© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:
10.82
1082 		10.84
1084
    10.83
KnotPlot 		This page is passe. Go here instead!

   The Alternating Knot 1083   

Visit 1083's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 1083's page at Knotilus!

Acknowledgement

10.83
KnotPlot

Warning. There is a mixup in the original (1976) Rolfsen table between the pictures and the invariants of the knots 1083 and 1086. That mixup lead to a similar mixup here. In the new (2003) edition of Rolfsen's book the mixup was corrected and on August 17, 2004, it was corrected here consistently with Rolfsen's correction. In the years between 1976 and 2003 other authors fixed the problem in different ways and our enumeration here may be different than theirs. I wish to thank Z-X. Tao for telling me about the (now corrected) mixup here and A. Stoimenow for telling me about the mixup in Rolfsen's original table.

PD Presentation: X1627 X5,16,6,17 X13,1,14,20 X7,15,8,14 X3948 X9,5,10,4 X19,11,20,10 X11,19,12,18 X17,13,18,12 X15,2,16,3

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

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

Minimum Braid Representative:


Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

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

Alexander Polynomial: 2t-3 - 9t-2 + 19t-1 - 23 + 19t - 9t2 + 2t3

Conway Polynomial: 1 + z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial: {K11a307, K11a323, ...}

Determinant and Signature: {83, 2}

Jones Polynomial: - q-2 + 4q-1 - 7 + 11q - 13q2 + 14q3 - 13q4 + 10q5 - 6q6 + 3q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial: {1073, ...}

A2 (sl(3)) Invariant: - q-6 + 2q-4 + 3q2 - 3q4 + 2q6 - q8 + 2q12 - 2q14 + 3q16 - q18 - q20 + q22 - q24

HOMFLY-PT Polynomial: - a-6 - 2a-6z2 - a-6z4 + 2a-4 + 4a-4z2 + 3a-4z4 + a-4z6 - a-2 + 2a-2z4 + a-2z6 + 1 - z2 - z4

Kauffman Polynomial: - 2a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 3a-8z6 - 3a-7z + 9a-7z3 - 11a-7z5 + 5a-7z7 + a-6 - 4a-6z2 + 10a-6z4 - 10a-6z6 + 5a-6z8 - 6a-5z + 20a-5z3 - 18a-5z5 + 5a-5z7 + 2a-5z9 + 2a-4 - 10a-4z2 + 22a-4z4 - 22a-4z6 + 10a-4z8 - 4a-3z + 13a-3z3 - 17a-3z5 + 6a-3z7 + 2a-3z9 + a-2 - 2a-2z2 - a-2z4 - 5a-2z6 + 5a-2z8 - a-1z + 3a-1z3 - 10a-1z5 + 6a-1z7 + 1 + 2z2 - 7z4 + 4z6 - az3 + az5

V2 and V3, the type 2 and 3 Vassiliev invariants: {1, 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 1083. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 17          1
j = 15         2 
j = 13        41 
j = 11       62  
j = 9      74   
j = 7     76    
j = 5    67     
j = 3   57      
j = 1  37       
j = -1 14        
j = -3 3         
j = -51          

