(KnotPlot image)

See the full Rolfsen Knot Table.
Visit 4 1's page
at the Knot Server
(KnotPlot driven, includes 3D interactive images!)
Visit http://srankin.math.uwo.ca/cgibin/retrieve.cgi/1,4,3,1,2,3,4,2/goTop.html at Knotilus!
Visit 4 1's page at the original Knot Atlas!

4_1 is 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] .
For two 4_1 knots along a closed loop, see 10_59, 10_60, K12a975, and K12a991.


Alternate square depiction



A NeliKolam with 3x2 dot array [1]

In curved symmetrical form


Symmetrical from parametric equation


Nonprime (compound) versions
PseudoCeltic ornamental knot pattern, with three figure8 knots along a closed triangular loop.
Coat of arms surrounded by figure8 knots
Knot presentations
Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 4, width is 3,
Braid index is 3


[{3, 5}, {6, 4}, {5, 2}, {1, 3}, {2, 6}, {4, 1}]

[edit Notes on presentations of 4 1]
Computer Talk
The above data is available with the
Mathematica package
KnotTheory`
. Your input (in
red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`


KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X_{4251} X_{8615} X_{6374} X_{2738}

Out[5]=

1, 4, 3, 1, 2, 3, 4, 2

(The path below may be different on your system)
In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]


KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=


In[10]:=

{First[br], Crossings[br], BraidIndex[K]}


KnotTheory::loading: Loading precomputed data in IndianaData`.

In[11]:=

Show[BraidPlot[br]]

In[12]:=

Show[DrawMorseLink[K]]


KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."


KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{3, 5}, {6, 4}, {5, 2}, {1, 3}, {2, 6}, {4, 1}]

Four dimensional invariants
Polynomial invariants
Further Quantum Invariants
Further quantum knot invariants for 4_1.
A1 Invariants.
Weight

Invariant

1


2


3


4


5


6


A2 Invariants.
Weight

Invariant

1,0


1,1


2,0


3,0


A3 Invariants.
Weight

Invariant

0,1,0


1,0,0


1,0,1


A4 Invariants.
Weight

Invariant

0,1,0,0


1,0,0,0


B2 Invariants.
Weight

Invariant

0,1


1,0


B3 Invariants.
Weight

Invariant

1,0,0


B4 Invariants.
Weight

Invariant

1,0,0,0


C3 Invariants.
Weight

Invariant

1,0,0


C4 Invariants.
Weight

Invariant

1,0,0,0


D4 Invariants.
Weight

Invariant

0,1,0,0


1,0,0,0


G2 Invariants.
Weight

Invariant

1,0


.
</div></div>
Computer Talk
The above data is available with the
Mathematica package
KnotTheory`
, as shown in the (simulated) Mathematica session below. Your input (in
red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot
5_2) as the notebook
PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`


KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=


Out[5]=


In[6]:=

Alexander[K, 2][t]


KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=


In[7]:=

{KnotDet[K], KnotSignature[K]}


KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=


In[9]:=

HOMFLYPT[K][a, z]


KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=


In[10]:=

Kauffman[K][a, z]


KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=


"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial:
{}
Same Jones Polynomial (up to mirroring, ):
{K11n19,}
Computer Talk
The above data is available with the
Mathematica package
KnotTheory`
. Your input (in
red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}


KnotTheory::loading: Loading precomputed data in PD4Knots`.


KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ , }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]


KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.


KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

In[6]:=

DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q]  (J /. q > 1/q) === Jones[#][q]) &
],
K
]


KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

V_{2,1} through V_{6,9}:

V_{2,1}

V_{3,1}

V_{4,1}

V_{4,2}

V_{4,3}

V_{5,1}

V_{5,2}

V_{5,3}

V_{5,4}

V_{6,1}

V_{6,2}

V_{6,3}

V_{6,4}

V_{6,5}

V_{6,6}

V_{6,7}

V_{6,8}

V_{6,9}




















V_{2,1} through V_{6,9} were provided by Petr DuninBarkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V_{2} and V_{3}.
The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 0 is the signature of 4 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.



2  1  0  1  2  χ 
5      1  1 
3       0 
1    1  1   0 
1   1  1    0 
3       0 
5  1      1 

The Coloured Jones Polynomials