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   The Alternating Knot 41   

Also known as "The Figure Eight Knot", as some people think it looks like a figure `8' in one of its common projections. See e.g. 1, 2.

Visit 41's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 41's page at Knotilus!

Acknowledgement

4.1
KnotPlot

Further views:   Thurston's Trefoil - Figure Eight Trick
Thurston's Trick

PD Presentation: X4251 X8615 X6374 X2738

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

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

Minimum Braid Representative:


Length is 4, 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
FullyAmphicheiral 1 1 2 / 3 1

Alexander Polynomial: - t-1 + 3 - t

Conway Polynomial: 1 - z2

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

Determinant and Signature: {5, 0}

Jones Polynomial: q-2 - q-1 + 1 - q + q2

Other knots (up to mirrors) with the same Jones Polynomial: {K11n19, ...}

A2 (sl(3)) Invariant: q-8 + q-6 - 1 + q6 + q8

HOMFLY-PT Polynomial: a-2 - 1 - z2 + a2

Kauffman Polynomial: - a-2 + a-2z2 - a-1z + a-1z3 - 1 + 2z2 - az + az3 - a2 + a2z2

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 41. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -2r = -1r = 0r = 1r = 2
j = 5    1
j = 3     
j = 1  11 
j = -1 11  
j = -3     
j = -51    

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-6 - q-5 - q-4 + 2q-3 - q-2 - q-1 + 3 - q - q2 + 2q3 - q4 - q5 + q6
3 q-12 - q-11 - q-10 + 2q-8 - 2q-6 + 3q-4 - 3q-2 + 3 - 3q2 + 3q4 - 2q6 + 2q8 - q10 - q11 + q12
4 q-20 - q-19 - q-18 + 3q-15 - q-14 - q-13 - q-12 - q-11 + 5q-10 - q-9 - 2q-8 - 2q-7 - q-6 + 6q-5 - q-4 - 2q-3 - 2q-2 - q-1 + 7 - q - 2q2 - 2q3 - q4 + 6q5 - q6 - 2q7 - 2q8 - q9 + 5q10 - q11 - q12 - q13 - q14 + 3q15 - q18 - q19 + q20
5 q-30 - q-29 - q-28 + q-25 + 2q-24 - 2q-22 - q-21 - q-20 + q-19 + 3q-18 + q-17 - 2q-16 - 3q-15 - 2q-14 + 2q-13 + 4q-12 + 2q-11 - 2q-10 - 4q-9 - 2q-8 + 2q-7 + 5q-6 + 2q-5 - 2q-4 - 5q-3 - 2q-2 + 2q-1 + 5 + 2q - 2q2 - 5q3 - 2q4 + 2q5 + 5q6 + 2q7 - 2q8 - 4q9 - 2q10 + 2q11 + 4q12 + 2q13 - 2q14 - 3q15 - 2q16 + q17 + 3q18 + q19 - q20 - q21 - 2q22 + 2q24 + q25 - q28 - q29 + q30
6 q-42 - q-41 - q-40 + q-37 + 3q-35 - q-34 - 2q-33 - q-32 - q-31 + 6q-28 - q-27 - 2q-26 - 2q-25 - 2q-24 - q-23 + 9q-21 - 2q-19 - 3q-18 - 3q-17 - 2q-16 + 11q-14 - 2q-12 - 4q-11 - 4q-10 - 2q-9 + 12q-7 - 2q-5 - 4q-4 - 4q-3 - 2q-2 + 13 - 2q2 - 4q3 - 4q4 - 2q5 + 12q7 - 2q9 - 4q10 - 4q11 - 2q12 + 11q14 - 2q16 - 3q17 - 3q18 - 2q19 + 9q21 - q23 - 2q24 - 2q25 - 2q26 - q27 + 6q28 - q31 - q32 - 2q33 - q34 + 3q35 + q37 - q40 - q41 + q42
7 q-56 - q-55 - q-54 + q-51 + q-49 + 2q-48 - q-47 - 2q-46 - q-45 - 2q-44 + q-43 + q-41 + 5q-40 - 2q-38 - 2q-37 - 4q-36 + 2q-33 + 7q-32 + q-31 - q-30 - 2q-29 - 7q-28 - 2q-27 + 2q-25 + 9q-24 + 2q-23 - 3q-21 - 9q-20 - 3q-19 + 3q-17 + 10q-16 + 3q-15 - 3q-13 - 10q-12 - 3q-11 + 3q-9 + 11q-8 + 3q-7 - 3q-5 - 11q-4 - 3q-3 + 3q-1 + 11 + 3q - 3q3 - 11q4 - 3q5 + 3q7 + 11q8 + 3q9 - 3q11 - 10q12 - 3q13 + 3q15 + 10q16 + 3q17 - 3q19 - 9q20 - 3q21 + 2q23 + 9q24 + 2q25 - 2q27 - 7q28 - 2q29 - q30 + q31 + 7q32 + 2q33 - 4q36 - 2q37 - 2q38 + 5q40 + q41 + q43 - 2q44 - q45 - 2q46 - q47 + 2q48 + q49 + q51 - q54 - q55 + q56

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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 41
3.1
31 		5.1
51