# 9 35

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 35's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 35 at Knotilus! 9_35 is also known as the pretzel knot P(3,3,3).

 Three-fold symmetric decorative knot Another three-fold symmetric decorative form

### Knot presentations

 Planar diagram presentation X1829 X7,14,8,15 X5,16,6,17 X9,18,10,1 X15,6,16,7 X17,10,18,11 X13,2,14,3 X3,12,4,13 X11,4,12,5 Gauss code -1, 7, -8, 9, -3, 5, -2, 1, -4, 6, -9, 8, -7, 2, -5, 3, -6, 4 Dowker-Thistlethwaite code 8 12 16 14 18 4 2 6 10 Conway Notation [3,3,3]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 14, width is 5,

Braid index is 5

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 3 3-genus 1 Bridge index 3 Super bridge index ${\displaystyle \{4,6\}}$ Nakanishi index 2 Maximal Thurston-Bennequin number [-12][1] Hyperbolic Volume 7.94058 A-Polynomial See Data:9 35/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 ${\displaystyle 7t-13+7t^{-1}}$ Conway polynomial ${\displaystyle 7z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{3,t+1\}}$ Determinant and Signature { 27, -2 } Jones polynomial ${\displaystyle q^{-1}-2q^{-2}+3q^{-3}-4q^{-4}+5q^{-5}-3q^{-6}+4q^{-7}-3q^{-8}+q^{-9}-q^{-10}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -a^{10}+z^{2}a^{8}-a^{8}+3z^{2}a^{6}+3a^{6}+2z^{2}a^{4}+z^{2}a^{2}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{7}a^{11}-6z^{5}a^{11}+12z^{3}a^{11}-8za^{11}+z^{8}a^{10}-4z^{6}a^{10}+3z^{4}a^{10}+z^{2}a^{10}+a^{10}+4z^{7}a^{9}-18z^{5}a^{9}+23z^{3}a^{9}-9za^{9}+z^{8}a^{8}+z^{6}a^{8}-15z^{4}a^{8}+16z^{2}a^{8}-a^{8}+3z^{7}a^{7}-8z^{5}a^{7}+3z^{3}a^{7}-za^{7}+5z^{6}a^{6}-15z^{4}a^{6}+12z^{2}a^{6}-3a^{6}+4z^{5}a^{5}-6z^{3}a^{5}+3z^{4}a^{4}-2z^{2}a^{4}+2z^{3}a^{3}+z^{2}a^{2}}$ The A2 invariant ${\displaystyle -q^{32}-q^{30}-2q^{26}-q^{24}+q^{22}+q^{20}+3q^{18}+2q^{16}+q^{14}-q^{10}+q^{8}-q^{4}+q^{2}}$ The G2 invariant ${\displaystyle q^{156}+3q^{152}-3q^{150}+2q^{148}-q^{146}-2q^{144}+7q^{142}-9q^{140}+6q^{138}-2q^{136}-2q^{134}+8q^{132}-12q^{130}+5q^{128}-2q^{126}-3q^{124}+3q^{122}-10q^{120}-2q^{118}+4q^{116}-2q^{114}+q^{112}-8q^{110}-2q^{108}+6q^{106}-6q^{104}+5q^{102}-11q^{100}+6q^{98}+8q^{96}-3q^{94}+8q^{92}-10q^{90}+12q^{88}+4q^{86}-5q^{84}+7q^{82}-5q^{80}+5q^{78}+7q^{76}-3q^{74}+2q^{72}+q^{70}-2q^{68}+4q^{66}-6q^{64}+3q^{62}-2q^{60}-2q^{58}+4q^{56}-4q^{54}+3q^{52}-2q^{50}+q^{48}-q^{46}-q^{44}+2q^{42}-3q^{40}+3q^{38}+q^{36}+q^{34}-q^{30}+2q^{28}-2q^{26}+2q^{24}-q^{22}-q^{16}+q^{14}-q^{12}+q^{10}}$

### "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: (7, -18)
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 ${\displaystyle 28}$ ${\displaystyle -144}$ ${\displaystyle 392}$ ${\displaystyle {\frac {3026}{3}}}$ ${\displaystyle {\frac {574}{3}}}$ ${\displaystyle -4032}$ ${\displaystyle -7584}$ ${\displaystyle -1344}$ ${\displaystyle -1296}$ ${\displaystyle {\frac {10976}{3}}}$ ${\displaystyle 10368}$ ${\displaystyle {\frac {84728}{3}}}$ ${\displaystyle {\frac {16072}{3}}}$ ${\displaystyle {\frac {1720297}{30}}}$ ${\displaystyle -{\frac {25154}{15}}}$ ${\displaystyle {\frac {1232834}{45}}}$ ${\displaystyle {\frac {8471}{18}}}$ ${\displaystyle {\frac {119977}{30}}}$

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 35. 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        21-1
-5       1  1
-7      32  -1
-9     21   1
-11    13    2
-13   32     1
-15   1      1
-17 13       -2
-19          0
-211         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$