© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: |
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The Alternating Knot 1030Visit 1030's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 1030's page at Knotilus! |
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PD Presentation: | X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X9,18,10,19 X13,20,14,1 X19,14,20,15 X17,6,18,7 X7,16,8,17 X15,8,16,9 |
Gauss Code: | {-1, 4, -3, 1, -2, 8, -9, 10, -5, 3, -4, 2, -6, 7, -10, 9, -8, 5, -7, 6} |
DT (Dowker-Thistlethwaite) Code: | 4 10 12 16 18 2 20 8 6 14 |
Minimum Braid Representative:
Length is 12, width is 5 Braid index is 5 |
A Morse Link Presentation:
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3D Invariants: |
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Alexander Polynomial: | - 4t-2 + 17t-1 - 25 + 17t - 4t2 |
Conway Polynomial: | 1 + z2 - 4z4 |
Other knots with the same Alexander/Conway Polynomial: | {K11a154, ...} |
Determinant and Signature: | {67, -2} |
Jones Polynomial: | q-9 - 3q-8 + 5q-7 - 8q-6 + 10q-5 - 11q-4 + 11q-3 - 8q-2 + 6q-1 - 3 + q |
Other knots (up to mirrors) with the same Jones Polynomial: | {...} |
A2 (sl(3)) Invariant: | q-28 - q-26 - q-24 + 2q-22 - 2q-20 + q-16 - 2q-14 + q-12 - q-10 + 2q-8 + 2q-6 - q-4 + 3q-2 - 1 - q2 + q4 |
HOMFLY-PT Polynomial: | z2 + 2a2 + a2z2 - a2z4 - a4 - 2a4z2 - 2a4z4 - a6z4 + a8z2 |
Kauffman Polynomial: | - z2 + z4 - 3az3 + 3az5 - 2a2 + 5a2z2 - 7a2z4 + 5a2z6 - a3z + 4a3z3 - 6a3z5 + 5a3z7 - a4 + 9a4z2 - 11a4z4 + 2a4z6 + 3a4z8 - 5a5z + 16a5z3 - 19a5z5 + 7a5z7 + a5z9 + 2a6z2 + 2a6z4 - 11a6z6 + 6a6z8 - 6a7z + 18a7z3 - 20a7z5 + 5a7z7 + a7z9 + a8z2 + 2a8z4 - 7a8z6 + 3a8z8 - 2a9z + 9a9z3 - 10a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6 |
V2 and V3, the type 2 and 3 Vassiliev invariants: | {1, -1} |
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 1030. 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-26 - 3q-25 + 10q-23 - 12q-22 - 8q-21 + 32q-20 - 20q-19 - 30q-18 + 61q-17 - 20q-16 - 61q-15 + 84q-14 - 9q-13 - 87q-12 + 91q-11 + 6q-10 - 94q-9 + 77q-8 + 15q-7 - 74q-6 + 49q-5 + 14q-4 - 42q-3 + 23q-2 + 7q-1 - 16 + 7q + 2q2 - 3q3 + q4 |
3 | q-51 - 3q-50 + 5q-48 + 5q-47 - 12q-46 - 14q-45 + 19q-44 + 31q-43 - 22q-42 - 58q-41 + 17q-40 + 92q-39 + 2q-38 - 131q-37 - 32q-36 + 159q-35 + 84q-34 - 187q-33 - 135q-32 + 192q-31 + 201q-30 - 191q-29 - 258q-28 + 170q-27 + 317q-26 - 144q-25 - 364q-24 + 108q-23 + 398q-22 - 67q-21 - 420q-20 + 30q-19 + 413q-18 + 17q-17 - 401q-16 - 36q-15 + 346q-14 + 69q-13 - 301q-12 - 67q-11 + 229q-10 + 73q-9 - 177q-8 - 53q-7 + 118q-6 + 45q-5 - 83q-4 - 26q-3 + 50q-2 + 18q-1 - 32 - 9q + 18q2 + 5q3 - 10q4 - q5 + 3q6 + 2q7 - 3q8 + q9 |
4 | q-84 - 3q-83 + 5q-81 + 5q-79 - 19q-78 - 7q-77 + 20q-76 + 11q-75 + 37q-74 - 60q-73 - 55q-72 + 22q-71 + 40q-70 + 155q-69 - 85q-68 - 161q-67 - 70q-66 + 17q-65 + 394q-64 + 16q-63 - 236q-62 - 294q-61 - 201q-60 + 636q-59 + 286q-58 - 94q-57 - 516q-56 - 669q-55 + 672q-54 + 579q-53 + 330q-52 - 520q-51 - 1228q-50 + 414q-49 + 686q-48 + 902q-47 - 238q-46 - 1669q-45 - 27q-44 + 554q-43 + 1431q-42 + 214q-41 - 1904q-40 - 499q-39 + 269q-38 + 1824q-37 + 691q-36 - 1945q-35 - 906q-34 - 79q-33 + 2023q-32 + 1110q-31 - 1774q-30 - 1173q-29 - 448q-28 + 1946q-27 + 1379q-26 - 1371q-25 - 1180q-24 - 757q-23 + 1542q-22 + 1377q-21 - 831q-20 - 891q-19 - 865q-18 + 958q-17 + 1072q-16 - 382q-15 - 451q-14 - 711q-13 + 452q-12 + 631q-11 - 155q-10 - 110q-9 - 429q-8 + 175q-7 + 282q-6 - 90q-5 + 28q-4 - 198q-3 + 71q-2 + 104q-1 - 63 + 37q - 75q2 + 33q3 + 35q4 - 34q5 + 18q6 - 24q7 + 12q8 + 11q9 - 12q10 + 5q11 - 5q12 + 3q13 + 2q14 - 3q15 + q16 |
