6 5 4 6 3 4 t t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t |
In[21]:= | ColouredJones[Knot[10, 25], 2][q] |
Out[21]= | -28 3 2 7 17 7 26 46 11 58 78 8 86 q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 27 26 25 24 23 22 21 20 19 18 17 16 q q q q q q q q q q q q 93 4 97 82 18 85 54 24 56 23 19 26 5 9 > --- - --- + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - -- - -- + 15 14 13 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q q q 7 2 > - - 2 q + q q |
Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1025 |
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