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10.40
1040
10.42
1042
    10.41
KnotPlot
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   The Alternating Knot 1041   

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Acknowledgement

10.41
KnotPlot

PD Presentation: X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X9,20,10,1 X15,19,16,18 X13,8,14,9 X17,6,18,7 X7,16,8,17 X19,15,20,14

Gauss Code: {-1, 4, -3, 1, -2, 8, -9, 7, -5, 3, -4, 2, -7, 10, -6, 9, -8, 6, -10, 5}

DT (Dowker-Thistlethwaite) Code: 4 10 12 16 20 2 8 18 6 14

Minimum Braid Representative:


Length is 10, width is 5
Braid index is 5

A Morse Link Presentation:

3D Invariants:
Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index
Reversible 2 3 2 / NotAvailable 1

Alexander Polynomial: t-3 - 7t-2 + 17t-1 - 21 + 17t - 7t2 + t3

Conway Polynomial: 1 - 2z2 - z4 + z6

Other knots with the same Alexander/Conway Polynomial: {K11n5, ...}

Determinant and Signature: {71, -2}

Jones Polynomial: q-7 - 3q-6 + 6q-5 - 9q-4 + 11q-3 - 12q-2 + 11q-1 - 8 + 6q - 3q2 + q3

Other knots (up to mirrors) with the same Jones Polynomial: {1094, ...}

A2 (sl(3)) Invariant: q-22 - q-18 + 2q-16 - 2q-14 + q-10 - 2q-8 + 2q-6 - 2q-4 + 2q-2 + 1 - q2 + 2q4 - q6 + q10

HOMFLY-PT Polynomial: a-2 + a-2z2 - 1 - 4z2 - 2z4 + 2a2 + 4a2z2 + 3a2z4 + a2z6 - 2a4 - 4a4z2 - 2a4z4 + a6 + a6z2

Kauffman Polynomial: - a-2 + 3a-2z2 - 3a-2z4 + a-2z6 - a-1z + 7a-1z3 - 9a-1z5 + 3a-1z7 - 1 + 7z2 - 4z4 - 5z6 + 3z8 - 2az + 13az3 - 20az5 + 6az7 + az9 - 2a2 + 9a2z2 - 8a2z4 - 7a2z6 + 6a2z8 - 2a3z + 10a3z3 - 18a3z5 + 8a3z7 + a3z9 - 2a4 + 10a4z2 - 14a4z4 + 4a4z6 + 3a4z8 + a5z3 - 4a5z5 + 5a5z7 - a6 + 4a6z2 - 6a6z4 + 5a6z6 + a7z - 3a7z3 + 3a7z5 - a8z2 + a8z4

V2 and V3, the type 2 and 3 Vassiliev invariants: {-2, 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=-2 is the signature of 1041. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 7          1
j = 5         2 
j = 3        41 
j = 1       42  
j = -1      74   
j = -3     65    
j = -5    56     
j = -7   46      
j = -9  25       
j = -11 14        
j = -13 2         
j = -151          

