# 10 132

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 132's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_132's page at Knotilus! Visit 10 132's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X4251 X8493 X5,12,6,13 X15,18,16,19 X9,16,10,17 X17,10,18,11 X13,20,14,1 X19,14,20,15 X11,6,12,7 X2837 Gauss code 1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7 Dowker-Thistlethwaite code 4 8 -12 2 -16 -6 -20 -18 -10 -14 Conway Notation [23,3,2-]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 11, width is 4,

Braid index is 4

[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 8}, {7, 9}, {8, 5}, {4, 6}, {5, 7}, {6, 13}, {12, 2}, {9, 1}]

### 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 [-8][-1] Hyperbolic Volume 4.05686 A-Polynomial See Data:10 132/A-polynomial

[edit Notes for 10 132's three dimensional invariants] 10 132 is a very interesting knot from the point of view of contact geometry. In particular, it is a transversely nonsimple knot, and it was the last knot with at most 10 crossings for which the maximal Thurston-Bennequin number was calculated.

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial t2−t + 1−t−1 + t−2 Conway polynomial z4 + 3z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 5, 0 } Jones polynomial q−2 + q−4−q−5 + q−6−q−7 HOMFLY-PT polynomial (db, data sources) −z2a6−2a6 + z4a4 + 4z2a4 + 3a4 Kauffman polynomial (db, data sources) a6z8 + a4z8 + a7z7 + 2a5z7 + a3z7−6a6z6−6a4z6−6a7z5−12a5z5−6a3z5 + 10a6z4 + 10a4z4 + 10a7z3 + 19a5z3 + 9a3z3−6a6z2−7a4z2−a2z2−5a7z−8a5z−4a3z−az + 2a6 + 3a4 The A2 invariant −q22−q20−q18 + q14 + q12 + 2q10 + q8 + q6 The G2 invariant q108 + q104−q100−q92−q90−q86−q84−q82−2q80−q78−q76−2q74−q68−q64 + q62 + q60 + q58 + q56 + q54 + 2q52 + 3q50 + q48 + q46 + 2q44 + q42 + 2q40 + q38 + q34 + q32−q28 + q26−q24−q18 + q16−q12 + q4−q2 + 1 + q−6−q−8

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

Same Alexander/Conway Polynomial: {5_1,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {5_1,}

### Vassiliev invariants

 V2 and V3: (3, -5)
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 12 −40 72 174 26 −480 $-\frac{2608}{3}$ $-\frac{352}{3}$ −168 288 800 2088 312 $\frac{42751}{10}$ $-\frac{3506}{15}$ $\frac{10034}{5}$ $\frac{203}{2}$ $\frac{2751}{10}$

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 10 132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-7-6-5-4-3-2-10χ
-1      110
-3       11
-5    12  1
-7   1    1
-9   11   0
-11 11     0
-13        0
-151       -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 i = −1 i = 1 r = −7 ${\mathbb Z}$ r = −6 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −5 ${\mathbb Z}$ r = −4 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}_2$ ${\mathbb Z}^{2}$ 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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