# 6 2

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 6 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 6_2's page at Knotilus! Visit 6 2's page at the original Knot Atlas! Dror likes to call 6_2 "The Miller Institute Knot", as it is the logo of the Miller Institute for Basic Research.
 The Miller Institute Mug [1] Simple square depiction 3D depiction

### Knot presentations

 Planar diagram presentation X1425 X5,10,6,11 X3948 X9,3,10,2 X7,12,8,1 X11,6,12,7 Gauss code -1, 4, -3, 1, -2, 6, -5, 3, -4, 2, -6, 5 Dowker-Thistlethwaite code 4 8 10 12 2 6 Conway Notation [312]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 6, width is 3,

Braid index is 3

[{8, 2}, {1, 6}, {7, 3}, {2, 4}, {6, 8}, {3, 5}, {4, 7}, {5, 1}]
 Knot 6_2. A graph, knot 6_2.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index {3,4} Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-1] Hyperbolic Volume 4.40083 A-Polynomial See Data:6 2/A-polynomial

### Four dimensional invariants

 Smooth 4 genus 1 Topological 4 genus 1 Concordance genus 2 Rasmussen s-Invariant -2

### Polynomial invariants

 Alexander polynomial −t2 + 3t−3 + 3t−1−t−2 Conway polynomial −z4−z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 11, -2 } Jones polynomial q−1 + 2q−1−2q−2 + 2q−3−2q−4 + q−5 HOMFLY-PT polynomial (db, data sources) z2a4 + a4−z4a2−3z2a2−2a2 + z2 + 2 Kauffman polynomial (db, data sources) z2a6 + 2z3a5−za5 + 2z4a4−2z2a4 + a4 + z5a3−za3 + 3z4a2−6z2a2 + 2a2 + z5a−2z3a + z4−3z2 + 2 The A2 invariant q16−q8−q4 + q2 + 1 + q−2 + q−4 The G2 invariant q86−q84 + q82−q80−q78−q74 + 3q72−2q70 + q68 + 2q62−q60 + q58 + q56 + q52 + 3q46−3q44 + q42−q40−q38 + 2q36−4q34 + q32−2q30−3q24 + q22−q20−q14 + q12 + q10 + 2q6−q4 + 2q2 + 1−q−2 + 3q−4−q−6 + 2q−8−q−12 + 2q−14 + q−18

### "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 $\frac{34}{3}$ $\frac{38}{3}$ −32 $-\frac{208}{3}$ $-\frac{64}{3}$ −24 $-\frac{32}{3}$ 32 $-\frac{136}{3}$ $-\frac{152}{3}$ $\frac{2129}{30}$ $\frac{662}{15}$ $-\frac{1862}{45}$ $\frac{463}{18}$ $-\frac{751}{30}$

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 6 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-1012χ
3      11
1       0
-1    21 1
-3   11  0
-5  11   0
-7 11    0
-9 1     -1
-111      1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 i = −1 r = −4 ${\mathbb Z}$ r = −3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −1 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 1 ${\mathbb Z}$ r = 2 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.