# 10 124

 (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 at Knotilus! 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 ${\displaystyle Q_{30}}$. See [1].

10_124 is not ${\displaystyle k}$-colourable for any ${\displaystyle k}$. See The Determinant and the Signature.

 Torus knot T(5,3) form

### 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 ${\displaystyle 4}$ Topological 4 genus ${\displaystyle 4}$ Concordance genus ${\displaystyle 4}$ Rasmussen s-Invariant -8

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{4}-t^{3}+t-1+t^{-1}-t^{-3}+t^{-4}}$ Conway polynomial ${\displaystyle z^{8}+7z^{6}+14z^{4}+8z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 1, 8 } Jones polynomial ${\displaystyle -q^{10}+q^{6}+q^{4}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-7z^{4}a^{-10}+21z^{2}a^{-8}-14z^{2}a^{-10}+z^{2}a^{-12}+7a^{-8}-8a^{-10}+2a^{-12}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-8z^{6}a^{-10}-7z^{5}a^{-9}-7z^{5}a^{-11}+21z^{4}a^{-8}+21z^{4}a^{-10}+14z^{3}a^{-9}+14z^{3}a^{-11}-21z^{2}a^{-8}-22z^{2}a^{-10}-z^{2}a^{-12}-8za^{-9}-8za^{-11}+7a^{-8}+8a^{-10}+2a^{-12}}$ The A2 invariant ${\displaystyle q^{-14}+q^{-16}+2q^{-18}+2q^{-20}+2q^{-22}+q^{-24}-2q^{-28}-2q^{-30}-2q^{-32}-q^{-34}+q^{-40}}$ The G2 invariant ${\displaystyle q^{-70}+q^{-72}+q^{-74}+q^{-76}+q^{-78}+2q^{-80}+2q^{-82}+q^{-84}+2q^{-86}+2q^{-88}+2q^{-90}+2q^{-92}+2q^{-94}+q^{-96}+2q^{-98}+q^{-100}+q^{-104}+q^{-106}-q^{-112}-q^{-114}-q^{-118}-2q^{-120}-2q^{-122}-q^{-124}-q^{-126}-2q^{-128}-2q^{-130}-2q^{-132}-q^{-134}-q^{-136}-2q^{-138}-2q^{-140}-q^{-142}-q^{-146}-q^{-148}+q^{-160}+q^{-162}+q^{-168}+q^{-170}+q^{-180}}$

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

Same Alexander/Conway Polynomial: {}

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

### 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 ${\displaystyle 32}$ ${\displaystyle 160}$ ${\displaystyle 512}$ ${\displaystyle {\frac {3184}{3}}}$ ${\displaystyle {\frac {416}{3}}}$ ${\displaystyle 5120}$ ${\displaystyle {\frac {23680}{3}}}$ ${\displaystyle {\frac {4000}{3}}}$ ${\displaystyle 832}$ ${\displaystyle {\frac {16384}{3}}}$ ${\displaystyle 12800}$ ${\displaystyle {\frac {101888}{3}}}$ ${\displaystyle {\frac {13312}{3}}}$ ${\displaystyle {\frac {910684}{15}}}$ ${\displaystyle {\frac {59264}{15}}}$ ${\displaystyle {\frac {869536}{45}}}$ ${\displaystyle {\frac {2756}{9}}}$ ${\displaystyle {\frac {34684}{15}}}$

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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle 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
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=5}$ ${\displaystyle i=7}$ ${\displaystyle i=9}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$