© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:
9.38
938 		9.40
940
    9.39
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   The Alternating Knot 939   

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Acknowledgement

9.39
KnotPlot

PD Presentation: X1627 X3,11,4,10 X7,18,8,1 X17,13,18,12 X9,17,10,16 X5,15,6,14 X15,5,16,4 X11,3,12,2 X13,9,14,8

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

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

Minimum Braid Representative:


Length is 12, 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 1 2 3 / 4--6 1

Alexander Polynomial: - 3t-2 + 14t-1 - 21 + 14t - 3t2

Conway Polynomial: 1 + 2z2 - 3z4

Other knots with the same Alexander/Conway Polynomial: {K11n162, ...}

Determinant and Signature: {55, 2}

Jones Polynomial: q-1 - 3 + 6q - 8q2 + 10q3 - 9q4 + 8q5 - 6q6 + 3q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial: {K11n11, K11n112, ...}

A2 (sl(3)) Invariant: q-4 - q-2 - 1 + 3q2 - q4 + 2q6 + q8 - q10 + q12 - 2q14 + 2q16 - q20 + 2q22 - q24 - q26

HOMFLY-PT Polynomial: - a-8 + 2a-6 + 3a-6z2 - 2a-4 - 3a-4z2 - 2a-4z4 + 2a-2 + a-2z2 - a-2z4 + z2

Kauffman Polynomial: a-9z - 2a-9z3 + a-9z5 - a-8 + 3a-8z2 - 6a-8z4 + 3a-8z6 - a-7z + 2a-7z3 - 7a-7z5 + 4a-7z7 - 2a-6 + 9a-6z2 - 13a-6z4 + 3a-6z6 + 2a-6z8 - 3a-5z + 12a-5z3 - 18a-5z5 + 9a-5z7 - 2a-4 + 12a-4z2 - 15a-4z4 + 5a-4z6 + 2a-4z8 - a-3z + 5a-3z3 - 7a-3z5 + 5a-3z7 - 2a-2 + 5a-2z2 - 7a-2z4 + 5a-2z6 - 3a-1z3 + 3a-1z5 - z2 + z4

V2 and V3, the type 2 and 3 Vassiliev invariants: {2, 4}

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 939. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 17         1
j = 15        2 
j = 13       41 
j = 11      42  
j = 9     54   
j = 7    54    
j = 5   35     
j = 3  35      
j = 1 14       
j = -1 2        
j = -31         