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-20 - 3q-19 + 2q-18 + 7q-17 - 17q-16 + 8q-15 + 24q-14 - 47q-13 + 18q-12 + 53q-11 - 86q-10 + 22q-9 + 85q-8 - 110q-7 + 12q-6 + 103q-5 - 104q-4 - 6q-3 + 97q-2 - 74q-1 - 19 + 71q - 37q2 - 22q3 + 37q4 - 10q5 - 13q6 + 11q7 - 3q9 + q10
3 q-39 - 3q-38 + 2q-37 + 3q-36 - q-35 - 11q-34 + 6q-33 + 20q-32 - 12q-31 - 36q-30 + 24q-29 + 59q-28 - 40q-27 - 97q-26 + 66q-25 + 146q-24 - 91q-23 - 213q-22 + 114q-21 + 293q-20 - 130q-19 - 373q-18 + 124q-17 + 452q-16 - 105q-15 - 510q-14 + 67q-13 + 546q-12 - 19q-11 - 553q-10 - 36q-9 + 535q-8 + 93q-7 - 500q-6 - 137q-5 + 436q-4 + 187q-3 - 375q-2 - 207q-1 + 287 + 230q - 214q2 - 216q3 + 129q4 + 198q5 - 67q6 - 157q7 + 15q8 + 117q9 + 9q10 - 71q11 - 22q12 + 39q13 + 20q14 - 17q15 - 13q16 + 6q17 + 5q18 - 3q20 + q21
4 q-64 - 3q-63 + 2q-62 + 3q-61 - 5q-60 + 5q-59 - 13q-58 + 12q-57 + 14q-56 - 26q-55 + 16q-54 - 37q-53 + 44q-52 + 41q-51 - 92q-50 + 27q-49 - 63q-48 + 144q-47 + 91q-46 - 266q-45 - 12q-44 - 74q-43 + 409q-42 + 230q-41 - 609q-40 - 235q-39 - 119q-38 + 929q-37 + 600q-36 - 1057q-35 - 749q-34 - 361q-33 + 1608q-32 + 1293q-31 - 1359q-30 - 1432q-29 - 916q-28 + 2127q-27 + 2135q-26 - 1285q-25 - 1940q-24 - 1656q-23 + 2217q-22 + 2774q-21 - 874q-20 - 2026q-19 - 2279q-18 + 1883q-17 + 2980q-16 - 320q-15 - 1710q-14 - 2619q-13 + 1288q-12 + 2791q-11 + 235q-10 - 1157q-9 - 2665q-8 + 577q-7 + 2302q-6 + 713q-5 - 475q-4 - 2422q-3 - 122q-2 + 1579q-1 + 981 + 210q - 1864q2 - 604q3 + 746q4 + 898q5 + 682q6 - 1095q7 - 692q8 + 69q9 + 517q10 + 757q11 - 398q12 - 445q13 - 231q14 + 119q15 + 507q16 - 28q17 - 141q18 - 196q19 - 67q20 + 206q21 + 45q22 + 8q23 - 73q24 - 64q25 + 48q26 + 15q27 + 20q28 - 10q29 - 20q30 + 6q31 + 5q33 - 3q35 + q36
5 q-95 - 3q-94 + 2q-93 + 3q-92 - 5q-91 + q-90 + 3q-89 - 7q-88 + 6q-87 + 10q-86 - 15q-85 - 5q-84 + 11q-83 - 4q-82 + 11q-81 + 5q-80 - 33q-79 - 15q-78 + 40q-77 + 49q-76 + 8q-75 - 73q-74 - 131q-73 - 23q-72 + 195q-71 + 279q-70 + 44q-69 - 378q-68 - 563q-67 - 136q-66 + 667q-65 + 1045q-64 + 366q-63 - 1041q-62 - 1829q-61 - 813q-60 + 1475q-59 + 2906q-58 + 1630q-57 - 1827q-56 - 4354q-55 - 2896q-54 + 2020q-53 + 6007q-52 + 4654q-51 - 1824q-50 - 7734q-49 - 6881q-48 + 1143q-47 + 9335q-46 + 9366q-45 + 49q-44 - 10492q-43 - 11907q-42 - 1751q-41 + 11134q-40 + 14226q-39 + 3685q-38 - 11104q-37 - 16046q-36 - 5751q-35 + 10522q-34 + 17257q-33 + 7641q-32 - 9471q-31 - 17800q-30 - 9242q-29 + 8139q-28 + 17727q-27 + 10462q-26 - 6620q-25 - 17182q-24 - 11311q-23 + 5044q-22 + 16226q-21 + 11872q-20 - 3440q-19 - 14968q-18 - 12163q-17 + 1751q-16 + 13474q-15 + 12281q-14 - 108q-13 - 11696q-12 - 12094q-11 - 1663q-10 + 9709q-9 + 11732q-8 + 3196q-7 - 7467q-6 - 10866q-5 - 4708q-4 + 5110q-3 + 9762q-2 + 5675q-1 - 2747 - 8086q - 6309q2 + 563q3 + 6272q4 + 6215q5 + 1215q6 - 4183q7 - 5669q8 - 2444q9 + 2290q10 + 4575q11 + 3036q12 - 616q13 - 3320q14 - 3034q15 - 495q16 + 1963q17 + 2580q18 + 1169q19 - 890q20 - 1893q21 - 1294q22 + 98q23 + 1143q24 + 1162q25 + 309q26 - 557q27 - 822q28 - 440q29 + 156q30 + 488q31 + 388q32 + 48q33 - 231q34 - 274q35 - 98q36 + 81q37 + 136q38 + 98q39 - 3q40 - 73q41 - 57q42 - 3q43 + 15q44 + 24q45 + 20q46 - 10q47 - 13q48 - q49 + 5q52 - 3q54 + q55
