# 8 9

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 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 9's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 9 at Knotilus!

### Knot presentations

 Planar diagram presentation X6271 X14,8,15,7 X10,3,11,4 X2,13,3,14 X12,5,13,6 X4,11,5,12 X16,10,1,9 X8,16,9,15 Gauss code 1, -4, 3, -6, 5, -1, 2, -8, 7, -3, 6, -5, 4, -2, 8, -7 Dowker-Thistlethwaite code 6 10 12 14 16 4 2 8 Conway Notation [3113]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 8, width is 3,

Braid index is 3

[{2, 10}, {1, 5}, {9, 4}, {10, 6}, {5, 3}, {4, 2}, {3, 7}, {6, 8}, {7, 9}, {8, 1}]
 Knot 8_9. A graph, knot 8_9.

### Three dimensional invariants

 Symmetry type Fully amphicheiral Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index ${\displaystyle \{3,6\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-5] Hyperbolic Volume 7.58818 A-Polynomial See Data:8 9/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial ${\displaystyle -t^{3}+3t^{2}-5t+7-5t^{-1}+3t^{-2}-t^{-3}}$ Conway polynomial ${\displaystyle -z^{6}-3z^{4}-2z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 25, 0 } Jones polynomial ${\displaystyle q^{4}-2q^{3}+3q^{2}-4q+5-4q^{-1}+3q^{-2}-2q^{-3}+q^{-4}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{6}+a^{2}z^{4}+z^{4}a^{-2}-5z^{4}+3a^{2}z^{2}+3z^{2}a^{-2}-8z^{2}+2a^{2}+2a^{-2}-3}$ Kauffman polynomial (db, data sources) ${\displaystyle az^{7}+z^{7}a^{-1}+2a^{2}z^{6}+2z^{6}a^{-2}+4z^{6}+2a^{3}z^{5}+2z^{5}a^{-3}+a^{4}z^{4}-4a^{2}z^{4}-4z^{4}a^{-2}+z^{4}a^{-4}-10z^{4}-4a^{3}z^{3}-az^{3}-z^{3}a^{-1}-4z^{3}a^{-3}-2a^{4}z^{2}+4a^{2}z^{2}+4z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}+a^{3}z+az+za^{-1}+za^{-3}-2a^{2}-2a^{-2}-3}$ The A2 invariant ${\displaystyle q^{12}+q^{8}-q^{4}+q^{2}-1+q^{-2}-q^{-4}+q^{-8}+q^{-12}}$ The G2 invariant ${\displaystyle q^{66}-q^{64}+2q^{62}-3q^{60}+q^{58}-3q^{54}+6q^{52}-6q^{50}+7q^{48}-5q^{46}+6q^{42}-10q^{40}+12q^{38}-8q^{36}+4q^{34}+3q^{32}-6q^{30}+11q^{28}-6q^{26}+2q^{24}+4q^{22}-7q^{20}+5q^{18}-8q^{14}+11q^{12}-10q^{10}+9q^{8}-3q^{6}-10q^{4}+14q^{2}-17+14q^{-2}-10q^{-4}-3q^{-6}+9q^{-8}-10q^{-10}+11q^{-12}-8q^{-14}+5q^{-18}-7q^{-20}+4q^{-22}+2q^{-24}-6q^{-26}+11q^{-28}-6q^{-30}+3q^{-32}+4q^{-34}-8q^{-36}+12q^{-38}-10q^{-40}+6q^{-42}-5q^{-46}+7q^{-48}-6q^{-50}+6q^{-52}-3q^{-54}+q^{-58}-3q^{-60}+2q^{-62}-q^{-64}+q^{-66}}$

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

Same Alexander/Conway Polynomial: {10_155, K11n37,}

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

### Vassiliev invariants

 V2 and V3: (-2, 0)
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 ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle {\frac {212}{3}}}$ ${\displaystyle {\frac {124}{3}}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {256}{3}}}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {1696}{3}}}$ ${\displaystyle -{\frac {992}{3}}}$ ${\displaystyle -{\frac {8071}{15}}}$ ${\displaystyle {\frac {1668}{5}}}$ ${\displaystyle -{\frac {37804}{45}}}$ ${\displaystyle {\frac {1063}{9}}}$ ${\displaystyle -{\frac {3271}{15}}}$

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=}$0 is the signature of 8 9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-101234χ
9        11
7       1 -1
5      21 1
3     21  -1
1    32   1
-1   23    1
-3  12     -1
-5 12      1
-7 1       -1
-91        1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$