# 9 2

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 2 at Knotilus!

### Knot presentations

 Planar diagram presentation X1425 X3,12,4,13 X5,18,6,1 X7,16,8,17 X9,14,10,15 X13,10,14,11 X15,8,16,9 X17,6,18,7 X11,2,12,3 Gauss code -1, 9, -2, 1, -3, 8, -4, 7, -5, 6, -9, 2, -6, 5, -7, 4, -8, 3 Dowker-Thistlethwaite code 4 12 18 16 14 2 10 8 6 Conway Notation [72]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 12, width is 5,

Braid index is 5

[{11, 8}, {7, 9}, {8, 6}, {5, 7}, {6, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 10}, {9, 11}, {10, 1}]
 Knot 9_2. A graph, knot 9_2.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index ${\displaystyle \{4,7\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-12][1] Hyperbolic Volume 3.48666 A-Polynomial See Data:9 2/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial ${\displaystyle 4t-7+4t^{-1}}$ Conway polynomial ${\displaystyle 4z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 15, -2 } Jones polynomial ${\displaystyle q^{-1}-q^{-2}+2q^{-3}-2q^{-4}+2q^{-5}-2q^{-6}+2q^{-7}-q^{-8}+q^{-9}-q^{-10}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -a^{10}+z^{2}a^{8}+a^{8}+z^{2}a^{6}+z^{2}a^{4}+z^{2}a^{2}+a^{2}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{7}a^{11}-6z^{5}a^{11}+10z^{3}a^{11}-4za^{11}+z^{8}a^{10}-6z^{6}a^{10}+11z^{4}a^{10}-7z^{2}a^{10}+a^{10}+2z^{7}a^{9}-10z^{5}a^{9}+13z^{3}a^{9}-4za^{9}+z^{8}a^{8}-5z^{6}a^{8}+8z^{4}a^{8}-6z^{2}a^{8}+a^{8}+z^{7}a^{7}-3z^{5}a^{7}+z^{3}a^{7}+z^{6}a^{6}-2z^{4}a^{6}+z^{5}a^{5}-z^{3}a^{5}+z^{4}a^{4}+z^{3}a^{3}+z^{2}a^{2}-a^{2}}$ The A2 invariant ${\displaystyle -q^{32}-q^{30}+q^{24}+q^{22}+q^{8}+q^{6}+q^{2}}$ The G2 invariant ${\displaystyle q^{156}+q^{152}-q^{150}+q^{142}-2q^{140}+q^{138}-q^{136}-q^{134}-2q^{130}-q^{128}-q^{126}-q^{124}-q^{118}+q^{112}-q^{108}+q^{106}+q^{104}+2q^{102}+q^{98}+q^{94}+q^{92}-2q^{90}+q^{88}+q^{86}+q^{76}-q^{72}+q^{66}-q^{62}-q^{52}+q^{48}+q^{38}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10}}$

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

Same Alexander/Conway Polynomial: {7_4,}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11n13,}

### Vassiliev invariants

 V2 and V3: (4, -10)
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 ${\displaystyle 16}$ ${\displaystyle -80}$ ${\displaystyle 128}$ ${\displaystyle {\frac {1304}{3}}}$ ${\displaystyle {\frac {184}{3}}}$ ${\displaystyle -1280}$ ${\displaystyle -{\frac {8000}{3}}}$ ${\displaystyle -{\frac {1280}{3}}}$ ${\displaystyle -400}$ ${\displaystyle {\frac {2048}{3}}}$ ${\displaystyle 3200}$ ${\displaystyle {\frac {20864}{3}}}$ ${\displaystyle {\frac {2944}{3}}}$ ${\displaystyle {\frac {249422}{15}}}$ ${\displaystyle -{\frac {856}{5}}}$ ${\displaystyle {\frac {315368}{45}}}$ ${\displaystyle {\frac {2482}{9}}}$ ${\displaystyle {\frac {13742}{15}}}$

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 ${\displaystyle s=}$-2 is the signature of 9 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-9-8-7-6-5-4-3-2-10χ
-1         11
-3        110
-5       1  1
-7      11  0
-9     11   0
-11    11    0
-13   11     0
-15   1      1
-17 11       0
-19          0
-211         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$