6 q-132 - 3q-131 + 2q-130 + 3q-129 - 5q-128 + q-127 - q-126 + 9q-125 - 13q-124 + 2q-123 + 21q-122 - 26q-121 + 2q-120 + 3q-119 + 29q-118 - 40q-117 - 10q-116 + 67q-115 - 70q-114 + 12q-113 + 36q-112 + 83q-111 - 132q-110 - 92q-109 + 128q-108 - 142q-107 + 123q-106 + 219q-105 + 249q-104 - 392q-103 - 487q-102 + 10q-101 - 302q-100 + 595q-99 + 1025q-98 + 891q-97 - 920q-96 - 1814q-95 - 1077q-94 - 997q-93 + 1822q-92 + 3610q-91 + 3269q-90 - 1321q-89 - 5025q-88 - 5031q-87 - 3947q-86 + 3611q-85 + 9688q-84 + 10205q-83 + 543q-82 - 10274q-81 - 14492q-80 - 12899q-79 + 3455q-78 + 19731q-77 + 25188q-76 + 9323q-75 - 14530q-74 - 30078q-73 - 31958q-72 - 4180q-71 + 29880q-70 + 48458q-69 + 29527q-68 - 11158q-67 - 46714q-66 - 60527q-65 - 24051q-64 + 32174q-63 + 73183q-62 + 59611q-61 + 4988q-60 - 55055q-59 - 90038q-58 - 53732q-57 + 21109q-56 + 88627q-55 + 89805q-54 + 30992q-53 - 49394q-52 - 109059q-51 - 82953q-50 - 124q-49 + 89073q-48 + 109001q-47 + 56857q-46 - 33077q-45 - 112526q-44 - 101680q-43 - 22271q-42 + 77868q-41 + 113419q-40 + 74179q-39 - 14318q-38 - 104025q-37 - 107583q-36 - 38712q-35 + 62045q-34 + 107014q-33 + 82006q-32 + 1943q-31 - 89622q-30 - 104730q-29 - 49560q-28 + 45142q-27 + 94960q-26 + 84162q-25 + 16414q-24 - 71826q-23 - 97096q-22 - 57964q-21 + 26256q-20 + 78682q-19 + 83078q-18 + 31243q-17 - 49669q-16 - 84666q-15 - 64503q-14 + 4383q-13 + 56713q-12 + 76917q-11 + 45146q-10 - 22978q-9 - 65037q-8 - 65382q-7 - 17439q-6 + 29106q-5 + 62043q-4 + 52563q-3 + 3894q-2 - 38220q-1 - 55786 - 32044q + 1043q2 + 38313q3 + 47836q4 + 22642q5 - 10156q6 - 35689q7 - 33135q8 - 18421q9 + 12391q10 + 31357q11 + 26828q12 + 9494q13 - 12620q14 - 21653q15 - 23111q16 - 5657q17 + 11488q18 + 18223q19 + 14993q20 + 3169q21 - 6530q22 - 15749q23 - 10728q24 - 1711q25 + 6133q26 + 9801q27 + 7427q28 + 2853q29 - 5672q30 - 6815q31 - 5025q32 - 1036q33 + 2707q34 + 4486q35 + 4402q36 - 12q37 - 1764q38 - 2865q39 - 2167q40 - 733q41 + 1026q42 + 2312q43 + 974q44 + 398q45 - 608q46 - 932q47 - 953q48 - 255q49 + 601q50 + 367q51 + 456q52 + 110q53 - 95q54 - 367q55 - 242q56 + 63q57 + 9q58 + 135q59 + 89q60 + 62q61 - 73q62 - 68q63 + 4q64 - 27q65 + 15q66 + 15q67 + 29q68 - 10q69 - 13q70 + 6q71 - 7q72 + 5q75 - 3q77 + q78
