# 10 137

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 137's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 137 at Knotilus!

### Knot presentations

 Planar diagram presentation X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X20,18,1,17 X18,13,19,14 X12,19,13,20 Gauss code -1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8 Dowker-Thistlethwaite code 4 8 10 -14 2 -16 -18 -6 -20 -12 Conway Notation [22,211,2-]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 10, width is 5,

Braid index is 5

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

### 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 [-7][-3] Hyperbolic Volume 9.25056 A-Polynomial See Data:10 137/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 0}$ Topological 4 genus ${\displaystyle 0}$ Concordance genus ${\displaystyle 0}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{2}-6t+11-6t^{-1}+t^{-2}}$ Conway polynomial ${\displaystyle z^{4}-2z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 25, 0 } Jones polynomial ${\displaystyle q^{2}-2q+4-4q^{-1}+4q^{-2}-4q^{-3}+3q^{-4}-2q^{-5}+q^{-6}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle a^{6}-2z^{2}a^{4}-2a^{4}+z^{4}a^{2}+2z^{2}a^{2}+2a^{2}-2z^{2}-1+a^{-2}}$ Kauffman polynomial (db, data sources) ${\displaystyle a^{4}z^{8}+a^{2}z^{8}+2a^{5}z^{7}+4a^{3}z^{7}+2az^{7}+a^{6}z^{6}-a^{4}z^{6}-a^{2}z^{6}+z^{6}-8a^{5}z^{5}-15a^{3}z^{5}-7az^{5}-4a^{6}z^{4}-7a^{4}z^{4}-5a^{2}z^{4}-2z^{4}+8a^{5}z^{3}+15a^{3}z^{3}+9az^{3}+2z^{3}a^{-1}+4a^{6}z^{2}+8a^{4}z^{2}+7a^{2}z^{2}+z^{2}a^{-2}+4z^{2}-3a^{5}z-5a^{3}z-3az-za^{-1}-a^{6}-2a^{4}-2a^{2}-a^{-2}-1}$ The A2 invariant ${\displaystyle q^{20}+q^{18}-q^{16}-q^{12}-q^{10}+q^{8}+q^{4}+q^{-2}-q^{-4}+q^{-6}+q^{-8}}$ The G2 invariant ${\displaystyle q^{94}-q^{92}+3q^{90}-4q^{88}+3q^{86}-q^{84}-4q^{82}+10q^{80}-10q^{78}+10q^{76}-4q^{74}-4q^{72}+11q^{70}-12q^{68}+8q^{66}-q^{64}-6q^{62}+8q^{60}-6q^{58}-2q^{56}+9q^{54}-13q^{52}+10q^{50}-5q^{48}-6q^{46}+12q^{44}-15q^{42}+13q^{40}-9q^{38}+3q^{36}+5q^{34}-9q^{32}+11q^{30}-9q^{28}+5q^{26}+3q^{24}-6q^{22}+6q^{20}-2q^{18}-3q^{16}+10q^{14}-11q^{12}+7q^{10}+q^{8}-10q^{6}+14q^{4}-13q^{2}+7+q^{-2}-7q^{-4}+7q^{-6}-5q^{-8}+3q^{-10}+q^{-12}-2q^{-14}+q^{-16}-2q^{-20}+3q^{-22}+2q^{-28}-q^{-30}+q^{-32}+q^{-38}}$

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {10_155, K11n37,}

### Vassiliev invariants

 V2 and V3: (-2, 2)
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 ${\displaystyle -8}$ ${\displaystyle 16}$ ${\displaystyle 32}$ ${\displaystyle {\frac {20}{3}}}$ ${\displaystyle {\frac {28}{3}}}$ ${\displaystyle -128}$ ${\displaystyle -{\frac {416}{3}}}$ ${\displaystyle -{\frac {128}{3}}}$ ${\displaystyle -16}$ ${\displaystyle -{\frac {256}{3}}}$ ${\displaystyle 128}$ ${\displaystyle -{\frac {160}{3}}}$ ${\displaystyle -{\frac {224}{3}}}$ ${\displaystyle {\frac {4409}{15}}}$ ${\displaystyle {\frac {308}{5}}}$ ${\displaystyle {\frac {1556}{45}}}$ ${\displaystyle {\frac {151}{9}}}$ ${\displaystyle -{\frac {151}{15}}}$

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=}$0 is the signature of 10 137. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-1012χ
5        11
3       1 -1
1      31 2
-1     22  0
-3    22   0
-5   22    0
-7  12     -1
-9 12      1
-11 1       -1
-131        1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$