# 9 23

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 23's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 23 at Knotilus!

 Symmetrical decorative knot With crossings on 3x3 grid Depiction with two axes of symmetry Mongolian ornament (two crossings are unnecessary) Mongolian ornament, sum of two 9.23 Logo of the ICMC-USP, Brazil Other depicture with central symmetry by Alain Esculier

### Knot presentations

 Planar diagram presentation X1425 X3,10,4,11 X5,12,6,13 X7,16,8,17 X13,18,14,1 X17,14,18,15 X15,6,16,7 X11,8,12,9 X9,2,10,3 Gauss code -1, 9, -2, 1, -3, 7, -4, 8, -9, 2, -8, 3, -5, 6, -7, 4, -6, 5 Dowker-Thistlethwaite code 4 10 12 16 2 8 18 6 14 Conway Notation [22122]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 11, width is 4,

Braid index is 4

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index ${\displaystyle \{4,7\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-14][3] Hyperbolic Volume 10.6113 A-Polynomial See Data:9 23/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial ${\displaystyle 4t^{2}-11t+15-11t^{-1}+4t^{-2}}$ Conway polynomial ${\displaystyle 4z^{4}+5z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 45, -4 } Jones polynomial ${\displaystyle q^{-2}-2q^{-3}+5q^{-4}-6q^{-5}+8q^{-6}-8q^{-7}+6q^{-8}-5q^{-9}+3q^{-10}-q^{-11}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{2}a^{10}+z^{4}a^{8}-2a^{8}+2z^{4}a^{6}+4z^{2}a^{6}+2a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-7z^{4}a^{12}+3z^{2}a^{12}+3z^{7}a^{11}-5z^{5}a^{11}+za^{11}+z^{8}a^{10}+4z^{6}a^{10}-10z^{4}a^{10}+3z^{2}a^{10}+5z^{7}a^{9}-6z^{5}a^{9}-2z^{3}a^{9}+4za^{9}+z^{8}a^{8}+4z^{6}a^{8}-8z^{4}a^{8}+6z^{2}a^{8}-2a^{8}+2z^{7}a^{7}+2z^{5}a^{7}-6z^{3}a^{7}+4za^{7}+3z^{6}a^{6}-4z^{4}a^{6}+4z^{2}a^{6}-2a^{6}+2z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-2z^{2}a^{4}+a^{4}}$ The A2 invariant ${\displaystyle -q^{34}+q^{32}+q^{30}-2q^{28}-2q^{24}-q^{22}+q^{20}+3q^{16}+q^{12}+2q^{10}-q^{8}+q^{6}}$ The G2 invariant ${\displaystyle q^{176}-2q^{174}+5q^{172}-8q^{170}+7q^{168}-3q^{166}-6q^{164}+20q^{162}-27q^{160}+30q^{158}-24q^{156}+q^{154}+23q^{152}-47q^{150}+55q^{148}-43q^{146}+18q^{144}+14q^{142}-38q^{140}+48q^{138}-38q^{136}+13q^{134}+12q^{132}-30q^{130}+30q^{128}-10q^{126}-16q^{124}+40q^{122}-43q^{120}+36q^{118}-13q^{116}-28q^{114}+56q^{112}-73q^{110}+66q^{108}-39q^{106}-6q^{104}+43q^{102}-64q^{100}+61q^{98}-44q^{96}+7q^{94}+20q^{92}-36q^{90}+30q^{88}-7q^{86}-16q^{84}+35q^{82}-29q^{80}+11q^{78}+13q^{76}-34q^{74}+47q^{72}-41q^{70}+28q^{68}-2q^{66}-19q^{64}+35q^{62}-35q^{60}+31q^{58}-17q^{56}+4q^{54}+7q^{52}-15q^{50}+16q^{48}-12q^{46}+9q^{44}-2q^{42}-q^{40}+3q^{38}-3q^{36}+3q^{34}-q^{32}+q^{30}}$

### "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: (5, -11)
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 ${\displaystyle 20}$ ${\displaystyle -88}$ ${\displaystyle 200}$ ${\displaystyle {\frac {1462}{3}}}$ ${\displaystyle {\frac {194}{3}}}$ ${\displaystyle -1760}$ ${\displaystyle -{\frac {9040}{3}}}$ ${\displaystyle -{\frac {1504}{3}}}$ ${\displaystyle -344}$ ${\displaystyle {\frac {4000}{3}}}$ ${\displaystyle 3872}$ ${\displaystyle {\frac {29240}{3}}}$ ${\displaystyle {\frac {3880}{3}}}$ ${\displaystyle {\frac {115087}{6}}}$ ${\displaystyle {\frac {2986}{3}}}$ ${\displaystyle {\frac {58214}{9}}}$ ${\displaystyle {\frac {2389}{18}}}$ ${\displaystyle {\frac {4687}{6}}}$

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 ${\displaystyle s=}$-4 is the signature of 9 23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-9-8-7-6-5-4-3-2-10χ
-3         11
-5        21-1
-7       3  3
-9      32  -1
-11     53   2
-13    33    0
-15   35     -2
-17  23      1
-19 13       -2
-21 2        2
-231         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$