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The Alternating Knot 1087Visit 1087's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 1087's page at Knotilus! |
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PD Presentation: | X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X16,7,17,8 X6,19,7,20 X8,12,9,11 X18,13,19,14 X12,17,13,18 X2,10,3,9 |
Gauss Code: | {1, -10, 2, -1, 3, -6, 5, -7, 10, -2, 7, -9, 8, -3, 4, -5, 9, -8, 6, -4} |
DT (Dowker-Thistlethwaite) Code: | 4 10 14 16 2 8 18 20 12 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 + 9t-2 - 18t-1 + 23 - 18t + 9t2 - 2t3 |
Conway Polynomial: | 1 - 3z4 - 2z6 |
Other knots with the same Alexander/Conway Polynomial: | {1098, K11a58, K11a165, K11n72, ...} |
Determinant and Signature: | {81, 0} |
Jones Polynomial: | q-4 - 3q-3 + 6q-2 - 10q-1 + 13 - 13q + 13q2 - 10q3 + 7q4 - 4q5 + q6 |
Other knots (up to mirrors) with the same Jones Polynomial: | {...} |
A2 (sl(3)) Invariant: | q-12 - q-10 + q-8 + q-6 - 3q-4 + 2q-2 - 2 + q2 + 2q4 + 4q8 - 2q10 - 2q16 + q18 |
HOMFLY-PT Polynomial: | - a-4 + a-4z2 + a-4z4 + 3a-2 + a-2z2 - 2a-2z4 - a-2z6 - 2 - 4z2 - 3z4 - z6 + a2 + 2a2z2 + a2z4 |
Kauffman Polynomial: | a-6z2 - 2a-6z4 + a-6z6 + 7a-5z3 - 11a-5z5 + 4a-5z7 - a-4 + a-4z2 + 5a-4z4 - 12a-4z6 + 5a-4z8 - a-3z + 15a-3z3 - 23a-3z5 + 5a-3z7 + 2a-3z9 - 3a-2 + 3a-2z2 + 8a-2z4 - 21a-2z6 + 10a-2z8 - a-1z + 13a-1z3 - 21a-1z5 + 7a-1z7 + 2a-1z9 - 2 + 7z2 - 5z4 - 3z6 + 5z8 + az + 2az3 - 6az5 + 6az7 - a2 + 3a2z2 - 5a2z4 + 5a2z6 + a3z - 3a3z3 + 3a3z5 - a4z2 + a4z4 |
V2 and V3, the type 2 and 3 Vassiliev invariants: | {0, 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=0 is the signature of 1087. 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-12 - 3q-11 + 2q-10 + 6q-9 - 16q-8 + 12q-7 + 18q-6 - 51q-5 + 33q-4 + 47q-3 - 105q-2 + 46q-1 + 90 - 143q + 35q2 + 120q3 - 141q4 + 7q5 + 123q6 - 105q7 - 20q8 + 97q9 - 54q10 - 31q11 + 53q12 - 13q13 - 20q14 + 15q15 + q16 - 4q17 + q18 |
3 | q-24 - 3q-23 + 2q-22 + 2q-21 - 8q-19 + 6q-18 + 10q-17 - 16q-16 - 13q-15 + 39q-14 + 20q-13 - 83q-12 - 39q-11 + 154q-10 + 80q-9 - 245q-8 - 154q-7 + 336q-6 + 272q-5 - 419q-4 - 410q-3 + 464q-2 + 549q-1 - 453 - 685q + 419q2 + 767q3 - 329q4 - 830q5 + 242q6 + 827q7 - 116q8 - 815q9 + 11q10 + 747q11 + 113q12 - 669q13 - 205q14 + 546q15 + 287q16 - 413q17 - 326q18 + 271q19 + 317q20 - 135q21 - 275q22 + 36q23 + 202q24 + 26q25 - 127q26 - 44q27 + 60q28 + 43q29 - 25q30 - 23q31 + 5q32 + 9q33 + q34 - 4q35 + q36 |
