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   The Alternating Knot 31   

Also known as "The Trefoil Knot", after plants of the genus Trifolium, which have compound trifoliate leaves, and as the "Overhand Knot".

Visit 31's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 31's page at Knotilus!

Acknowledgement

3.1
KnotPlot

Further views:   Mike Hutchings' Rope Trick
Mike Hutchings' Rope Trick 		Thurston's Trefoil - Figure Eight Trick
Thurston's Trick 		A Kenyan Stone
A Kenyan Stone 		A Knotted Box
A Knotted Box
A trefoil near the Hollander York Gallery
A trefoil near the Hollander York Gallery 		A Knotted Pencil
A Knotted Pencil 		The Auryn of The NeverEnding Story
The Auryn of The NeverEnding Story is a connected sum of two trefoils. 		Caixa Geral
Logo of Caixa Geral de Depositos, Lisboa
Banco Do Brasil
Banco Do Brasil

PD Presentation: X1425 X3641 X5263

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

DT (Dowker-Thistlethwaite) Code: 4 6 2

Minimum Braid Representative:


Length is 3, width is 2
Braid index is 2

A Morse Link Presentation:

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

Alexander Polynomial: t-1 - 1 + t

Conway Polynomial: 1 + z2

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

Determinant and Signature: {3, -2}

Jones Polynomial: - q-4 + q-3 + q-1

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

A2 (sl(3)) Invariant: - q-14 - q-12 + q-8 + 2q-6 + q-4 + q-2

HOMFLY-PT Polynomial: 2a2 + a2z2 - a4

Kauffman Polynomial: - 2a2 + a2z2 + a3z - a4 + a4z2 + a5z

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 31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -3r = -2r = -1r = 0
j = -1   1
j = -3   1
j = -5 1  
j = -7    
j = -91   

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-11 - q-10 - q-9 + q-8 - q-7 + q-5 + q-2
3 - q-21 + q-20 + q-19 - q-17 + q-15 - q-14 - q-13 + q-11 - q-10 + q-7 + q-3
4 q-34 - q-33 - q-32 + 2q-29 - q-28 + 2q-24 - q-23 - q-22 + q-19 - q-18 - q-17 + q-14 - q-13 + q-9 + q-4
5 - q-50 + q-49 + q-48 - q-45 - q-44 + q-42 - q-39 + q-36 + q-35 - q-33 + q-30 + q-29 - q-28 - q-27 + q-23 - q-22 - q-21 + q-17 - q-16 + q-11 + q-5
6 q-69 - q-68 - q-67 + q-64 + 2q-62 - q-61 - q-60 + 2q-55 - 2q-54 - q-53 + 2q-48 - q-47 - q-46 + q-43 + 2q-41 - q-40 - q-39 + q-36 + q-34 - q-33 - q-32 + q-27 - q-26 - q-25 + q-20 - q-19 + q-13 + q-6
7 - q-91 + q-90 + q-89 - q-86 - q-84 - q-83 + q-82 + q-81 + q-79 - q-78 - q-75 + q-74 + q-73 + q-71 - q-70 - q-69 - q-67 + q-66 + q-63 - q-62 - q-61 - q-59 + q-58 + q-56 + q-55 - q-54 - q-53 + q-50 + q-48 + q-47 - q-46 - q-45 + q-42 + q-39 - q-38 - q-37 + q-31 - q-30 - q-29 + q-23 - q-22 + q-15 + 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[3, 1]]
Out[2]=   
PD[X[1, 4, 2, 5], X[3, 6, 4, 1], X[5, 2, 6, 3]]
In[3]:=
GaussCode[Knot[3, 1]]
Out[3]=   
GaussCode[-1, 3, -2, 1, -3, 2]
In[4]:=
DTCode[Knot[3, 1]]
Out[4]=   
DTCode[4, 6, 2]
In[5]:=
br = BR[Knot[3, 1]]
Out[5]=   
BR[2, {-1, -1, -1}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=   
{2, 3}
In[7]:=
BraidIndex[Knot[3, 1]]
Out[7]=   
2
In[8]:=
Show[DrawMorseLink[Knot[3, 1]]]
Out[8]=   
 -Graphics- 
In[9]:=
#[Knot[3, 1]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=   
{Reversible, 1, 1, 2, 3, 1}
In[10]:=
alex = Alexander[Knot[3, 1]][t]
Out[10]=   
     1
-1 + - + t
     t
In[11]:=
Conway[Knot[3, 1]][z]
Out[11]=   
     2
1 + z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=   
{Knot[3, 1]}
In[13]:=
{KnotDet[Knot[3, 1]], KnotSignature[Knot[3, 1]]}
Out[13]=   
{3, -2}
In[14]:=
Jones[Knot[3, 1]][q]
Out[14]=   
  -4    -3   1
-q   + q   + -
             q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=   
{Knot[3, 1]}
In[16]:=
A2Invariant[Knot[3, 1]][q]
Out[16]=   
  -14    -12    -8   2     -4    -2
-q    - q    + q   + -- + q   + q
                      6
                     q
In[17]:=
HOMFLYPT[Knot[3, 1]][a, z]
Out[17]=   
   2    4    2  2
2 a  - a  + a  z
In[18]:=
Kauffman[Knot[3, 1]][a, z]
Out[18]=   
    2    4    3      5      2  2    4  2
-2 a  - a  + a  z + a  z + a  z  + a  z
In[19]:=
{Vassiliev[2][Knot[3, 1]], Vassiliev[3][Knot[3, 1]]}
Out[19]=   
{1, -1}
In[20]:=
Kh[Knot[3, 1]][q, t]
Out[20]=   
 -3   1     1       1
q   + - + ----- + -----
      q    9  3    5  2
          q  t    q  t
In[21]:=
ColouredJones[Knot[3, 1], 2][q]
Out[21]=   
 -11    -10    -9    -8    -7    -5    -2
q    - q    - q   + q   - q   + q   + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 31
0.1
01 		4.1
41