# 5 2

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 5 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 5 2's page at the original Knot Atlas! 5_2 is also known as the 3-twist knot.

 3D depiction Simple square depiction

### Knot presentations

 Planar diagram presentation X1425 X3849 X5,10,6,1 X9,6,10,7 X7283 Gauss code -1, 5, -2, 1, -3, 4, -5, 2, -4, 3 Dowker-Thistlethwaite code 4 8 10 2 6 Conway Notation [32]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 6, width is 3,

Braid index is 3

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index {3,4} Nakanishi index 1 Maximal Thurston-Bennequin number [-8][1] Hyperbolic Volume 2.82812 A-Polynomial See Data:5 2/A-polynomial

### Four dimensional invariants

 Smooth 4 genus 1 Topological 4 genus 1 Concordance genus ConcordanceGenus(Knot(5,2)) Rasmussen s-Invariant -2

### Polynomial invariants

 Alexander polynomial 2t + 2t−1−3 Conway polynomial 2z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 7, -2 } Jones polynomial −q−6 + q−5−q−4 + 2q−3−q−2 + q−1 HOMFLY-PT polynomial (db, data sources) −a6 + a4z2 + a4 + a2z2 + a2 Kauffman polynomial (db, data sources) a7z3−2a7z + a6z4−2a6z2 + a6 + 2a5z3−2a5z + a4z4−a4z2 + a4 + a3z3 + a2z2−a2 The A2 invariant −q20−q18 + q12 + q10 + q8 + q6 + q2 The G2 invariant q100 + q96−q94−q92 + q90−q88−q84−q82−q78−q76−q74−q72−q68−q66 + q64 + q60 + q56 + q54 + 2q50−q48 + 2q46 + q44 + q40 + q34 + 2q24 + q20 + q14 + q10

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n57,}

### Vassiliev invariants

 V2 and V3: (2, -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 8 −24 32 $\frac{268}{3}$ $\frac{44}{3}$ −192 −368 −64 −56 $\frac{256}{3}$ 288 $\frac{2144}{3}$ $\frac{352}{3}$ $\frac{22951}{15}$ $-\frac{28}{5}$ $\frac{29764}{45}$ $\frac{137}{9}$ $\frac{1351}{15}$

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 5 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-10χ
-1     11
-3    110
-5   1  1
-7   1  1
-9 11   0
-11      0
-131     -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 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 ${\mathbb Z}$ ${\mathbb Z}$

### 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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