4 | q-40 - 3q-39 + 2q-38 + 2q-37 - 4q-36 + 8q-35 - 14q-34 + 8q-33 + 5q-32 - 20q-31 + 40q-30 - 29q-29 + 16q-28 - 22q-27 - 88q-26 + 145q-25 + 37q-24 + 60q-23 - 188q-22 - 385q-21 + 326q-20 + 411q-19 + 396q-18 - 537q-17 - 1302q-16 + 251q-15 + 1209q-14 + 1544q-13 - 655q-12 - 2969q-11 - 709q-10 + 1882q-9 + 3592q-8 + 189q-7 - 4632q-6 - 2600q-5 + 1575q-4 + 5610q-3 + 1987q-2 - 5236q-1 - 4448 + 234q + 6521q2 + 3810q3 - 4613q4 - 5337q5 - 1368q6 + 6185q7 + 4901q8 - 3322q9 - 5219q10 - 2692q11 + 5060q12 + 5267q13 - 1756q14 - 4449q15 - 3695q16 + 3417q17 + 5059q18 - 28q19 - 3117q20 - 4268q21 + 1389q22 + 4126q23 + 1478q24 - 1275q25 - 3959q26 - 500q27 + 2401q28 + 2056q29 + 512q30 - 2611q31 - 1411q32 + 544q33 + 1450q34 + 1348q35 - 948q36 - 1099q37 - 470q38 + 406q39 + 1047q40 + 20q41 - 343q42 - 466q43 - 146q44 + 391q45 + 156q46 + 41q47 - 149q48 - 142q49 + 59q50 + 38q51 + 50q52 - 10q53 - 35q54 + 2q55 - q56 + 9q57 + q58 - 4q59 + q60 |
5 | q-60 - 3q-59 + 2q-58 + 2q-57 - 4q-56 + 4q-55 + 2q-54 - 12q-53 + 3q-52 + 11q-51 - 7q-50 + 14q-49 + 9q-48 - 41q-47 - 27q-46 + 11q-45 + 40q-44 + 92q-43 + 55q-42 - 129q-41 - 255q-40 - 137q-39 + 213q-38 + 563q-37 + 453q-36 - 306q-35 - 1170q-34 - 1133q-33 + 284q-32 + 2115q-31 + 2460q-30 + 174q-29 - 3377q-28 - 4774q-27 - 1508q-26 + 4732q-25 + 8230q-24 + 4252q-23 - 5576q-22 - 12742q-21 - 8907q-20 + 5172q-19 + 17807q-18 + 15585q-17 - 2828q-16 - 22464q-15 - 23822q-14 - 1989q-13 + 25672q-12 + 32799q-11 + 9036q-10 - 26662q-9 - 41084q-8 - 17662q-7 + 25016q-6 + 47626q-5 + 26730q-4 - 21076q-3 - 51744q-2 - 34918q-1 + 15615 + 53054q + 41535q2 - 9434q3 - 52288q4 - 46021q5 + 3628q6 + 49590q7 + 48605q8 + 1776q9 - 46135q10 - 49591q11 - 6259q12 + 41817q13 + 49507q14 + 10504q15 - 37297q16 - 48648q17 - 14339q18 + 31969q19 + 47190q20 + 18380q21 - 26171q22 - 44870q23 - 22173q24 + 19260q25 + 41502q26 + 25764q27 - 11811q28 - 36688q29 - 28199q30 + 3785q31 + 30312q32 + 29233q33 + 3740q34 - 22616q35 - 28002q36 - 10120q37 + 14131q38 + 24602q39 + 14406q40 - 5945q41 - 19243q42 - 16028q43 - 987q44 + 12790q45 + 15075q46 + 5733q47 - 6459q48 - 12002q49 - 7936q50 + 1232q51 + 7945q52 + 7801q53 + 2142q54 - 3944q55 - 6123q56 - 3570q57 + 944q58 + 3784q59 + 3444q60 + 845q61 - 1744q62 - 2524q63 - 1364q64 + 362q65 + 1361q66 + 1236q67 + 300q68 - 571q69 - 767q70 - 395q71 + 81q72 + 351q73 + 320q74 + 63q75 - 129q76 - 151q77 - 65q78 + 9q79 + 57q80 + 53q81 - 3q82 - 20q83 - 10q84 - 4q85 - q86 + 9q87 + q88 - 4q89 + q90 |
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, 87]] |
Out[2]= | PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 16, 1, 15], > X[16, 7, 17, 8], X[6, 19, 7, 20], X[8, 12, 9, 11], X[18, 13, 19, 14], > X[12, 17, 13, 18], X[2, 10, 3, 9]] |
In[3]:= | GaussCode[Knot[10, 87]] |
Out[3]= | GaussCode[1, -10, 2, -1, 3, -6, 5, -7, 10, -2, 7, -9, 8, -3, 4, -5, 9, -8, 6, > -4] |
In[4]:= | DTCode[Knot[10, 87]] |
Out[4]= | DTCode[4, 10, 14, 16, 2, 8, 18, 20, 12, 6] |
In[5]:= | br = BR[Knot[10, 87]] |
Out[5]= | BR[4, {1, 1, 1, 2, -1, -3, 2, -3, 2, -3, -3}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {4, 11} |
In[7]:= | BraidIndex[Knot[10, 87]] |
