# 9 40

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 40's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 40 at Knotilus!
 In three-fold symmetrical form Symmetrical triangular form (less open) (alternate) Variant Obtained by an epitrochoid. Cylindrical depiction. 9.40 as a geodesic line of the oblate spheroid Photo of an alsatian chair, musée de l'oeuvre Notre Dame, Strasbourg, France.

### Knot presentations

 Planar diagram presentation X1627 X7,12,8,13 X5,15,6,14 X11,3,12,2 X15,10,16,11 X3,16,4,17 X9,4,10,5 X17,9,18,8 X13,18,14,1 Gauss code -1, 4, -6, 7, -3, 1, -2, 8, -7, 5, -4, 2, -9, 3, -5, 6, -8, 9 Dowker-Thistlethwaite code 6 16 14 12 4 2 18 10 8 Conway Notation [9*]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 9, width is 4,

Braid index is 4

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index 4 Nakanishi index 2 Maximal Thurston-Bennequin number [-9][-2] Hyperbolic Volume 15.0183 A-Polynomial See Data:9 40/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle [1,3]}$ Rasmussen s-Invariant -2

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{3}-7t^{2}+18t-23+18t^{-1}-7t^{-2}+t^{-3}}$ Conway polynomial ${\displaystyle z^{6}-z^{4}-z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \left\{t^{2}-3t+1\right\}}$ Determinant and Signature { 75, -2 } Jones polynomial ${\displaystyle -q^{2}+5q-8+11q^{-1}-13q^{-2}+13q^{-3}-11q^{-4}+8q^{-5}-4q^{-6}+q^{-7}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{2}a^{6}-2z^{4}a^{4}-2z^{2}a^{4}+a^{4}+z^{6}a^{2}+2z^{4}a^{2}-2a^{2}-z^{4}+2}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{4}a^{8}+4z^{5}a^{7}-2z^{3}a^{7}+8z^{6}a^{6}-9z^{4}a^{6}+4z^{2}a^{6}+9z^{7}a^{5}-12z^{5}a^{5}+6z^{3}a^{5}-za^{5}+4z^{8}a^{4}+7z^{6}a^{4}-20z^{4}a^{4}+7z^{2}a^{4}+a^{4}+17z^{7}a^{3}-32z^{5}a^{3}+14z^{3}a^{3}-za^{3}+4z^{8}a^{2}+4z^{6}a^{2}-17z^{4}a^{2}+3z^{2}a^{2}+2a^{2}+8z^{7}a-15z^{5}a+6z^{3}a+5z^{6}-7z^{4}+2+z^{5}a^{-1}}$ The A2 invariant ${\displaystyle q^{22}-q^{20}-2q^{18}+3q^{16}-q^{14}+2q^{12}+q^{10}-3q^{8}+q^{6}-4q^{4}+3q^{2}+1+3q^{-4}-q^{-6}}$ The G2 invariant ${\displaystyle q^{114}-3q^{112}+6q^{110}-10q^{108}+10q^{106}-8q^{104}+q^{102}+17q^{100}-35q^{98}+57q^{96}-69q^{94}+58q^{92}-26q^{90}-41q^{88}+121q^{86}-182q^{84}+197q^{82}-139q^{80}+14q^{78}+135q^{76}-248q^{74}+274q^{72}-196q^{70}+38q^{68}+122q^{66}-223q^{64}+212q^{62}-79q^{60}-87q^{58}+218q^{56}-237q^{54}+135q^{52}+47q^{50}-232q^{48}+337q^{46}-328q^{44}+209q^{42}-7q^{40}-197q^{38}+334q^{36}-361q^{34}+269q^{32}-104q^{30}-93q^{28}+225q^{26}-263q^{24}+194q^{22}-39q^{20}-123q^{18}+217q^{16}-194q^{14}+58q^{12}+116q^{10}-252q^{8}+282q^{6}-192q^{4}+31q^{2}+141-245q^{-2}+261q^{-4}-179q^{-6}+57q^{-8}+53q^{-10}-121q^{-12}+126q^{-14}-87q^{-16}+44q^{-18}-2q^{-20}-19q^{-22}+24q^{-24}-20q^{-26}+10q^{-28}-4q^{-30}+q^{-32}}$

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

Same Alexander/Conway Polynomial: {10_59, K11n66,}

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

### Vassiliev invariants

 V2 and V3: (-1, 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 ${\displaystyle -4}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle {\frac {34}{3}}}$ ${\displaystyle {\frac {38}{3}}}$ ${\displaystyle -32}$ ${\displaystyle -{\frac {208}{3}}}$ ${\displaystyle -{\frac {160}{3}}}$ ${\displaystyle 8}$ ${\displaystyle -{\frac {32}{3}}}$ ${\displaystyle 32}$ ${\displaystyle -{\frac {136}{3}}}$ ${\displaystyle -{\frac {152}{3}}}$ ${\displaystyle {\frac {2129}{30}}}$ ${\displaystyle {\frac {2102}{15}}}$ ${\displaystyle -{\frac {4742}{45}}}$ ${\displaystyle -{\frac {113}{18}}}$ ${\displaystyle -{\frac {751}{30}}}$

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 40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-10123χ
5         1-1
3        4 4
1       41 -3
-1      74  3
-3     75   -2
-5    66    0
-7   57     2
-9  36      -3
-11 15       4
-13 3        -3
-151         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$