# 10 13

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 13's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_13's page at Knotilus! Visit 10 13's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,18,6,19 X13,1,14,20 X19,15,20,14 X7,16,8,17 X15,8,16,9 X17,6,18,7 Gauss code -1, 4, -3, 1, -5, 10, -8, 9, -2, 3, -4, 2, -6, 7, -9, 8, -10, 5, -7, 6 Dowker-Thistlethwaite code 4 10 18 16 12 2 20 8 6 14 Conway Notation [4222]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 11, width is 6,

Braid index is 6

[{12, 9}, {10, 8}, {9, 11}, {3, 10}, {7, 1}, {8, 6}, {5, 7}, {6, 4}, {2, 5}, {4, 12}, {1, 3}, {11, 2}]

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-5] Hyperbolic Volume 10.5785 A-Polynomial See Data:10 13/A-polynomial

### Four dimensional invariants

 Smooth 4 genus 1 Topological 4 genus 1 Concordance genus 2 Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial 2t2−13t + 23−13t−1 + 2t−2 Conway polynomial 2z4−5z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 53, 0 } Jones polynomial q4−2q3 + 5q2−7q + 8−9q−1 + 8q−2−6q−3 + 4q−4−2q−5 + q−6 HOMFLY-PT polynomial (db, data sources) a6−2z2a4−a4 + z4a2 + a2 + z4−z2−1−2z2a−2 + a−4 Kauffman polynomial (db, data sources) a3z9 + az9 + 2a4z8 + 4a2z8 + 2z8 + 2a5z7 + az7 + 3z7a−1 + a6z6−5a4z6−9a2z6 + 3z6a−2−7a5z5−4a3z5−2az5−3z5a−1 + 2z5a−3−4a6z4 + a4z4 + 6a2z4−3z4a−2 + z4a−4−3z4 + 6a5z3 + a3z3 + 3z3a−1−2z3a−3 + 4a6z2 + 2a4z2−a2z2 + z2a−2−2z2a−4 + 4z2−a5z + a3z−2za−1−a6−a4−a2 + a−4−1 The A2 invariant q20 + q18−q16 + q14−2q10 + 2q8−2 + q−2−2q−4 + q−6 + 2q−8−q−10 + q−12 + q−14 The G2 invariant Data:10 13/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (-5, 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 −20 16 200 $\frac{698}{3}$ $\frac{262}{3}$ −320 $-\frac{1376}{3}$ $-\frac{320}{3}$ −80 $-\frac{4000}{3}$ 128 $-\frac{13960}{3}$ $-\frac{5240}{3}$ $-\frac{21823}{6}$ $\frac{1582}{3}$ $-\frac{27422}{9}$ $\frac{7067}{18}$ $-\frac{4063}{6}$

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 10 13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-101234χ
9          11
7         1 -1
5        41 3
3       31  -2
1      54   1
-1     54    -1
-3    34     -1
-5   35      2
-7  13       -2
-9 13        2
-11 1         -1
-131          1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −1 i = 1 r = −6 ${\mathbb Z}$ r = −5 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = −2 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = −1 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = 0 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ r = 1 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = 2 ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = 3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 4 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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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