# 10 153

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 153's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_153's page at Knotilus! Visit 10 153's page at the original Knot Atlas!

10_153 is not k-colourable for any k. See The Determinant and the Signature.

### Knot presentations

 Planar diagram presentation X4251 X8493 X12,6,13,5 X13,18,14,19 X9,16,10,17 X17,10,18,11 X15,20,16,1 X19,14,20,15 X6,12,7,11 X2837 Gauss code 1, -10, 2, -1, 3, -9, 10, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7 Dowker-Thistlethwaite code 4 8 12 2 -16 6 -18 -20 -10 -14 Conway Notation [(3,2)-(21,2)]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 11, width is 4,

Braid index is 4

[{3, 9}, {2, 4}, {1, 3}, {10, 5}, {9, 2}, {11, 6}, {5, 7}, {4, 10}, {6, 8}, {7, 11}, {8, 1}]
 Knot 10_153. A graph, knot 10_153. A part of a knot and a part of a graph.

### Three dimensional invariants

 Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-4] Hyperbolic Volume 7.37434 A-Polynomial See Data:10 153/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial t3−t2−t + 3−t−1−t−2 + t−3 Conway polynomial z6 + 5z4 + 4z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 1, 0 } Jones polynomial −q4 + q3−q2 + q + 1 + q−2−q−3 + q−4−q−5 HOMFLY-PT polynomial (db, data sources) z6−z4a−2 + 6z4−a4z2−a2z2−4z2a−2 + 10z2−a4−a2−3a−2 + 6 Kauffman polynomial (db, data sources) z8a−2 + z8 + az7 + 2z7a−1 + z7a−3 + a4z6−6z6a−2−7z6 + a5z5 + a3z5−7az5−13z5a−1−6z5a−3−4a4z4 + 10z4a−2 + 14z4−4a5z3−4a3z3 + 12az3 + 22z3a−1 + 10z3a−3 + 3a4z2−2a2z2−7z2a−2−12z2 + 3a5z + 2a3z−6az−10za−1−5za−3−a4 + a2 + 3a−2 + 6 The A2 invariant −q16−q12−q10 + 2q4 + 2q2 + 3 + 2q−2−q−8−q−10−q−12 The G2 invariant q80 + q76−q74 + q70−2q68 + q64−3q62 + q60−q58−4q56 + 5q54−5q52 + q50 + q48−5q46 + 5q44−3q42−2q40 + 2q38−4q36 + q34 + 3q32−5q30 + 3q28−q24 + 2q22−2q20 + 2q18 + 4q14−2q12 + 3q10 + 4q8−q6 + 5q4−q2 + 2 + 6q−2−q−4 + 2q−6 + 4q−8−2q−10 + 6q−12−q−14−3q−16 + 4q−18−4q−20 + 2q−22−3q−26 + q−28−q−30−3q−32−2q−36−q−38−3q−42−q−48−q−52 + q−56 + q−60

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (4, -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 16 −8 128 $\frac{488}{3}$ $\frac{64}{3}$ −128 $-\frac{560}{3}$ $-\frac{128}{3}$ −8 $\frac{2048}{3}$ 32 $\frac{7808}{3}$ $\frac{1024}{3}$ $\frac{41222}{15}$ $\frac{464}{5}$ $\frac{44528}{45}$ $\frac{442}{9}$ $\frac{1622}{15}$

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 10 153. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-1012345χ
9          1-1
7           0
5        11 0
3      11   0
1     1 1   2
-1    131    1
-3   1       1
-5   11      0
-7 11        0
-9           0
-111          -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 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 ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}_2$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ r = 1 ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ r = 2 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 3 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 4 ${\mathbb Z}$ r = 5 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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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