© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:
10.155
10155 		10.157
10157
    10.156
KnotPlot 		This page is passe. Go here instead!

   The Non Alternating Knot 10156   

Visit 10156's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10156's page at Knotilus!

Acknowledgement

10.156
KnotPlot

PD Presentation: X4251 X12,4,13,3 X7,14,8,15 X18,9,19,10 X6,19,7,20 X16,5,17,6 X10,17,11,18 X13,8,14,9 X20,15,1,16 X2,12,3,11

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

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

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 3 3 / NotAvailable 1

Alexander Polynomial: t-3 - 4t-2 + 8t-1 - 9 + 8t - 4t2 + t3

Conway Polynomial: 1 + z2 + 2z4 + z6

Other knots with the same Alexander/Conway Polynomial: {816, K11n15, K11n56, K11n58, ...}

Determinant and Signature: {35, -2}

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

Other knots (up to mirrors) with the same Jones Polynomial: {816, ...}

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

HOMFLY-PT Polynomial: - 2z2 - z4 + 2a2 + 5a2z2 + 4a2z4 + a2z6 - a4 - 2a4z2 - a4z4

Kauffman Polynomial: - 2a-1z3 + a-1z5 + 4z2 - 8z4 + 3z6 - az + 3az3 - 7az5 + 3az7 - 2a2 + 7a2z2 - 9a2z4 + 2a2z6 + a2z8 - 2a3z + 8a3z3 - 9a3z5 + 4a3z7 - a4 + a4z2 + 2a4z4 - a4z6 + a4z8 - 2a5z + 4a5z3 - a5z5 + a5z7 - 2a6z2 + 3a6z4 - a7z + a7z3

V2 and V3, the type 2 and 3 Vassiliev invariants: {1, -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=-2 is the signature of 10156. 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 = 5        1
j = 3       2 
j = 1      21 
j = -1     42  
j = -3    33   
j = -5   33    
j = -7  23     
j = -9 13      
j = -11 2       
j = -131        

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 - 2q-16 + 4q-15 + 3q-14 - 14q-13 + 9q-12 + 15q-11 - 30q-10 + 9q-9 + 29q-8 - 38q-7 + 4q-6 + 35q-5 - 34q-4 - 4q-3 + 33q-2 - 21q-1 - 10 + 23q - 7q2 - 10q3 + 9q4 - 3q6 + q7
3 q-34 - q-33 - q-32 - 4q-31 + 5q-30 + 11q-29 - 3q-28 - 23q-27 - 9q-26 + 41q-25 + 26q-24 - 54q-23 - 53q-22 + 59q-21 + 87q-20 - 59q-19 - 115q-18 + 51q-17 + 137q-16 - 38q-15 - 151q-14 + 24q-13 + 154q-12 - 7q-11 - 154q-10 - 6q-9 + 140q-8 + 27q-7 - 128q-6 - 39q-5 + 101q-4 + 61q-3 - 81q-2 - 63q-1 + 46 + 69q - 22q2 - 58q3 - q4 + 44q5 + 12q6 - 25q7 - 16q8 + 13q9 + 10q10 - 4q11 - 5q12 + 3q14 - q15
4 - q-56 + q-55 + 3q-54 - 2q-52 - 11q-51 - 7q-50 + 19q-49 + 25q-48 + 16q-47 - 44q-46 - 78q-45 + 8q-44 + 86q-43 + 135q-42 - 22q-41 - 228q-40 - 132q-39 + 83q-38 + 357q-37 + 163q-36 - 332q-35 - 378q-34 - 89q-33 + 533q-32 + 458q-31 - 289q-30 - 572q-29 - 348q-28 + 566q-27 + 689q-26 - 159q-25 - 624q-24 - 549q-23 + 498q-22 + 782q-21 - 38q-20 - 572q-19 - 646q-18 + 390q-17 + 763q-16 + 66q-15 - 453q-14 - 677q-13 + 238q-12 + 667q-11 + 180q-10 - 267q-9 - 646q-8 + 34q-7 + 477q-6 + 273q-5 - 21q-4 - 510q-3 - 148q-2 + 208q-1 + 253 + 183q - 268q2 - 189q3 - 26q4 + 108q5 + 222q6 - 50q7 - 89q8 - 96q9 - 25q10 + 121q11 + 27q12 + 5q13 - 47q14 - 46q15 + 31q16 + 11q17 + 17q18 - 6q19 - 17q20 + 4q21 + 5q23 - 3q25 + q26

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, 156]]
Out[2]=   
PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[7, 14, 8, 15], X[18, 9, 19, 10], 
 
