© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:
10.163
10163 		10.165
10165
    10.164
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   The Non Alternating Knot 10164   

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Acknowledgement

10.164
KnotPlot

Warning. In 1974 K. Perko noticed that the knots labeled 10161 and 10162 in Rolfsen's tables are in fact the same. In our table we removed his 10162 and renumbered the subsequent knots, so that our 10 crossings total is 165, one less than Rolfsen's 166. Read more: 1 2 3 4.

PD Presentation: X6271 X14,7,15,8 X15,2,16,3 X5,12,6,13 X9,19,10,18 X3,11,4,10 X17,5,18,4 X19,9,20,8 X11,16,12,17 X20,13,1,14

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

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

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
Reversible 1 2 3 / NotAvailable 1

Alexander Polynomial: 3t-2 - 11t-1 + 17 - 11t + 3t2

Conway Polynomial: 1 + z2 + 3z4

Other knots with the same Alexander/Conway Polynomial: {1010, ...}

Determinant and Signature: {45, 0}

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

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

A2 (sl(3)) Invariant: - q-16 + q-14 + q-12 - 2q-10 + q-8 - q-6 + 2q-2 + 3q2 - q4 + q6 + q8 - 2q10

HOMFLY-PT Polynomial: - a-2 - 2a-2z2 + 3 + 4z2 + 2z4 - a2 + a2z4 - a4z2

Kauffman Polynomial: - 2a-3z + 3a-3z3 + a-2 - 6a-2z2 + 4a-2z4 + a-2z6 - 5a-1z + 10a-1z3 - 6a-1z5 + 3a-1z7 + 3 - 9z2 + 8z4 - 3z6 + 2z8 - 5az + 16az3 - 17az5 + 7az7 + a2 - 3a2z4 - a2z6 + 2a2z8 - 2a3z + 7a3z3 - 10a3z5 + 4a3z7 + 3a4z2 - 7a4z4 + 3a4z6 - 2a5z3 + a5z5

V2 and V3, the type 2 and 3 Vassiliev invariants: {1, 0}

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 10164. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 7        2
j = 5       3 
j = 3      32 
j = 1     53  
j = -1    44   
j = -3   34    
j = -5  24     
j = -7 13      
j = -9 2       
j = -111        

