# 10 136

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 136's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 136 at Knotilus!

### Knot presentations

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

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 10, width is 5,

Braid index is 4

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

[edit Notes on presentations of 10 136] The knot 10_136 is the only knot in the Rolfsen Knot Table whose braid index is smaller than the width of its minimum braid.

The next such knot is K11n8.

### 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 [-3][-6] Hyperbolic Volume 7.74627 A-Polynomial See Data:10 136/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 2}$ Rasmussen s-Invariant 0

### Polynomial invariants

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

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

Same Alexander/Conway Polynomial: {8_21,}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11n92,}

### Vassiliev invariants

 V2 and V3: (0, 1)
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 ${\displaystyle 0}$ ${\displaystyle 8}$ ${\displaystyle 0}$ ${\displaystyle 16}$ ${\displaystyle 8}$ ${\displaystyle 0}$ ${\displaystyle {\frac {80}{3}}}$ ${\displaystyle -{\frac {64}{3}}}$ ${\displaystyle 40}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 88}$ ${\displaystyle -88}$ ${\displaystyle {\frac {328}{3}}}$ ${\displaystyle {\frac {88}{3}}}$ ${\displaystyle 24}$

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=}$2 is the signature of 10 136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-10123χ
9       1-1
7      1 1
5     11 0
3    21  1
1   12   1
-1  121   0
-3 11     0
-5 1      -1
-71       1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$