Out[7]= | 4 |
In[8]:= | Show[DrawMorseLink[Knot[10, 87]]] |
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Out[8]= | -Graphics- |
In[9]:= | #[Knot[10, 87]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Chiral, 2, 3, 3, NotAvailable, 1} |
In[10]:= | alex = Alexander[Knot[10, 87]][t] |
Out[10]= | 2 9 18 2 3 23 - -- + -- - -- - 18 t + 9 t - 2 t 3 2 t t t |
In[11]:= | Conway[Knot[10, 87]][z] |
Out[11]= | 4 6 1 - 3 z - 2 z |
In[12]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[12]= | {Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], > Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]} |
In[13]:= | {KnotDet[Knot[10, 87]], KnotSignature[Knot[10, 87]]} |
Out[13]= | {81, 0} |
In[14]:= | Jones[Knot[10, 87]][q] |
Out[14]= | -4 3 6 10 2 3 4 5 6 13 + q - -- + -- - -- - 13 q + 13 q - 10 q + 7 q - 4 q + q 3 2 q q q |
In[15]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[15]= | {Knot[10, 87]} |
In[16]:= | A2Invariant[Knot[10, 87]][q] |
Out[16]= | -12 -10 -8 -6 3 2 2 4 8 10 16 18 -2 + q - q + q + q - -- + -- + q + 2 q + 4 q - 2 q - 2 q + q 4 2 q q |
In[17]:= | HOMFLYPT[Knot[10, 87]][a, z] |
Out[17]= | 2 2 4 4 -4 3 2 2 z z 2 2 4 z 2 z 2 4 6 -2 - a + -- + a - 4 z + -- + -- + 2 a z - 3 z + -- - ---- + a z - z - 2 4 2 4 2 a a a a a 6 z > -- 2 a |
In[18]:= | Kauffman[Knot[10, 87]][a, z] |
Out[18]= | 2 2 2 -4 3 2 z z 3 2 z z 3 z 2 2 -2 - a - -- - a - -- - - + a z + a z + 7 z + -- + -- + ---- + 3 a z - 2 3 a 6 4 2 a a a a a 3 3 3 4 4 4 2 7 z 15 z 13 z 3 3 3 4 2 z 5 z > a z + ---- + ----- + ----- + 2 a z - 3 a z - 5 z - ---- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 8 z 2 4 4 4 11 z 23 z 21 z 5 3 5 6 > ---- - 5 a z + a z - ----- - ----- - ----- - 6 a z + 3 a z - 3 z + 2 5 3 a a a a 6 6 6 7 7 7 8 z 12 z 21 z 2 6 4 z 5 z 7 z 7 8 5 z > -- - ----- - ----- + 5 a z + ---- + ---- + ---- + 6 a z + 5 z + ---- + 6 4 2 5 3 a 4 a a a a a a 8 9 9 10 z 2 z 2 z > ----- + ---- + ---- 2 3 a a a |
In[19]:= | {Vassiliev[2][Knot[10, 87]], Vassiliev[3][Knot[10, 87]]} |
Out[19]= | {0, 1} |
In[20]:= | Kh[Knot[10, 87]][q, t] |
Out[20]= | 7 1 2 1 4 2 6 4 3 - + 7 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + 6 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 6 q t + 7 q t + 4 q t + 6 q t + 3 q t + 4 q t + q t + 11 5 13 6 > 3 q t + q t |
In[21]:= | ColouredJones[Knot[10, 87], 2][q] |
Out[21]= | -12 3 2 6 16 12 18 51 33 47 105 46 90 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - --- + -- - 143 q + 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 10 > 35 q + 120 q - 141 q + 7 q + 123 q - 105 q - 20 q + 97 q - 54 q - 11 12 13 14 15 16 17 18 > 31 q + 53 q - 13 q - 20 q + 15 q + q - 4 q + q |
Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1087 |
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