# 10 125

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## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 125's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_125's page at Knotilus! Visit 10 125's page at the original Knot Atlas! 10_125 is also known as the pretzel knot P(5,-3,2).

### Knot presentations

 Planar diagram presentation X1425 X3849 X5,14,6,15 X20,16,1,15 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,20,13,19 X13,6,14,7 X7283 Gauss code -1, 10, -2, 1, -3, 9, -10, 2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4 Dowker-Thistlethwaite code 4 8 14 2 -16 -18 6 -20 -10 -12 Conway Notation [5,21,2-]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 10, width is 3,

Braid index is 3

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-6] Hyperbolic Volume 4.61196 A-Polynomial See Data:10 125/A-polynomial

### Four dimensional invariants

 Smooth 4 genus 1 Topological 4 genus 1 Concordance genus 3 Rasmussen s-Invariant 2

### Polynomial invariants

 Alexander polynomial t3−2t2 + 2t−1 + 2t−1−2t−2 + t−3 Conway polynomial z6 + 4z4 + 3z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 11, 2 } Jones polynomial −q4 + q3−q2 + 2q−1 + 2q−1−q−2 + q−3−q−4 HOMFLY-PT polynomial (db, data sources) z6−a2z4−z4a−2 + 6z4−4a2z2−4z2a−2 + 11z2−3a2−3a−2 + 7 Kauffman polynomial (db, data sources) a2z8 + z8 + a3z7 + 2az7 + z7a−1−6a2z6−6z6−6a3z5−11az5−5z5a−1 + 11a2z4 + 2z4a−2 + 13z4 + 10a3z3 + 17az3 + 8z3a−1 + z3a−3−8a2z2−6z2a−2 + z2a−4−15z2−4a3z−8az−6za−1−za−3 + za−5 + 3a2 + 3a−2 + 7 The A2 invariant −q12−q10−q8 + q4 + 2q2 + 3 + 2q−2 + q−4−q−8−q−10−q−12 The G2 invariant q60 + q56−q54−q48−2q44−q40−2q38−q36−q34−2q32−2q28−q26 + q24−q22 + q20 + q16 + 2q14 + q12 + 2q10 + 2q8 + 2q6 + 2q4 + 3q2 + 1 + 2q−2 + 2q−4 + q−6 + 2q−8 + 2q−10 + q−14 + 2q−16−q−18 + q−20−q−24 + q−26−q−28−q−30−q−34−q−36−q−38−2q−40−q−44−q−46−q−50−q−56 + q−72

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (3, 0)
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 0 72 78 2 0 −32 32 −64 288 0 936 24 $\frac{9151}{10}$ $\frac{2374}{15}$ $\frac{474}{5}$ $\frac{75}{2}$ $-\frac{289}{10}$

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 = 2 is the signature of 10 125. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-10123χ
9        1-1
7         0
5      11 0
3     1   1
1    12   1
-1   1     1
-3   1     1
-5 11      0
-7         0
-91        -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 1 i = 3 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 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}^{2}$ ${\mathbb Z}$ r = 1 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 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.

###  Modifying This Page

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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