# 9 2

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9_2's page at Knotilus! Visit 9 2's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X1425 X3,12,4,13 X5,18,6,1 X7,16,8,17 X9,14,10,15 X13,10,14,11 X15,8,16,9 X17,6,18,7 X11,2,12,3 Gauss code -1, 9, -2, 1, -3, 8, -4, 7, -5, 6, -9, 2, -6, 5, -7, 4, -8, 3 Dowker-Thistlethwaite code 4 12 18 16 14 2 10 8 6 Conway Notation [72]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 12, width is 5,

Braid index is 5

[{11, 8}, {7, 9}, {8, 6}, {5, 7}, {6, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 10}, {9, 11}, {10, 1}]
 Knot 9_2. A graph, knot 9_2.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index {4,7} Nakanishi index 1 Maximal Thurston-Bennequin number [-12][1] Hyperbolic Volume 3.48666 A-Polynomial See Data:9 2/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial 4t−7 + 4t−1 Conway polynomial 4z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 15, -2 } Jones polynomial q−1−q−2 + 2q−3−2q−4 + 2q−5−2q−6 + 2q−7−q−8 + q−9−q−10 HOMFLY-PT polynomial (db, data sources) −a10 + z2a8 + a8 + z2a6 + z2a4 + z2a2 + a2 Kauffman polynomial (db, data sources) z7a11−6z5a11 + 10z3a11−4za11 + z8a10−6z6a10 + 11z4a10−7z2a10 + a10 + 2z7a9−10z5a9 + 13z3a9−4za9 + z8a8−5z6a8 + 8z4a8−6z2a8 + a8 + z7a7−3z5a7 + z3a7 + z6a6−2z4a6 + z5a5−z3a5 + z4a4 + z3a3 + z2a2−a2 The A2 invariant −q32−q30 + q24 + q22 + q8 + q6 + q2 The G2 invariant q156 + q152−q150 + q142−2q140 + q138−q136−q134−2q130−q128−q126−q124−q118 + q112−q108 + q106 + q104 + 2q102 + q98 + q94 + q92−2q90 + q88 + q86 + q76−q72 + q66−q62−q52 + q48 + q38 + q34 + q28 + q24 + q20 + q14 + q10

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

Same Alexander/Conway Polynomial: {7_4,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n13,}

### Vassiliev invariants

 V2 and V3: (4, -10)
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 16 −80 128 $\frac{1304}{3}$ $\frac{184}{3}$ −1280 $-\frac{8000}{3}$ $-\frac{1280}{3}$ −400 $\frac{2048}{3}$ 3200 $\frac{20864}{3}$ $\frac{2944}{3}$ $\frac{249422}{15}$ $-\frac{856}{5}$ $\frac{315368}{45}$ $\frac{2482}{9}$ $\frac{13742}{15}$

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 9 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-9-8-7-6-5-4-3-2-10χ
-1         11
-3        110
-5       1  1
-7      11  0
-9     11   0
-11    11    0
-13   11     0
-15   1      1
-17 11       0
-19          0
-211         -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 i = −1 r = −9 ${\mathbb Z}$ r = −8 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −7 ${\mathbb Z}$ r = −6 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −5 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −1 ${\mathbb Z}_2$ ${\mathbb Z}$ 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.