# 3 1

## Contents

 (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'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 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

[{5, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 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 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 ConcordanceGenus(Knot(3,1)) Rasmussen s-Invariant -2

### Polynomial invariants

 Alexander polynomial t + t−1−1 Conway polynomial z2 + 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) −a4 + a2z2 + 2a2 Kauffman polynomial (db, data sources) a5z + a4z2−a4 + a3z + a2z2−2a2 The A2 invariant −q14−q12 + q8 + 2q6 + q4 + q2 The G2 invariant q72−q64−q62−q56−2q54−q52 + q50−q46−2q44 + 2q40 + q38−q36 + 2q32 + 2q30 + q28 + 2q22 + 2q20 + q14 + q12 + q10

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

### Vassiliev invariants

 V2 and V3: (1, -1)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 4 −8 8 $\frac{62}{3}$ $\frac{10}{3}$ −32 $-\frac{176}{3}$ $-\frac{32}{3}$ −8 $\frac{32}{3}$ 32 $\frac{248}{3}$ $\frac{40}{3}$ $\frac{5071}{30}$ $\frac{58}{15}$ $\frac{3062}{45}$ $\frac{17}{18}$ $\frac{271}{30}$

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <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 V2 and V3.

### Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). 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 3 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-3-2-10χ
-1   11
-3   11
-5 1  1
-7    0
-91   -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 i = −1 r = −3 ${\mathbb Z}$ r = −2 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −1 r = 0 ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session, or any of the Computer Talk sections above.

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.