(KnotPlot image)

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

3_1 is also known as "The Trefoil Knot", after plants of the genus Trifolium, which have compound trifoliate leaves, and as the "Overhand Knot". See also T(3,2).

The trefoil is perhaps the easiest knot to find in "nature", and is topologically equivalent to the interlaced form of the common Christian and pagan "triquetra" symbol [12]:
Logo of Caixa Geral de Depositos, Lisboa [1]

A knot consists of two harts in Kolam [2]

A basic form of the interlaced Triquetra; as a Christian symbol, it refers to the Trinity


Further images...
Trefoil/triquetra without outside corners (made from straight lines and 240° circular arcs)

Triquetra made from circular arc ribbons




A trefoil near the Hollander York Gallery [4]

Trefoil of three intersecting circles

Trefoil depicted in nonthreefold form

3D depiction in nonthreefold form

A hagfish tying itself in a knot to escape capture. [5]

One version of the Germanic "Valknut" symbol



In the form of an architectural trefoil


Alternate Valknut depiction

Simple overhand knot of practical knottying

Tightly folded pentagonal overhand knot

Visually fancier square trefoil

Trefoil knot as impossible object

Logo of the Caixa Geral de Depósitos with white background

The NeverEnding Story logo is a connected sum of two trefoils. [7]

Mike Hutchings' Rope Trick [8]

Thurston's Trefoil  Figure Eight Trick [9]



Nonprime (compound) versions
Two trefoils (singleclosedloop version of the "granny knot" of practical knottying).
Two trefoils (singleclosedloop version of the "square knot" of practical knottying)
Three trefoils (symmetrical).
Four trefoils (Celtic or pseudoCeltic decorative knot which fits in square)
Three trefoils along a closed loop which itself is knotted as a trefoil.
For configurations of two trefoils along a closed loop which are prime, see 8_15 and 10_120. For a configuration of three trefoils along a closed loop which is prime, see K13a248. For a prime link consisting of two joined trefoils, see L10a108.
Knot presentations
Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 3, width is 2,
Braid index is 2


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

[edit Notes on presentations of 3 1]


A part of a knot and a part of a graph.

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_{1425} X_{3641} X_{5263}

Out[5]=

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

Four dimensional invariants
Polynomial invariants
Further Quantum Invariants
Further quantum knot invariants for 3_1.
The braid index of 3_1 is only 2, so it's easy to calculate lots of quantum invariants.
A1 Invariants.
Weight

Invariant

1


2


3


4


5


6


8


A2 Invariants.
Weight

Invariant

0,1


0,2


1,0


1,1


2,0


3,0


A3 Invariants.
Weight

Invariant

0,0,1


0,1,0


1,0,0


1,0,1


A4 Invariants.
Weight

Invariant

0,0,0,1


0,1,0,0


1,0,0,0


A5 Invariants.
Weight

Invariant

0,0,0,0,1


1,0,0,0,0


A6 Invariants.
Weight

Invariant

0,0,0,0,0,1


1,0,0,0,0,0


B2 Invariants.
Weight

Invariant

0,1


1,0


B3 Invariants.
Weight

Invariant

1,0,0


B4 Invariants.
Weight

Invariant

1,0,0,0


B5 Invariants.
Weight

Invariant

1,0,0,0,0


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

0,1


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, ):
{}
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 2 is the signature of 3 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.



3  2  1  0  χ 
1     1  1 
3     1  1 
5   1    1 
7      0 
9  1     1 

The Coloured Jones Polynomials