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-7 - 4q-6 + 2q-5 + 12q-4 - 23q-3 + q-2 + 48q-1 - 56 - 19q + 107q2 - 82q3 - 59q4 + 159q5 - 83q6 - 97q7 + 175q8 - 61q9 - 110q10 + 145q11 - 26q12 - 91q13 + 85q14 + q15 - 52q16 + 32q17 + 7q18 - 18q19 + 7q20 + 2q21 - 3q22 + q23
3 - q-15 + 4q-14 - 2q-13 - 7q-12 + 19q-10 + 4q-9 - 45q-8 - 8q-7 + 76q-6 + 39q-5 - 132q-4 - 88q-3 + 186q-2 + 186q-1 - 243 - 308q + 255q2 + 481q3 - 251q4 - 640q5 + 184q6 + 806q7 - 100q8 - 925q9 - 18q10 + 1015q11 + 129q12 - 1047q13 - 243q14 + 1036q15 + 339q16 - 972q17 - 422q18 + 867q19 + 481q20 - 725q21 - 507q22 + 555q23 + 498q24 - 378q25 - 457q26 + 226q27 + 372q28 - 97q29 - 279q30 + 20q31 + 184q32 + 16q33 - 105q34 - 25q35 + 54q36 + 17q37 - 24q38 - 9q39 + 11q40 + 2q41 - 3q42 - 2q43 + 3q44 - q45
4 q-26 - 4q-25 + 2q-24 + 7q-23 - 5q-22 + 4q-21 - 24q-20 + 10q-19 + 38q-18 - 20q-17 + 10q-16 - 101q-15 + 30q-14 + 160q-13 - 13q-12 - 3q-11 - 371q-10 - 19q-9 + 472q-8 + 245q-7 + 117q-6 - 1028q-5 - 530q-4 + 770q-3 + 1045q-2 + 940q-1 - 1794 - 1883q + 296q2 + 2051q3 + 2909q4 - 1787q5 - 3648q6 - 1433q7 + 2331q8 + 5457q9 - 530q10 - 4793q11 - 3823q12 + 1463q13 + 7463q14 + 1397q15 - 4833q16 - 5817q17 - 40q18 + 8329q19 + 3122q20 - 4044q21 - 6895q22 - 1549q23 + 8120q24 + 4292q25 - 2771q26 - 7048q27 - 2870q28 + 6944q29 + 4893q30 - 1085q31 - 6263q32 - 3919q33 + 4842q34 + 4730q35 + 760q36 - 4474q37 - 4266q38 + 2245q39 + 3557q40 + 2023q41 - 2140q42 - 3487q43 + 188q44 + 1750q45 + 2055q46 - 303q47 - 1952q48 - 577q49 + 313q50 + 1205q51 + 370q52 - 670q53 - 390q54 - 196q55 + 411q56 + 270q57 - 127q58 - 91q59 - 145q60 + 83q61 + 81q62 - 23q63 + 4q64 - 41q65 + 14q66 + 15q67 - 10q68 + 5q69 - 6q70 + 3q71 + 2q72 - 3q73 + q74

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, 83]]
Out[2]=   
PD[X[1, 6, 2, 7], X[5, 16, 6, 17], X[13, 1, 14, 20], X[7, 15, 8, 14], 
 
>   X[3, 9, 4, 8], X[9, 5, 10, 4], X[19, 11, 20, 10], X[11, 19, 12, 18], 
 
>   X[17, 13, 18, 12], X[15, 2, 16, 3]]
In[3]:=
GaussCode[Knot[10, 83]]
Out[3]=   
GaussCode[-1, 10, -5, 6, -2, 1, -4, 5, -6, 7, -8, 9, -3, 4, -10, 2, -9, 8, -7, 
 
>   3]
In[4]:=
DTCode[Knot[10, 83]]
Out[4]=   
DTCode[6, 8, 16, 14, 4, 18, 20, 2, 12, 10]
In[5]:=
br = BR[Knot[10, 83]]
Out[5]=   
BR[4, {1, 1, 2, -1, 2, -3, 2, 2, -3, 2, -3}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=   
{4, 11}
In[7]:=
BraidIndex[Knot[10, 83]]
Out[7]=   
4
In[8]:=
Show[DrawMorseLink[Knot[10, 83]]]
Out[8]=   
 -Graphics- 
In[9]:=
#[Knot[10, 83]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=   
{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=
alex = Alexander[Knot[10, 83]][t]
Out[10]=   
      2    9    19             2      3
-23 + -- - -- + -- + 19 t - 9 t  + 2 t
       3    2   t
      t    t
In[11]:=
Conway[Knot[10, 83]][z]
Out[11]=   
     2      4      6
1 + z  + 3 z  + 2 z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=   
{Knot[10, 83], Knot[11, Alternating, 307], Knot[11, Alternating, 323]}
In[13]:=
{KnotDet[Knot[10, 83]], KnotSignature[Knot[10, 83]]}
Out[13]=   
{83, 2}
In[14]:=
Jones[Knot[10, 83]][q]
Out[14]=   
      -2   4              2       3       4       5      6      7    8
-7 - q   + - + 11 q - 13 q  + 14 q  - 13 q  + 10 q  - 6 q  + 3 q  - q
           q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=   
{Knot[10, 73], Knot[10, 83]}
In[16]:=
A2Invariant[Knot[10, 83]][q]
Out[16]=   
  -6   2       2      4      6    8      12      14      16    18    20    22
-q   + -- + 3 q  - 3 q  + 2 q  - q  + 2 q   - 2 q   + 3 q   - q   - q   + q   - 
        4
       q
 