>   X[6, 19, 7, 20], X[16, 5, 17, 6], X[10, 17, 11, 18], X[13, 8, 14, 9], 
 
>   X[20, 15, 1, 16], X[2, 12, 3, 11]]
In[3]:=
GaussCode[Knot[10, 156]]
Out[3]=   
GaussCode[1, -10, 2, -1, 6, -5, -3, 8, 4, -7, 10, -2, -8, 3, 9, -6, 7, -4, 5, 
 
>   -9]
In[4]:=
DTCode[Knot[10, 156]]
Out[4]=   
DTCode[4, 12, 16, -14, 18, 2, -8, 20, 10, 6]
In[5]:=
br = BR[Knot[10, 156]]
Out[5]=   
BR[4, {-1, -1, -1, 2, 1, 1, -3, -2, 1, -2, -3}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=   
{4, 11}
In[7]:=
BraidIndex[Knot[10, 156]]
Out[7]=   
4
In[8]:=
Show[DrawMorseLink[Knot[10, 156]]]
Out[8]=   
 -Graphics- 
In[9]:=
#[Knot[10, 156]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=   
{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=
alex = Alexander[Knot[10, 156]][t]
Out[10]=   
      -3   4    8            2    3
-9 + t   - -- + - + 8 t - 4 t  + t
            2   t
           t
In[11]:=
Conway[Knot[10, 156]][z]
Out[11]=   
     2      4    6
1 + z  + 2 z  + z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=   
{Knot[8, 16], Knot[10, 156], Knot[11, NonAlternating, 15], 
 
>   Knot[11, NonAlternating, 56], Knot[11, NonAlternating, 58]}
In[13]:=
{KnotDet[Knot[10, 156]], KnotSignature[Knot[10, 156]]}
Out[13]=   
{35, -2}
In[14]:=
Jones[Knot[10, 156]][q]
Out[14]=   
      -6   3    5    6    6    6          2
-4 - q   + -- - -- + -- - -- + - + 3 q - q
            5    4    3    2   q
           q    q    q    q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=   
{Knot[8, 16], Knot[10, 156]}
In[16]:=
A2Invariant[Knot[10, 156]][q]
Out[16]=   
     -18    -16    -14    -10    -8   2     -4   2     4    6
1 - q    + q    - q    + q    - q   + -- - q   + -- + q  - q
                                       6          2
                                      q          q
In[17]:=
HOMFLYPT[Knot[10, 156]][a, z]
Out[17]=   
   2    4      2      2  2      4  2    4      2  4    4  4    2  6
2 a  - a  - 2 z  + 5 a  z  - 2 a  z  - z  + 4 a  z  - a  z  + a  z
In[18]:=
Kauffman[Knot[10, 156]][a, z]
Out[18]=   
    2    4            3        5      7        2      2  2    4  2      6  2
-2 a  - a  - a z - 2 a  z - 2 a  z - a  z + 4 z  + 7 a  z  + a  z  - 2 a  z  - 
 
       3
    2 z         3      3  3      5  3    7  3      4      2  4      4  4
>   ---- + 3 a z  + 8 a  z  + 4 a  z  + a  z  - 8 z  - 9 a  z  + 2 a  z  + 
     a
 
               5
       6  4   z         5      3  5    5  5      6      2  6    4  6        7
>   3 a  z  + -- - 7 a z  - 9 a  z  - a  z  + 3 z  + 2 a  z  - a  z  + 3 a z  + 
              a
 
       3  7    5  7    2  8    4  8
>   4 a  z  + a  z  + a  z  + a  z
In[19]:=
{Vassiliev[2][Knot[10, 156]], Vassiliev[3][Knot[10, 156]]}
Out[19]=   
{1, -1}
In[20]:=
Kh[Knot[10, 156]][q, t]
Out[20]=   
3    4     1        2        1       3       2       3       3      3
-- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 
 3   q    13  5    11  4    9  4    9  3    7  3    7  2    5  2    5
q        q   t    q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
     3     2 t              2      3  2    5  3
>   ---- + --- + 2 q t + q t  + 2 q  t  + q  t
     3      q
    q  t
In[21]:=
ColouredJones[Knot[10, 156], 2][q]
Out[21]=   
       2     4     3    14     9    15    30    9    29   38   4    35   34
-10 - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + -- + -- - -- - 
       16    15    14    13    12    11    10    9    8    7    6    5    4
      q     q     q     q     q     q     q     q    q    q    q    q    q
 
    4    33   21             2       3      4      6    7
>   -- + -- - -- + 23 q - 7 q  - 10 q  + 9 q  - 3 q  + q
     3    2   q
    q    q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10156
10.155
10155 		10.157
10157