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-4 - 3q-3 + 2q-2 + 7q-1 - 17 + 7q + 28q2 - 45q3 + 4q4 + 62q5 - 68q6 - 9q7 + 88q8 - 72q9 - 23q10 + 89q11 - 55q12 - 32q13 + 69q14 - 27q15 - 30q16 + 37q17 - 5q18 - 16q19 + 10q20 + q21 - 3q22 + q23
3 q-9 - 3q-8 + 2q-7 + 3q-6 - 2q-5 - 10q-4 + 8q-3 + 23q-2 - 17q-1 - 49 + 25q + 91q2 - 18q3 - 156q4 + 2q5 + 220q6 + 44q7 - 288q8 - 100q9 + 336q10 + 172q11 - 373q12 - 227q13 + 372q14 + 291q15 - 370q16 - 321q17 + 331q18 + 353q19 - 293q20 - 357q21 + 229q22 + 356q23 - 162q24 - 336q25 + 91q26 + 298q27 - 25q28 - 245q29 - 25q30 + 179q31 + 58q32 - 120q33 - 59q34 + 62q35 + 51q36 - 26q37 - 33q38 + 8q39 + 16q40 - 2q41 - 5q42 - q43 + 3q44 - q45
4 q-16 - 3q-15 + 2q-14 + 3q-13 - 6q-12 + 5q-11 - 9q-10 + 14q-9 + 12q-8 - 38q-7 + 3q-6 - 20q-5 + 77q-4 + 72q-3 - 128q-2 - 89q-1 - 109 + 250q + 345q2 - 174q3 - 358q4 - 521q5 + 384q6 + 950q7 + 132q8 - 612q9 - 1364q10 + 137q11 + 1614q12 + 881q13 - 489q14 - 2296q15 - 533q16 + 1925q17 + 1701q18 + 26q19 - 2863q20 - 1260q21 + 1798q22 + 2205q23 + 634q24 - 2957q25 - 1747q26 + 1419q27 + 2332q28 + 1123q29 - 2683q30 - 1969q31 + 889q32 + 2169q33 + 1508q34 - 2105q35 - 1981q36 + 220q37 + 1727q38 + 1765q39 - 1256q40 - 1713q41 - 452q42 + 1002q43 + 1710q44 - 348q45 - 1105q46 - 799q47 + 209q48 + 1217q49 + 228q50 - 380q51 - 651q52 - 254q53 + 546q54 + 286q55 + 55q56 - 268q57 - 260q58 + 116q59 + 105q60 + 108q61 - 38q62 - 100q63 + 6q64 + 5q65 + 35q66 + 4q67 - 19q68 + 2q69 - 3q70 + 5q71 + q72 - 3q73 + q74
5 q-25 - 3q-24 + 2q-23 + 3q-22 - 6q-21 + q-20 + 6q-19 - 3q-18 + 3q-17 + 2q-16 - 25q-15 - 11q-14 + 35q-13 + 41q-12 + 25q-11 - 39q-10 - 129q-9 - 109q-8 + 78q-7 + 281q-6 + 280q-5 - 28q-4 - 506q-3 - 679q-2 - 192q-1 + 749 + 1346q + 768q2 - 849q3 - 2242q4 - 1862q5 + 517q6 + 3232q7 + 3563q8 + 414q9 - 3988q10 - 5645q11 - 2223q12 + 4195q13 + 7949q14 + 4709q15 - 3653q16 - 9909q17 - 7720q18 + 2243q19 + 11390q20 + 10743q21 - 208q22 - 12040q23 - 13495q24 - 2183q25 + 12021q26 + 15589q27 + 4551q28 - 11311q29 - 17086q30 - 6663q31 + 10381q32 + 17780q33 + 8360q34 - 9117q35 - 18085q36 - 9684q37 + 7990q38 + 17843q39 + 10627q40 - 6638q41 - 17433q42 - 11403q43 + 5371q44 + 16645q45 + 11986q46 - 3802q47 - 15624q48 - 12498q49 + 2110q50 + 14176q51 + 12791q52 - 134q53 - 12309q54 - 12793q55 - 1903q56 + 9944q57 + 12318q58 + 3854q59 - 7211q60 - 11228q61 - 5428q62 + 4321q63 + 9482q64 + 6351q65 - 1549q66 - 7254q67 - 6469q68 - 680q69 + 4783q70 + 5738q71 + 2201q72 - 2468q73 - 4485q74 - 2779q75 + 657q76 + 2919q77 + 2637q78 + 511q79 - 1554q80 - 2007q81 - 945q82 + 513q83 + 1245q84 + 918q85 + 50q86 - 610q87 - 649q88 - 236q89 + 212q90 + 349q91 + 211q92 - 11q93 - 159q94 - 132q95 - 20q96 + 52q97 + 47q98 + 29q99 - 8q100 - 30q101 - 6q102 + 7q103 + q104 + 3q105 + 3q106 - 5q107 - q108 + 3q109 - q110

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[9, 39]]
Out[2]=   
PD[X[1, 6, 2, 7], X[3, 11, 4, 10], X[7, 18, 8, 1], X[17, 13, 18, 12], 
 