7 q-175 - 3q-174 + 2q-173 + 3q-172 - 5q-171 + q-170 - q-169 + 5q-168 + 3q-167 - 17q-166 + 13q-165 + 10q-164 - 19q-163 + 4q-162 - 5q-161 + 18q-160 + 8q-159 - 58q-158 + 38q-157 + 34q-156 - 36q-155 + 24q-154 - 28q-153 + 32q-152 - 3q-151 - 168q-150 + 71q-149 + 107q-148 + 27q-147 + 162q-146 - 66q-145 - 48q-144 - 169q-143 - 530q-142 + 31q-141 + 325q-140 + 505q-139 + 862q-138 + 73q-137 - 426q-136 - 1089q-135 - 1911q-134 - 574q-133 + 852q-132 + 2379q-131 + 3622q-130 + 1531q-129 - 1171q-128 - 4360q-127 - 6875q-126 - 3923q-125 + 1268q-124 + 7721q-123 + 12522q-122 + 8471q-121 - 548q-120 - 12382q-119 - 21548q-118 - 17000q-117 - 2489q-116 + 18291q-115 + 35373q-114 + 31648q-113 + 9734q-112 - 24540q-111 - 54676q-110 - 54708q-109 - 24306q-108 + 28709q-107 + 79332q-106 + 88736q-105 + 50119q-104 - 27646q-103 - 108038q-102 - 134545q-101 - 90360q-100 + 16367q-99 + 136538q-98 + 191585q-97 + 148231q-96 + 10073q-95 - 160059q-94 - 256327q-93 - 223315q-92 - 55721q-91 + 171536q-90 + 322316q-89 + 313046q-88 + 122682q-87 - 165017q-86 - 382051q-85 - 411023q-84 - 208698q-83 + 136688q-82 + 426833q-81 + 507937q-80 + 308775q-79 - 85577q-78 - 450571q-77 - 594816q-76 - 414116q-75 + 15962q-74 + 450056q-73 + 662652q-72 + 514775q-71 + 66032q-70 - 426152q-69 - 706850q-68 - 602189q-67 - 151130q-66 + 384010q-65 + 725758q-64 + 669613q-63 + 231138q-62 - 329990q-61 - 722203q-60 - 714860q-59 - 299629q-58 + 271792q-57 + 701339q-56 + 738647q-55 + 353345q-54 - 215267q-53 - 669088q-52 - 744700q-51 - 392270q-50 + 163718q-49 + 630766q-48 + 738229q-47 + 418986q-46 - 118240q-45 - 590117q-44 - 723423q-43 - 437224q-42 + 76868q-41 + 548224q-40 + 704227q-39 + 451185q-38 - 37239q-37 - 504643q-36 - 681906q-35 - 463670q-34 - 3979q-33 + 457319q-32 + 656253q-31 + 476008q-30 + 49238q-29 - 403628q-28 - 625688q-27 - 487955q-26 - 99047q-25 + 341870q-24 + 586987q-23 + 496546q-22 + 153051q-21 - 270228q-20 - 537948q-19 - 499106q-18 - 207640q-17 + 190604q-16 + 475838q-15 + 489935q-14 + 258719q-13 - 104271q-12 - 400552q-11 - 466762q-10 - 300224q-9 + 18020q-8 + 313588q-7 + 425090q-6 + 326287q-5 + 63541q-4 - 219039q-3 - 366700q-2 - 332406q-1 - 131266 + 123886q + 292537q2 + 315818q3 + 180193q4 - 35602q5 - 209941q6 - 278016q7 - 204535q8 - 37376q9 + 125841q10 + 223127q11 + 204314q12 + 89082q13 - 49939q14 - 158894q15 - 181668q16 - 116323q17 - 10861q18 + 94365q19 + 143664q20 + 120100q21 + 50949q22 - 37985q23 - 98015q24 - 105251q25 - 70199q26 - 4146q27 + 54279q28 + 79120q29 + 70628q30 + 29217q31 - 18218q32 - 49299q33 - 58918q34 - 38811q35 - 5655q36 + 23044q37 + 40873q38 + 36288q39 + 17772q40 - 3654q41 - 22959q42 - 27579q43 - 20157q44 - 7103q45 + 8893q46 + 16829q47 + 16620q48 + 10960q49 - 50q50 - 7761q51 - 10979q52 - 10138q53 - 3848q54 + 1829q55 + 5569q56 + 7157q57 + 4471q58 + 1249q59 - 1785q60 - 4216q61 - 3479q62 - 1965q63 - 112q64 + 1827q65 + 1973q66 + 1732q67 + 908q68 - 557q69 - 1000q70 - 1064q71 - 756q72 + 6q73 + 218q74 + 500q75 + 605q76 + 170q77 - 34q78 - 229q79 - 269q80 - 83q81 - 88q82 + 8q83 + 148q84 + 89q85 + 53q86 - 19q87 - 57q88 + 4q89 - 31q90 - 27q91 + 15q92 + 15q93 + 20q94 - q95 - 13q96 + 6q97 - 7q99 + 5q102 - 3q104 + q105