     24
>   q
In[17]:=
HOMFLYPT[Knot[10, 83]][a, z]
Out[17]=   
                             2      2         4      4      4    6    6
     -6   2     -2    2   2 z    4 z     4   z    3 z    2 z    z    z
1 - a   + -- - a   - z  - ---- + ---- - z  - -- + ---- + ---- + -- + --
           4                6      4          6     4      2     4    2
          a                a      a          a     a      a     a    a
In[18]:=
Kauffman[Knot[10, 83]][a, z]
Out[18]=   
                                                     2      2       2      2
     -6   2     -2   3 z   6 z   4 z   z      2   2 z    4 z    10 z    2 z
1 + a   + -- + a   - --- - --- - --- - - + 2 z  + ---- - ---- - ----- - ---- - 
           4          7     5     3    a            8      6      4       2
          a          a     a     a                 a      a      a       a
 
       3      3       3       3      3                    4       4       4
    2 z    9 z    20 z    13 z    3 z       3      4   6 z    10 z    22 z
>   ---- + ---- + ----- + ----- + ---- - a z  - 7 z  - ---- + ----- + ----- - 
      9      7      5       3      a                     8      6       4
     a      a      a       a                            a      a       a
 
     4    5       5       5       5       5                    6       6
    z    z    11 z    18 z    17 z    10 z       5      6   3 z    10 z
>   -- + -- - ----- - ----- - ----- - ----- + a z  + 4 z  + ---- - ----- - 
     2    9     7       5       3       a                     8      6
    a    a     a       a       a                             a      a
 
        6      6      7      7      7      7      8       8      8      9      9
    22 z    5 z    5 z    5 z    6 z    6 z    5 z    10 z    5 z    2 z    2 z
>   ----- - ---- + ---- + ---- + ---- + ---- + ---- + ----- + ---- + ---- + ----
      4       2      7      5      3     a       6      4       2      5      3
     a       a      a      a      a             a      a       a      a      a
In[19]:=
{Vassiliev[2][Knot[10, 83]], Vassiliev[3][Knot[10, 83]]}
Out[19]=   
{1, 2}
In[20]:=
Kh[Knot[10, 83]][q, t]
Out[20]=   
         3     1       3      1      4    3 q      3        5        5  2
7 q + 5 q  + ----- + ----- + ---- + --- + --- + 7 q  t + 6 q  t + 7 q  t  + 
              5  3    3  2      2   q t    t
             q  t    q  t    q t
 
       7  2      7  3      9  3      9  4      11  4      11  5      13  5
>   7 q  t  + 6 q  t  + 7 q  t  + 4 q  t  + 6 q   t  + 2 q   t  + 4 q   t  + 
 
     13  6      15  6    17  7
>   q   t  + 2 q   t  + q   t
In[21]:=
ColouredJones[Knot[10, 83], 2][q]
Out[21]=   
       -7   4    2    12   23    -2   48               2       3       4
-56 + q   - -- + -- + -- - -- + q   + -- - 19 q + 107 q  - 82 q  - 59 q  + 
             6    5    4    3         q
            q    q    q    q
 
         5       6       7        8       9        10        11       12
>   159 q  - 83 q  - 97 q  + 175 q  - 61 q  - 110 q   + 145 q   - 26 q   - 
 
        13       14    15       16       17      18       19      20      21
>   91 q   + 85 q   + q   - 52 q   + 32 q   + 7 q   - 18 q   + 7 q   + 2 q   - 
 
       22    23
>   3 q   + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1083
10.82
1082 		10.84
1084