>   X[9, 17, 10, 16], X[5, 15, 6, 14], X[15, 5, 16, 4], X[11, 3, 12, 2], 
 
>   X[13, 9, 14, 8]]
In[3]:=
GaussCode[Knot[9, 39]]
Out[3]=   
GaussCode[-1, 8, -2, 7, -6, 1, -3, 9, -5, 2, -8, 4, -9, 6, -7, 5, -4, 3]
In[4]:=
DTCode[Knot[9, 39]]
Out[4]=   
DTCode[6, 10, 14, 18, 16, 2, 8, 4, 12]
In[5]:=
br = BR[Knot[9, 39]]
Out[5]=   
BR[5, {1, 1, 2, -1, -3, -2, 1, 4, 3, -2, 3, 4}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=   
{5, 12}
In[7]:=
BraidIndex[Knot[9, 39]]
Out[7]=   
5
In[8]:=
Show[DrawMorseLink[Knot[9, 39]]]
Out[8]=   
 -Graphics- 
In[9]:=
#[Knot[9, 39]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=   
{Reversible, 1, 2, 3, {4, 6}, 1}
In[10]:=
alex = Alexander[Knot[9, 39]][t]
Out[10]=   
      3    14             2
-21 - -- + -- + 14 t - 3 t
       2   t
      t
In[11]:=
Conway[Knot[9, 39]][z]
Out[11]=   
       2      4
1 + 2 z  - 3 z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=   
{Knot[9, 39], Knot[11, NonAlternating, 162]}
In[13]:=
{KnotDet[Knot[9, 39]], KnotSignature[Knot[9, 39]]}
Out[13]=   
{55, 2}
In[14]:=
Jones[Knot[9, 39]][q]
Out[14]=   
     1            2       3      4      5      6      7    8
-3 + - + 6 q - 8 q  + 10 q  - 9 q  + 8 q  - 6 q  + 3 q  - q
     q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=   
{Knot[9, 39], Knot[11, NonAlternating, 11], Knot[11, NonAlternating, 112]}
In[16]:=
A2Invariant[Knot[9, 39]][q]
Out[16]=   
      -4    -2      2    4      6    8    10    12      14      16    20
-1 + q   - q   + 3 q  - q  + 2 q  + q  - q   + q   - 2 q   + 2 q   - q   + 
 
       22    24    26
>   2 q   - q   - q
In[17]:=
HOMFLYPT[Knot[9, 39]][a, z]
Out[17]=   
                              2      2    2      4    4
  -8   2    2    2     2   3 z    3 z    z    2 z    z
-a   + -- - -- + -- + z  + ---- - ---- + -- - ---- - --
        6    4    2          6      4     2     4     2
       a    a    a          a      a     a     a     a
In[18]:=
Kauffman[Knot[9, 39]][a, z]
Out[18]=   
                                                   2      2       2      2
  -8   2    2    2    z    z    3 z   z     2   3 z    9 z    12 z    5 z
-a   - -- - -- - -- + -- - -- - --- - -- - z  + ---- + ---- + ----- + ---- - 
        6    4    2    9    7    5     3          8      6      4       2
       a    a    a    a    a    a     a          a      a      a       a
 
       3      3       3      3      3           4       4       4      4    5
    2 z    2 z    12 z    5 z    3 z     4   6 z    13 z    15 z    7 z    z
>   ---- + ---- + ----- + ---- - ---- + z  - ---- - ----- - ----- - ---- + -- - 
      9      7      5       3     a            8      6       4       2     9
     a      a      a       a                  a      a       a       a     a
 
       5       5      5      5      6      6      6      6      7      7
    7 z    18 z    7 z    3 z    3 z    3 z    5 z    5 z    4 z    9 z
>   ---- - ----- - ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + 
      7      5       3     a       8      6      4      2      7      5
     a      a       a             a      a      a      a      a      a
 
       7      8      8
    5 z    2 z    2 z
>   ---- + ---- + ----
      3      6      4
     a      a      a
In[19]:=
{Vassiliev[2][Knot[9, 39]], Vassiliev[3][Knot[9, 39]]}
Out[19]=   
{2, 4}
In[20]:=
Kh[Knot[9, 39]][q, t]
Out[20]=   
         3     1      2    q      3        5        5  2      7  2      7  3
4 q + 3 q  + ----- + --- + - + 5 q  t + 3 q  t + 5 q  t  + 5 q  t  + 4 q  t  + 
              3  2   q t   t
             q  t
 
       9  3      9  4      11  4      11  5      13  5    13  6      15  6
>   5 q  t  + 4 q  t  + 4 q   t  + 2 q   t  + 4 q   t  + q   t  + 2 q   t  + 
 
     17  7
>   q   t
In[21]:=
ColouredJones[Knot[9, 39], 2][q]
Out[21]=   
       -4   3    2    7             2       3      4       5       6      7
-17 + q   - -- + -- + - + 7 q + 28 q  - 45 q  + 4 q  + 62 q  - 68 q  - 9 q  + 
             3    2   q
            q    q
 
        8       9       10       11       12       13       14       15
>   88 q  - 72 q  - 23 q   + 89 q   - 55 q   - 32 q   + 69 q   - 27 q   - 
 
        16       17      18       19       20    21      22    23
>   30 q   + 37 q   - 5 q   - 16 q   + 10 q   + q   - 3 q   + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 939
9.38
938 		9.40
940