# 10 124

## Contents

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 124's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_124's page at Knotilus! Visit 10 124's page at the original Knot Atlas! 10_124 is also known as the torus knot T(5,3) or the pretzel knot P(5,3,-2). It is one of two knots which are both torus knots and pretzel knots, the other being 8_19 = T(4,3) = P(3,3,-2).

If one takes the symmetric diagram for 10_123 and makes it doubly alternating one gets a diagram for 10_124. That's the torus knot view. There is then a nice representation of the quandle of 10_124 into the dodecahedral quandle Q30. See [1].

10_124 is not k-colourable for any k. See The Determinant and the Signature.

### Knot presentations

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

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 10, width is 3,

Braid index is 3

[{4, 12}, {3, 5}, {1, 4}, {6, 11}, {5, 10}, {2, 6}, {12, 3}, {11, 9}, {10, 8}, {9, 7}, {8, 2}, {7, 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][-15] Hyperbolic Volume Not hyperbolic A-Polynomial See Data:10 124/A-polynomial

### Four dimensional invariants

 Smooth 4 genus 4 Topological 4 genus 4 Concordance genus 4 Rasmussen s-Invariant -8

### Polynomial invariants

 Alexander polynomial t4−t3 + t−1 + t−1−t−3 + t−4 Conway polynomial z8 + 7z6 + 14z4 + 8z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 1, 8 } Jones polynomial −q10 + q6 + q4 HOMFLY-PT polynomial (db, data sources) z8a−8 + 8z6a−8−z6a−10 + 21z4a−8−7z4a−10 + 21z2a−8−14z2a−10 + z2a−12 + 7a−8−8a−10 + 2a−12 Kauffman polynomial (db, data sources) z8a−8 + z8a−10 + z7a−9 + z7a−11−8z6a−8−8z6a−10−7z5a−9−7z5a−11 + 21z4a−8 + 21z4a−10 + 14z3a−9 + 14z3a−11−21z2a−8−22z2a−10−z2a−12−8za−9−8za−11 + 7a−8 + 8a−10 + 2a−12 The A2 invariant q−14 + q−16 + 2q−18 + 2q−20 + 2q−22 + q−24−2q−28−2q−30−2q−32−q−34 + q−40 The G2 invariant q−70 + q−72 + q−74 + q−76 + q−78 + 2q−80 + 2q−82 + q−84 + 2q−86 + 2q−88 + 2q−90 + 2q−92 + 2q−94 + q−96 + 2q−98 + q−100 + q−104 + q−106−q−112−q−114−q−118−2q−120−2q−122−q−124−q−126−2q−128−2q−130−2q−132−q−134−q−136−2q−138−2q−140−q−142−q−146−q−148 + q−160 + q−162 + q−168 + q−170 + q−180

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (8, 20)
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 32 160 512 $\frac{3184}{3}$ $\frac{416}{3}$ 5120 $\frac{23680}{3}$ $\frac{4000}{3}$ 832 $\frac{16384}{3}$ 12800 $\frac{101888}{3}$ $\frac{13312}{3}$ $\frac{910684}{15}$ $\frac{59264}{15}$ $\frac{869536}{45}$ $\frac{2756}{9}$ $\frac{34684}{15}$

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 = 8 is the signature of 10 124. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
01234567χ
21       1-1
19     1  -1
17     11 0
15   11   0
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 ${\mathbb Z}$ ${\mathbb Z}$ r = 6 ${\mathbb Z}$ r = 7 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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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