# 8 5

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8_5's page at Knotilus! Visit 8 5's page at the original Knot Atlas! 8 5 is also known as the pretzel knot P(3,3,2).

### Knot presentations

 Planar diagram presentation X6271 X8493 X2837 X14,10,15,9 X12,5,13,6 X4,13,5,14 X16,12,1,11 X10,16,11,15 Gauss code 1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 7, -5, 6, -4, 8, -7 Dowker-Thistlethwaite code 6 8 12 2 14 16 4 10 Conway Notation [3,3,2]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 8, width is 3,

Braid index is 3

[{6, 11}, {1, 10}, {11, 9}, {10, 4}, {8, 3}, {9, 7}, {5, 8}, {4, 2}, {3, 6}, {2, 5}, {7, 1}]
 Knot 8_5. A graph, knot 8_5.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index {4,6} Nakanishi index 1 Maximal Thurston-Bennequin number [1][-11] Hyperbolic Volume 6.99719 A-Polynomial See Data:8 5/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 + 3t2−4t + 5−4t−1 + 3t−2−t−3 Conway polynomial −z6−3z4−z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 21, 4 } Jones polynomial q8−2q7 + 3q6−4q5 + 3q4−3q3 + 3q2−q + 1 HOMFLY-PT polynomial (db, data sources) −z6a−4 + z4a−2−5z4a−4 + z4a−6 + 4z2a−2−8z2a−4 + 3z2a−6 + 4a−2−5a−4 + 2a−6 Kauffman polynomial (db, data sources) z7a−3 + z7a−5 + z6a−2 + 4z6a−4 + 3z6a−6−3z5a−3 + z5a−5 + 4z5a−7−5z4a−2−15z4a−4−7z4a−6 + 3z4a−8−10z3a−5−8z3a−7 + 2z3a−9 + 8z2a−2 + 15z2a−4 + 4z2a−6−2z2a−8 + z2a−10 + 3za−3 + 7za−5 + 4za−7−4a−2−5a−4−2a−6 The A2 invariant 1 + q−2 + 2q−4 + 2q−6−3q−12−q−14−q−16 + q−20 + q−24 The G2 invariant q−2 + 3q−6−2q−8 + 2q−10 + q−12−2q−14 + 7q−16−5q−18 + 5q−20 + q−22−2q−24 + 8q−26−5q−28 + 5q−30 + 2q−32−2q−34 + 3q−36−3q−38 + 2q−42−4q−44 + 2q−46−4q−48−2q−50 + 2q−52−9q−54 + 4q−56−7q−58 + q−62−7q−64 + 7q−66−7q−68 + 4q−70 + 2q−72−5q−74 + 5q−76−2q−78 + q−80 + 4q−82−2q−84 + 3q−86 + q−88−q−90 + 4q−92−4q−94 + 3q−96−2q−100 + 3q−102−3q−104 + 3q−106−q−108 + q−110−3q−114 + 2q−116−2q−118 + 2q−120−q−122−q−128 + q−130−q−132 + q−134

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

Same Alexander/Conway Polynomial: {10_141,}

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

### Vassiliev invariants

 V2 and V3: (-1, -3)
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 −24 8 $-\frac{62}{3}$ $\frac{86}{3}$ 96 208 96 104 $-\frac{32}{3}$ 288 $\frac{248}{3}$ $-\frac{344}{3}$ $\frac{31409}{30}$ $\frac{834}{5}$ $\frac{16618}{45}$ $\frac{2095}{18}$ $-\frac{1231}{30}$

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 = 4 is the signature of 8 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-2-10123456χ
17        11
15       1 -1
13      21 1
11     21  -1
9    12   -1
7   22    0
5  11     0
3 13      2
1         0
-11        1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 3 i = 5 r = −2 ${\mathbb Z}$ r = −1 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}^{3}$ ${\mathbb Z}$ r = 1 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 2 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 3 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 4 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 5 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 6 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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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