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, 41]]
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, 20, 10, 1], X[15, 19, 16, 18], X[13, 8, 14, 9], X[17, 6, 18, 7], 
 
>   X[7, 16, 8, 17], X[19, 15, 20, 14]]
In[3]:=
GaussCode[Knot[10, 41]]
Out[3]=   
GaussCode[-1, 4, -3, 1, -2, 8, -9, 7, -5, 3, -4, 2, -7, 10, -6, 9, -8, 6, -10, 
 
>   5]
In[4]:=
DTCode[Knot[10, 41]]
Out[4]=   
DTCode[4, 10, 12, 16, 20, 2, 8, 18, 6, 14]
In[5]:=
br = BR[Knot[10, 41]]
Out[5]=   
BR[5, {1, -2, 1, -2, -2, 3, -2, -4, 3, -4}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=   
{5, 10}
In[7]:=
BraidIndex[Knot[10, 41]]
Out[7]=   
5
In[8]:=
Show[DrawMorseLink[Knot[10, 41]]]
Out[8]=   
 -Graphics- 
In[9]:=
#[Knot[10, 41]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=   
{Reversible, 2, 3, 2, NotAvailable, 1}
In[10]:=
alex = Alexander[Knot[10, 41]][t]
Out[10]=   
       -3   7    17             2    3
-21 + t   - -- + -- + 17 t - 7 t  + t
             2   t
            t
In[11]:=
Conway[Knot[10, 41]][z]
Out[11]=   
       2    4    6
1 - 2 z  - z  + z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=   
{Knot[10, 41], Knot[11, NonAlternating, 5]}
In[13]:=
{KnotDet[Knot[10, 41]], KnotSignature[Knot[10, 41]]}
Out[13]=   
{71, -2}
In[14]:=
Jones[Knot[10, 41]][q]
Out[14]=   
      -7   3    6    9    11   12   11            2    3
-8 + q   - -- + -- - -- + -- - -- + -- + 6 q - 3 q  + q
            6    5    4    3    2   q
           q    q    q    q    q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=   
{Knot[10, 41], Knot[10, 94]}
In[16]:=
A2Invariant[Knot[10, 41]][q]
Out[16]=   
     -22    -18    2     2     -10   2    2    2    2     2      4    6    10
1 + q    - q    + --- - --- + q    - -- + -- - -- + -- - q  + 2 q  - q  + q
                   16    14           8    6    4    2
                  q     q            q    q    q    q
In[17]:=
HOMFLYPT[Knot[10, 41]][a, z]
Out[17]=   
                                      2
      -2      2      4    6      2   z       2  2      4  2    6  2      4
-1 + a   + 2 a  - 2 a  + a  - 4 z  + -- + 4 a  z  - 4 a  z  + a  z  - 2 z  + 
                                      2
                                     a
 
