# 10 139

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 139's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_139's page at Knotilus! Visit 10 139's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X4251 X10,4,11,3 X11,19,12,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8 X13,1,14,20 X19,13,20,12 X2,10,3,9 Gauss code 1, -10, 2, -1, -4, 6, -5, 7, 10, -2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8 Dowker-Thistlethwaite code 4 10 -14 -16 2 -18 -20 -6 -8 -12 Conway Notation [4,3,3-]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 10, width is 3,

Braid index is 3

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 4 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [7][-16] Hyperbolic Volume 4.85117 A-Polynomial See Data:10 139/A-polynomial

### Four dimensional invariants

 Smooth 4 genus 4 Topological 4 genus [3,4] Concordance genus 4 Rasmussen s-Invariant -8

### Polynomial invariants

 Alexander polynomial t4−t3 + 2t−3 + 2t−1−t−3 + t−4 Conway polynomial z8 + 7z6 + 14z4 + 9z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 3, 6 } Jones polynomial −q12 + q11−q10 + q9−q8 + q6 + q4 HOMFLY-PT polynomial (db, data sources) z8a−8 + 8z6a−8−z6a−10 + 21z4a−8−7z4a−10 + 21z2a−8−13z2a−10 + z2a−12 + 6a−8−6a−10 + a−12 Kauffman polynomial (db, data sources) z8a−8 + z8a−10 + z7a−9 + z7a−11−8z6a−8−8z6a−10−7z5a−9−7z5a−11 + 21z4a−8 + 20z4a−10 + z4a−14 + 13z3a−9 + 13z3a−11 + z3a−13 + z3a−15−21z2a−8−19z2a−10−2z2a−14−6za−9−5za−11−za−13−2za−15 + 6a−8 + 6a−10 + a−12 The A2 invariant q−14 + q−16 + 2q−18 + 2q−20 + q−22−q−28−q−32−q−34−q−36−q−38 + q−40 The G2 invariant q−70 + q−72 + q−74 + q−76 + 2q−80 + 3q−82 + q−84 + q−86 + q−88 + 3q−90 + 3q−92 + 2q−94−2q−96 + 2q−98 + 3q−100 + q−102−3q−106 + q−108 + 2q−110−2q−112−3q−114−3q−116−q−118 + 4q−120−3q−122−3q−124−q−126−q−128 + 2q−130−3q−132−q−134 + 2q−140−q−152−q−156−q−158 + 2q−160−2q−162−q−164−3q−168 + q−170−2q−172 + q−176−q−178 + 2q−180 + 2q−186−q−188−q−190 + q−192 + q−196

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (9, 25)
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 36 200 648 1466 206 7200 $\frac{35888}{3}$ $\frac{6272}{3}$ 1384 7776 20000 52776 7416 $\frac{1001773}{10}$ $\frac{77402}{15}$ $\frac{523826}{15}$ $\frac{3059}{6}$ $\frac{43373}{10}$

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 = 6 is the signature of 10 139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
0123456789χ
25         1-1
23          0
21       11 0
19     11   0
17     11   0
15   111    -1
13    1     1
11  1       1
91         1
71         1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 5 i = 7 i = 9 r = 0 ${\mathbb Z}$ ${\mathbb Z}$ r = 1 r = 2 ${\mathbb Z}$ r = 3 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 4 ${\mathbb Z}$ ${\mathbb Z}$ r = 5 ${\mathbb Z}$ ${\mathbb Z}$ ${\mathbb Z}$ r = 6 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 7 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 8 ${\mathbb Z}$ r = 9 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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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