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, 30]] |
Out[2]= | PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], > X[9, 18, 10, 19], X[13, 20, 14, 1], X[19, 14, 20, 15], X[17, 6, 18, 7], > X[7, 16, 8, 17], X[15, 8, 16, 9]] |
In[3]:= | GaussCode[Knot[10, 30]] |
Out[3]= | GaussCode[-1, 4, -3, 1, -2, 8, -9, 10, -5, 3, -4, 2, -6, 7, -10, 9, -8, 5, -7, > 6] |
In[4]:= | DTCode[Knot[10, 30]] |
Out[4]= | DTCode[4, 10, 12, 16, 18, 2, 20, 8, 6, 14] |
In[5]:= | br = BR[Knot[10, 30]] |
Out[5]= | BR[5, {-1, -1, -2, 1, -2, -2, -3, 2, -3, 4, -3, 4}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {5, 12} |
In[7]:= | BraidIndex[Knot[10, 30]] |
Out[7]= | 5 |
In[8]:= | Show[DrawMorseLink[Knot[10, 30]]] |
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Out[8]= | -Graphics- |
In[9]:= | #[Knot[10, 30]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Reversible, 1, 2, 2, NotAvailable, 1} |
In[10]:= | alex = Alexander[Knot[10, 30]][t] |
Out[10]= | 4 17 2 -25 - -- + -- + 17 t - 4 t 2 t t |
In[11]:= | Conway[Knot[10, 30]][z] |
Out[11]= | 2 4 1 + z - 4 z |
In[12]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[12]= | {Knot[10, 30], Knot[11, Alternating, 154]} |
In[13]:= | {KnotDet[Knot[10, 30]], KnotSignature[Knot[10, 30]]} |
Out[13]= | {67, -2} |
In[14]:= | Jones[Knot[10, 30]][q] |
Out[14]= | -9 3 5 8 10 11 11 8 6 -3 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 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, 30]} |
In[16]:= | A2Invariant[Knot[10, 30]][q] |
Out[16]= | -28 -26 -24 2 2 -16 2 -12 -10 2 2 -1 + q - q - q + --- - --- + q - --- + q - q + -- + -- - 22 20 14 8 6 q q q q q -4 3 2 4 > q + -- - q + q 2 q |
In[17]:= | HOMFLYPT[Knot[10, 30]][a, z] |
Out[17]= | 2 4 2 2 2 4 2 8 2 2 4 4 4 6 4 2 a - a + z + a z - 2 a z + a z - a z - 2 a z - a z |
In[18]:= | Kauffman[Knot[10, 30]][a, z] |
Out[18]= | 2 4 3 5 7 9 2 2 2 4 2 -2 a - a - a z - 5 a z - 6 a z - 2 a z - z + 5 a z + 9 a z + 6 2 8 2 10 2 3 3 3 5 3 7 3 > 2 a z + a z + 2 a z - 3 a z + 4 a z + 16 a z + 18 a z + 9 3 4 2 4 4 4 6 4 8 4 10 4 5 > 9 a z + z - 7 a z - 11 a z + 2 a z + 2 a z - 3 a z + 3 a z - 3 5 5 5 7 5 9 5 2 6 4 6 6 6 > 6 a z - 19 a z - 20 a z - 10 a z + 5 a z + 2 a z - 11 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 > 7 a z + a z + 5 a z + 7 a z + 5 a z + 3 a z + 3 a z + 6 8 8 8 5 9 7 9 > 6 a z + 3 a z + a z + a z |
In[19]:= | {Vassiliev[2][Knot[10, 30]], Vassiliev[3][Knot[10, 30]]} |
Out[19]= | {1, -1} |
In[20]:= | Kh[Knot[10, 30]][q, t] |
Out[20]= | 3 4 1 2 1 3 2 5 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 5 5 6 5 5 6 3 5 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + 2 q t + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 > q t |
In[21]:= | ColouredJones[Knot[10, 30], 2][q] |
Out[21]= | -26 3 10 12 8 32 20 30 61 20 61 84 -16 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q 9 87 91 6 94 77 15 74 49 14 42 23 7 > --- - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + -- + - + 7 q + 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 2 3 4 > 2 q - 3 q + q |
Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1030 |
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