       2  4      4  4    2  6
>   3 a  z  - 2 a  z  + a  z
In[18]:=
Kauffman[Knot[10, 41]][a, z]
Out[18]=   
                                                                    2
      -2      2      4    6   z              3      7        2   3 z
-1 - a   - 2 a  - 2 a  - a  - - - 2 a z - 2 a  z + a  z + 7 z  + ---- + 
                              a                                    2
                                                                  a
 
                                              3
       2  2       4  2      6  2    8  2   7 z          3       3  3    5  3
>   9 a  z  + 10 a  z  + 4 a  z  - a  z  + ---- + 13 a z  + 10 a  z  + a  z  - 
                                            a
 
                        4                                             5
       7  3      4   3 z       2  4       4  4      6  4    8  4   9 z
>   3 a  z  - 4 z  - ---- - 8 a  z  - 14 a  z  - 6 a  z  + a  z  - ---- - 
                       2                                            a
                      a
 
                                                     6
          5       3  5      5  5      7  5      6   z       2  6      4  6
>   20 a z  - 18 a  z  - 4 a  z  + 3 a  z  - 5 z  + -- - 7 a  z  + 4 a  z  + 
                                                     2
                                                    a
 
                 7
       6  6   3 z         7      3  7      5  7      8      2  8      4  8
>   5 a  z  + ---- + 6 a z  + 8 a  z  + 5 a  z  + 3 z  + 6 a  z  + 3 a  z  + 
               a
 
       9    3  9
>   a z  + a  z
In[19]:=
{Vassiliev[2][Knot[10, 41]], Vassiliev[3][Knot[10, 41]]}
Out[19]=   
{-2, 2}
In[20]:=
Kh[Knot[10, 41]][q, t]
Out[20]=   
5    7     1        2        1        4        2       5       4       6
-- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
 3   q    15  6    13  5    11  5    11  4    9  4    9  3    7  3    7  2
q        q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
      5      6      6     4 t                2      3  2    3  3      5  3
>   ----- + ---- + ---- + --- + 4 q t + 2 q t  + 4 q  t  + q  t  + 2 q  t  + 
     5  2    5      3      q
    q  t    q  t   q  t
 
     7  4
>   q  t
In[21]:=
ColouredJones[Knot[10, 41], 2][q]
Out[21]=   
       -20    3     2     7    17     8    24    47    18    53    86    22
-19 + q    - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + -- + 
              19    18    17    16    15    14    13    12    11    10    9
             q     q     q     q     q     q     q     q     q     q     q
 
    85   110   12   103   104   6    97   74              2       3       4
>   -- - --- + -- + --- - --- - -- + -- - -- + 71 q - 37 q  - 22 q  + 37 q  - 
     8    7     6    5     4     3    2   q
    q    q     q    q     q     q    q
 
        5       6       7      9    10
>   10 q  - 13 q  + 11 q  - 3 q  + q


Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1041
10.40
1040
10.42
1042