# 8 2

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8_2's page at Knotilus! Visit 8 2's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9 Gauss code -1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6 Dowker-Thistlethwaite code 4 10 12 14 16 2 6 8 Conway Notation [512]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 8, width is 3,

Braid index is 3

[{10, 2}, {1, 8}, {9, 3}, {2, 4}, {8, 10}, {3, 5}, {4, 6}, {5, 7}, {6, 9}, {7, 1}]
 Knot 8_2. A graph, knot 8_2.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index {4,5} Nakanishi index 1 Maximal Thurston-Bennequin number [-11][1] Hyperbolic Volume 4.93524 A-Polynomial See Data:8 2/A-polynomial

### Four dimensional invariants

 Smooth 4 genus 2 Topological 4 genus 2 Concordance genus 3 Rasmussen s-Invariant -4

### Polynomial invariants

 Alexander polynomial −t3 + 3t2−3t + 3−3t−1 + 3t−2−t−3 Conway polynomial −z6−3z4 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 17, -4 } Jones polynomial 1−q−1 + 2q−2−2q−3 + 3q−4−3q−5 + 2q−6−2q−7 + q−8 HOMFLY-PT polynomial (db, data sources) z4a6 + 3z2a6 + a6−z6a4−5z4a4−7z2a4−3a4 + z4a2 + 4z2a2 + 3a2 Kauffman polynomial (db, data sources) z2a10 + 2z3a9−za9 + 2z4a8−z2a8 + 2z5a7−2z3a7−za7 + 2z6a6−5z4a6 + 3z2a6−a6 + z7a5−2z5a5−z3a5 + za5 + 3z6a4−12z4a4 + 12z2a4−3a4 + z7a3−4z5a3 + 3z3a3 + za3 + z6a2−5z4a2 + 7z2a2−3a2 The A2 invariant q24−q18−q16−q12 + q10 + q6 + q4 + q2 + 1 The G2 invariant q134−q132 + q130−q128−q126−q122 + 2q120−2q118 + q116 + q110 + q106−q104 + q102 + q96 + 2q92−q84 + q82−2q78 + q76−q74 + q70−4q68 + 2q66−3q64−3q58 + 3q56−2q54 + q52−q50−q48 + q46−2q44 + q42−q38 + q36 + q32 + 2q30−2q28 + 3q26−q24 + q22 + 3q20−3q18 + 4q16 + q12 + q10−q8 + 2q6 + q2

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

Same Alexander/Conway Polynomial: {K11n6,}

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

### Vassiliev invariants

 V2 and V3: (0, 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 0 8 0 0 24 0 $-\frac{304}{3}$ $-\frac{160}{3}$ −88 0 32 0 0 432 −104 472 80 32

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 = -4 is the signature of 8 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-1012χ
1        11
-1         0
-3      21 1
-5     11  0
-7    21   1
-9   11    0
-11  12     -1
-13 11      0
-15 1       -1
-171        1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −5 i = −3 r = −6 ${\mathbb Z}$ r = −5 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ 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.