# 8 20

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8_20's page at Knotilus! Visit 8 20's page at the original Knot Atlas! 8_20 is also known as the pretzel knot P(3,-3,2). Its complement contains no complete totally geodesic immersed surfaces.[citation needed]

### Knot presentations

 Planar diagram presentation X4251 X8493 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X2837 Gauss code 1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4 Dowker-Thistlethwaite code 4 8 -12 2 -14 -6 -16 -10 Conway Notation [3,21,2-]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 8, width is 3,

Braid index is 3

[{3, 8}, {2, 4}, {1, 3}, {11, 9}, {8, 10}, {9, 5}, {4, 6}, {5, 7}, {6, 11}, {10, 2}, {7, 1}]
 Knot 8_20. A graph, knot 8_20.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-2] Hyperbolic Volume 4.1249 A-Polynomial See Data:8 20/A-polynomial

[edit Notes for 8 20's three dimensional invariants]
8_20 ribbon diagram from A. Kawauchi's text.

 Ribbon diagram for 8_20

### Four dimensional invariants

 Smooth 4 genus 0 Topological 4 genus 0 Concordance genus 0 Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial t2−2t + 3−2t−1 + t−2 Conway polynomial z4 + 2z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 9, 0 } Jones polynomial −q + 2−q−1 + 2q−2−q−3 + q−4−q−5 HOMFLY-PT polynomial (db, data sources) −z2a4−2a4 + z4a2 + 4z2a2 + 4a2−z2−1 Kauffman polynomial (db, data sources) a4z6 + a2z6 + a5z5 + 2a3z5 + az5−4a4z4−4a2z4−4a5z3−7a3z3−3az3 + 4a4z2 + 6a2z2 + 2z2 + 3a5z + 5a3z + 3az + za−1−2a4−4a2−1 The A2 invariant −q16−q14−q12 + 2q8 + 2q6 + 2q4 + q2−q−4 The G2 invariant q80 + q76−q74−2q68 + q66−q64−q62−q60−2q58−3q52−q50−q48 + q44−2q42 + q40 + q38 + 2q36 + q34 + 2q30 + 2q28 + 3q26 + 2q22 + 3q20 + q18 + q16 + q14 + 3q10−2q6 + q4 + 1−q−2−2q−4−q−12−q−14−q−20 + q−24

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

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

### Vassiliev invariants

 V2 and V3: (2, -2)
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 8 −16 32 $\frac{172}{3}$ $\frac{20}{3}$ −128 $-\frac{640}{3}$ $-\frac{64}{3}$ −48 $\frac{256}{3}$ 128 $\frac{1376}{3}$ $\frac{160}{3}$ $\frac{11911}{15}$ $-\frac{524}{15}$ $\frac{15484}{45}$ $\frac{233}{9}$ $\frac{631}{15}$

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 = 0 is the signature of 8 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-101χ
3      1-1
1     1 1
-1    12 1
-3   1   1
-5   1   1
-7 11    0
-9       0
-111      -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −1 i = 1 r = −5 ${\mathbb Z}$ r = −4 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}$ r = −2 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −1 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}^{2}$ ${\mathbb Z}$ r = 1 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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).

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