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-15 - 3q-14 + q-13 + 9q-12 - 14q-11 - 5q-10 + 30q-9 - 21q-8 - 24q-7 + 52q-6 - 17q-5 - 46q-4 + 62q-3 - 7q-2 - 57q-1 + 56 + 4q - 50q2 + 35q3 + 11q4 - 29q5 + 12q6 + 8q7 - 8q8 + q10
3 - q-30 + 3q-29 - q-28 - 5q-27 - q-26 + 14q-25 + 7q-24 - 28q-23 - 22q-22 + 38q-21 + 52q-20 - 38q-19 - 92q-18 + 21q-17 + 132q-16 + 16q-15 - 163q-14 - 70q-13 + 184q-12 + 124q-11 - 184q-10 - 182q-9 + 177q-8 + 229q-7 - 160q-6 - 265q-5 + 140q-4 + 286q-3 - 109q-2 - 301q-1 + 82 + 290q - 38q2 - 276q3 + 7q4 + 229q5 + 36q6 - 184q7 - 53q8 + 121q9 + 65q10 - 72q11 - 51q12 + 28q13 + 36q14 - 5q15 - 21q16 + q17 + 3q18 + 4q19 - 2q20
4 q-50 - 3q-49 + q-48 + 5q-47 - 3q-46 + q-45 - 17q-44 + 5q-43 + 31q-42 + 3q-41 + 5q-40 - 81q-39 - 33q-38 + 80q-37 + 78q-36 + 105q-35 - 170q-34 - 204q-33 - 11q-32 + 149q-31 + 424q-30 - 46q-29 - 381q-28 - 378q-27 - 76q-26 + 765q-25 + 413q-24 - 219q-23 - 780q-22 - 682q-21 + 775q-20 + 932q-19 + 323q-18 - 898q-17 - 1358q-16 + 442q-15 + 1218q-14 + 939q-13 - 744q-12 - 1820q-11 + 26q-10 + 1269q-9 + 1389q-8 - 505q-7 - 2034q-6 - 327q-5 + 1193q-4 + 1657q-3 - 246q-2 - 2039q-1 - 635 + 972q + 1746q2 + 98q3 - 1771q4 - 896q5 + 529q6 + 1568q7 + 485q8 - 1180q9 - 939q10 - 17q11 + 1046q12 + 664q13 - 456q14 - 643q15 - 333q16 + 404q17 + 480q18 + 4q19 - 221q20 - 268q21 + 26q22 + 168q23 + 77q24 - 6q25 - 80q26 - 31q27 + 16q28 + 18q29 + 12q30 - 5q31 - 6q32 + q34
5 - q-75 + 3q-74 - q-73 - 5q-72 + 3q-71 + 3q-70 + 2q-69 + 5q-68 - 7q-67 - 26q-66 - 7q-65 + 28q-64 + 42q-63 + 39q-62 - 17q-61 - 102q-60 - 127q-59 - 17q-58 + 149q-57 + 250q-56 + 183q-55 - 105q-54 - 419q-53 - 474q-52 - 114q-51 + 460q-50 + 842q-49 + 622q-48 - 218q-47 - 1144q-46 - 1349q-45 - 434q-44 + 1098q-43 + 2104q-42 + 1544q-41 - 517q-40 - 2612q-39 - 2898q-38 - 691q-37 + 2578q-36 + 4209q-35 + 2408q-34 - 1847q-33 - 5178q-32 - 4376q-31 + 507q-30 + 5563q-29 + 6246q-28 + 1318q-27 - 5366q-26 - 7815q-25 - 3267q-24 + 4681q-23 + 8900q-22 + 5146q-21 - 3692q-20 - 9584q-19 - 6731q-18 + 2623q-17 + 9897q-16 + 8002q-15 - 1618q-14 - 9989q-13 - 8957q-12 + 737q-11 + 9940q-10 + 9698q-9 + 10q-8 - 9816q-7 - 10229q-6 - 747q-5 + 9557q-4 + 10715q-3 + 1495q-2 - 9169q-1 - 10991 - 2384q + 8414q2 + 11191q3 + 3386q4 - 7379q5 - 10965q6 - 4454q7 + 5826q8 + 10402q9 + 5405q10 - 4024q11 - 9195q12 - 6043q13 + 1987q14 + 7564q15 + 6145q16 - 186q17 - 5470q18 - 5652q19 - 1263q20 + 3434q21 + 4596q22 + 1988q23 - 1583q24 - 3245q25 - 2134q26 + 325q27 + 1932q28 + 1728q29 + 354q30 - 874q31 - 1136q32 - 553q33 + 252q34 + 604q35 + 404q36 + 41q37 - 212q38 - 248q39 - 96q40 + 69q41 + 79q42 + 56q43 + 12q44 - 30q45 - 23q46 - q47 + 3q48 + 2q49 + 4q50 - 2q51

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, 164]]
Out[2]=   
PD[X[6, 2, 7, 1], X[14, 7, 15, 8], X[15, 2, 16, 3], X[5, 12, 6, 13], 
 
>   X[9, 19, 10, 18], X[3, 11, 4, 10], X[17, 5, 18, 4], X[19, 9, 20, 8], 
 
>   X[11, 16, 12, 17], X[20, 13, 1, 14]]
In[3]:=
GaussCode[Knot[10, 164]]
Out[3]=   
GaussCode[1, 3, -6, 7, -4, -1, 2, 8, -5, 6, -9, 4, 10, -2, -3, 9, -7, 5, -8, 
 
