3 1

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0 1.gif


4 1.gif



3 1.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 3 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 3 1 at Knotilus!

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
3D depiction

Knot presentations

Planar diagram presentation X1425 X3641 X5263
Gauss code -1, 3, -2, 1, -3, 2
Dowker-Thistlethwaite code 4 6 2
Conway Notation [3]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 3, width is 2,

Braid index is 2

3 1 ML.gif 3 1 AP.gif
[{5, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 1}]

[edit Notes on presentations of 3 1]

Knot 3_1.
A graph, knot 3_1.
A part of a knot and a part of a graph.

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index 3
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][1]
Hyperbolic Volume Not hyperbolic
A-Polynomial See Data:3 1/A-polynomial

[edit Notes for 3 1's three dimensional invariants] The rope length of the trefoil is known to be no more than 16.372, by numerical experiments, while the sharpest known lower bound (actually applicable to all non-trivial knots) is 15.66.

The Trefoil is Fibered.png
The trefoil is a fibered knot! A java applet demonstrating it, written by Robert Barrington Leigh at the University of Toronto, is here.

Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus \textrm{ConcordanceGenus}(\textrm{Knot}(3,1))
Rasmussen s-Invariant -2

[edit Notes for 3 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial t+ t^{-1} -1
Conway polynomial z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 3, -2 }
Jones polynomial - q^{-4} + q^{-3} + q^{-1}
HOMFLY-PT polynomial (db, data sources) -a^4+a^2 z^2+2 a^2
Kauffman polynomial (db, data sources) a^5 z+a^4 z^2-a^4+a^3 z+a^2 z^2-2 a^2
The A2 invariant -q^{14}-q^{12}+q^8+2 q^6+q^4+q^2
The G2 invariant q^{72}-q^{64}-q^{62}-q^{56}-2 q^{54}-q^{52}+q^{50}-q^{46}-2 q^{44}+2 q^{40}+q^{38}-q^{36}+2 q^{32}+2 q^{30}+q^{28}+2 q^{22}+2 q^{20}+q^{14}+q^{12}+q^{10}