© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:
5.1
51 		6.1
61
    5.2
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   The Alternating Knot 52   

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Acknowledgement

5.2
KnotPlot

PD Presentation: X1425 X3849 X5,10,6,1 X9,6,10,7 X7283

Gauss Code: {-1, 5, -2, 1, -3, 4, -5, 2, -4, 3}

DT (Dowker-Thistlethwaite) Code: 4 8 10 2 6

Minimum Braid Representative:


Length is 6, width is 3
Braid index is 3

A Morse Link Presentation:

3D Invariants:
Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index
Reversible 1 1 2 / 3--4 1

Alexander Polynomial: 2t-1 - 3 + 2t

Conway Polynomial: 1 + 2z2

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

Determinant and Signature: {7, -2}

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

Other knots (up to mirrors) with the same Jones Polynomial: {K11n57, ...}

A2 (sl(3)) Invariant: - q-20 - q-18 + q-12 + q-10 + q-8 + q-6 + q-2

HOMFLY-PT Polynomial: a2 + a2z2 + a4 + a4z2 - a6

Kauffman Polynomial: - a2 + a2z2 + a3z3 + a4 - a4z2 + a4z4 - 2a5z + 2a5z3 + a6 - 2a6z2 + a6z4 - 2a7z + a7z3

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

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 52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -5r = -4r = -3r = -2r = -1r = 0
j = -1     1
j = -3    11
j = -5   1  
j = -7   1  
j = -9 11   
j = -11      
j = -131     

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-17 - q-16 - q-15 + 2q-14 - q-13 - 2q-12 + 3q-11 - q-10 - 3q-9 + 4q-8 - q-7 - 2q-6 + 3q-5 - q-3 + q-2
3 - q-33 + q-32 + q-31 - 2q-29 + 2q-27 + q-26 - 3q-25 - q-24 + 3q-23 + 2q-22 - 3q-21 - 3q-20 + 4q-19 + 2q-18 - 4q-17 - 4q-16 + 5q-15 + 2q-14 - 3q-13 - 3q-12 + 4q-11 + 2q-10 - 2q-9 - 2q-8 + 2q-7 + q-6 - q-4 + q-3
4 q-54 - q-53 - q-52 + 3q-49 - q-48 - q-47 - q-46 - 2q-45 + 5q-44 - q-42 - 2q-41 - 4q-40 + 6q-39 + q-38 + q-37 - 3q-36 - 6q-35 + 7q-34 + 2q-33 + 2q-32 - 4q-31 - 8q-30 + 8q-29 + 2q-28 + 2q-27 - 5q-26 - 9q-25 + 9q-24 + 2q-23 + 2q-22 - 4q-21 - 8q-20 + 7q-19 + 2q-18 + 2q-17 - 3q-16 - 6q-15 + 5q-14 + q-13 + 2q-12 - q-11 - 3q-10 + 2q-9 + q-7 - q-5 + q-4
5 - q-80 + q-79 + q-78 - q-75 - 2q-74 + 2q-72 + q-71 + q-70 - 3q-68 - 2q-67 + q-66 + 2q-65 + 3q-64 + q-63 - 3q-62 - 4q-61 - q-60 + q-59 + 5q-58 + 3q-57 - 2q-56 - 5q-55 - 4q-54 + q-53 + 6q-52 + 5q-51 - 7q-49 - 6q-48 + q-47 + 7q-46 + 6q-45 + q-44 - 8q-43 - 8q-42 + 2q-41 + 8q-40 + 6q-39 + q-38 - 9q-37 - 9q-36 + 3q-35 + 7q-34 + 6q-33 + q-32 - 8q-31 - 9q-30 + 2q-29 + 6q-28 + 6q-27 + q-26 - 6q-25 - 7q-24 + q-23 + 4q-22 + 5q-21 + q-20 - 3q-19 - 5q-18 + q-17 + q-16 + 2q-15 + 2q-14 - q-13 - 2q-12 + q-11 + q-8 - q-6 + q-5
6 q-111 - q-110 - q-109 + q-106 + 3q-104 - q-103 - 2q-102 - q-101 - q-100 - q-98 + 6q-97 - q-95 - q-94 - 2q-93 - 2q-92 - 4q-91 + 8q-90 + 2q-89 + q-88 - q-86 - 5q-85 - 8q-84 + 8q-83 + 2q-82 + 4q-81 + 2q-80 + q-79 - 7q-78 - 12q-77 + 7q-76 + q-75 + 7q-74 + 5q-73 + 3q-72 - 9q-71 - 15q-70 + 6q-69 + 9q-67 + 7q-66 + 5q-65 - 11q-64 - 18q-63 + 6q-62 + 11q-60 + 8q-59 + 6q-58 - 12q-57 - 20q-56 + 8q-55 + 12q-53 + 8q-52 + 6q-51 - 13q-50 - 21q-49 + 8q-48 + 11q-46 + 8q-45 + 6q-44 - 12q-43 - 20q-42 + 6q-41 + 10q-39 + 8q-38 + 6q-37 - 10q-36 - 17q-35 + 4q-34 - q-33 + 8q-32 + 6q-31 + 6q-30 - 7q-29 - 12q-28 + 3q-27 - 2q-26 + 4q-25 + 4q-24 + 5q-23 - 3q-22 - 6q-21 + 2q-20 - 2q-19 + q-18 + q-17 + 3q-16 - q-15 - 2q-14 + 2q-13 - q-12 + q-9 - q-7 + q-6
7 - q-147 + q-146 + q-145 - q-142 - q-140 - 2q-139 + q-138 + 2q-137 + q-136 + 2q-135 - q-134 - 5q-131 - q-130 + q-129 + q-128 + 4q-127 + 2q-125 + 2q-124 - 6q-123 - 3q-122 - 2q-121 - 2q-120 + 5q-119 + q-118 + 5q-117 + 6q-116 - 5q-115 - 3q-114 - 5q-113 - 6q-112 + 2q-111 + q-110 + 7q-109 + 10q-108 - 2q-107 - q-106 - 6q-105 - 12q-104 - q-103 - q-102 + 8q-101 + 14q-100 + 2q-99 + 2q-98 - 7q-97 - 16q-96 - 4q-95 - 3q-94 + 7q-93 + 18q-92 + 6q-91 + 4q-90 - 8q-89 - 20q-88 - 6q-87 - 5q-86 + 7q-85 + 21q-84 + 8q-83 + 6q-82 - 10q-81 - 23q-80 - 7q-79 - 4q-78 + 7q-77 + 23q-76 + 9q-75 + 7q-74 - 11q-73 - 25q-72 - 6q-71 - 3q-70 + 7q-69 + 24q-68 + 9q-67 + 7q-66 - 12q-65 - 26q-64 - 5q-63 - 4q-62 + 7q-61 + 23q-60 + 10q-59 + 7q-58 - 11q-57 - 25q-56 - 6q-55 - 5q-54 + 7q-53 + 21q-52 + 10q-51 + 7q-50 - 9q-49 - 22q-48 - 7q-47 - 6q-46 + 5q-45 + 18q-44 + 9q-43 + 6q-42 - 5q-41 - 17q-40 - 5q-39 - 6q-38 + 2q-37 + 12q-36 + 6q-35 + 6q-34 - 2q-33 - 10q-32 - 2q-31 - 4q-30 - 2q-29 + 5q-28 + 4q-27 + 4q-26 - 5q-24 + 2q-23 - 2q-22 - 2q-21 + q-20 + q-19 + 2q-18 - 2q-16 + 2q-15 - q-13 + q-10 - q-8 + q-7

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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 52
5.1
51 		6.1
61