# 10 136

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 136's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_136's page at Knotilus! Visit 10 136's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13 Gauss code -1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7 Dowker-Thistlethwaite code 4 8 10 -14 2 -18 -6 -20 -12 -16 Conway Notation [22,22,2-]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 10, width is 5,

Braid index is 4

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

[edit Notes on presentations of 10 136] The knot 10_136 is the only knot in the Rolfsen Knot Table whose braid index is smaller than the width of its minimum braid.

The next such knot is K11n8.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-6] Hyperbolic Volume 7.74627 A-Polynomial See Data:10 136/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial −t2 + 4t−5 + 4t−1−t−2 Conway polynomial 1−z4 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 15, 2 } Jones polynomial −q4 + 2q3−2q2 + 3q−2 + 2q−1−2q−2 + q−3 HOMFLY-PT polynomial (db, data sources) −z4 + a2z2 + 2z2a−2−3z2 + a2 + 3a−2−a−4−2 Kauffman polynomial (db, data sources) z8a−2 + z8 + 2az7 + 3z7a−1 + z7a−3 + a2z6−4z6a−2−3z6−9az5−14z5a−1−5z5a−3−4a2z4 + 2z4a−2−2z4 + 9az3 + 16z3a−1 + 7z3a−3 + 3a2z2 + 4z2a−2 + z2a−4 + 6z2−2az−4za−1−2za−3−a2−3a−2−a−4−2 The A2 invariant q10−q2 + q−4 + 2q−6 + q−8 + q−10−q−12−q−14 The G2 invariant q46−q44 + 2q42−2q40 + q38−2q34 + 6q32−4q30 + 3q28−2q24 + 3q22−2q20−q18 + 3q16−3q14 + 3q10−6q8 + 6q6−7q4 + 3−5q−2 + 4q−4−3q−6 + 3q−8 + q−12−q−14 + 2q−18 + q−20 + q−24 + 3q−26 + 4q−30−6q−32 + 6q−34−2q−36 + 4q−40−7q−42 + 6q−44−2q−48 + q−50−2q−52−2q−54 + 3q−56−3q−58 + q−60 + q−62−3q−64 + 3q−66−3q−68 + q−70 + q−72−2q−74 + q−76

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

Same Alexander/Conway Polynomial: {8_21,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n92,}

### Vassiliev invariants

 V2 and V3: (0, 1)
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 0 8 0 16 8 0 $\frac{80}{3}$ $-\frac{64}{3}$ 40 0 32 0 0 88 −88 $\frac{328}{3}$ $\frac{88}{3}$ 24

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 10 136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-10123χ
9       1-1
7      1 1
5     11 0
3    21  1
1   12   1
-1  121   0
-3 11     0
-5 1      -1
-71       1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −1 i = 1 i = 3 r = −4 ${\mathbb Z}$ r = −3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −1 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 1 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 2 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 3 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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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