>   -10]
In[4]:=
DTCode[Knot[10, 164]]
Out[4]=   
DTCode[6, -10, -12, 14, -18, -16, 20, -2, -4, -8]
In[5]:=
br = BR[Knot[10, 164]]
Out[5]=   
BR[4, {1, 1, -2, 1, -2, -2, -3, 2, -1, 2, -3}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=   
{4, 11}
In[7]:=
BraidIndex[Knot[10, 164]]
Out[7]=   
4
In[8]:=
Show[DrawMorseLink[Knot[10, 164]]]
Out[8]=   
 -Graphics- 
In[9]:=
#[Knot[10, 164]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=   
{Reversible, 1, 2, 3, NotAvailable, 1}
In[10]:=
alex = Alexander[Knot[10, 164]][t]
Out[10]=   
     3    11             2
17 + -- - -- - 11 t + 3 t
      2   t
     t
In[11]:=
Conway[Knot[10, 164]][z]
Out[11]=   
     2      4
1 + z  + 3 z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=   
{Knot[10, 10], Knot[10, 164]}
In[13]:=
{KnotDet[Knot[10, 164]], KnotSignature[Knot[10, 164]]}
Out[13]=   
{45, 0}
In[14]:=
Jones[Knot[10, 164]][q]
Out[14]=   
     -5   3    5    7    8            2      3
8 - q   + -- - -- + -- - - - 6 q + 5 q  - 2 q
           4    3    2   q
          q    q    q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=   
{Knot[10, 164]}
In[16]:=
A2Invariant[Knot[10, 164]][q]
Out[16]=   
  -16    -14    -12    2     -8    -6   2       2    4    6    8      10
-q    + q    + q    - --- + q   - q   + -- + 3 q  - q  + q  + q  - 2 q
                       10                2
                      q                 q
In[17]:=
HOMFLYPT[Knot[10, 164]][a, z]
Out[17]=   
                         2
     -2    2      2   2 z     4  2      4    2  4
3 - a   - a  + 4 z  - ---- - a  z  + 2 z  + a  z
                        2
                       a
In[18]:=
Kauffman[Knot[10, 164]][a, z]
Out[18]=   
                                                      2                3
     -2    2   2 z   5 z              3        2   6 z       4  2   3 z
3 + a   + a  - --- - --- - 5 a z - 2 a  z - 9 z  - ---- + 3 a  z  + ---- + 
                3     a                              2                3
               a                                    a                a
 
        3                                           4
    10 z          3      3  3      5  3      4   4 z       2  4      4  4
>   ----- + 16 a z  + 7 a  z  - 2 a  z  + 8 z  + ---- - 3 a  z  - 7 a  z  - 
      a                                            2
                                                  a
 
       5                                        6                        7
    6 z          5       3  5    5  5      6   z     2  6      4  6   3 z
>   ---- - 17 a z  - 10 a  z  + a  z  - 3 z  + -- - a  z  + 3 a  z  + ---- + 
     a                                          2                      a
                                               a
 
         7      3  7      8      2  8
>   7 a z  + 4 a  z  + 2 z  + 2 a  z
In[19]:=
{Vassiliev[2][Knot[10, 164]], Vassiliev[3][Knot[10, 164]]}
Out[19]=   
{1, 0}
In[20]:=
Kh[Knot[10, 164]][q, t]
Out[20]=   
4           1        2       1       3       2       4       3      4      4
- + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 
q          11  5    9  4    7  4    7  3    5  3    5  2    3  2    3     q t
          q   t    q  t    q  t    q  t    q  t    q  t    q  t    q  t
 
               3        3  2      5  2      7  3
>   3 q t + 3 q  t + 2 q  t  + 3 q  t  + 2 q  t
In[21]:=
ColouredJones[Knot[10, 164], 2][q]
Out[21]=   
      -15    3     -13    9    14     5    30   21   24   52   17   46   62
56 + q    - --- + q    + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- - 
             14           12    11    10    9    8    7    6    5    4    3
            q            q     q     q     q    q    q    q    q    q    q
 
    7    57             2       3       4       5       6      7      8    10
>   -- - -- + 4 q - 50 q  + 35 q  + 11 q  - 29 q  + 12 q  + 8 q  - 8 q  + q
     2   q
    q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10164
10.163
10163 		10.165
10165