8 2

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 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 2 at Knotilus!

Knot presentations

 Planar diagram presentation X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X7,14,8,15 X9,16,10,1 X13,6,14,7 X15,8,16,9 Gauss code -1, 4, -3, 1, -2, 7, -5, 8, -6, 3, -4, 2, -7, 5, -8, 6 Dowker-Thistlethwaite code 4 10 12 14 16 2 6 8 Conway Notation [512]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 8, width is 3,

Braid index is 3

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

Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index ${\displaystyle \{4,5\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-11][1] Hyperbolic Volume 4.93524 A-Polynomial See Data:8 2/A-polynomial

Four dimensional invariants

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

Polynomial invariants

 Alexander polynomial ${\displaystyle -t^{3}+3t^{2}-3t+3-3t^{-1}+3t^{-2}-t^{-3}}$ Conway polynomial ${\displaystyle -z^{6}-3z^{4}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 17, -4 } Jones polynomial ${\displaystyle 1-q^{-1}+2q^{-2}-2q^{-3}+3q^{-4}-3q^{-5}+2q^{-6}-2q^{-7}+q^{-8}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{4}a^{6}+3z^{2}a^{6}+a^{6}-z^{6}a^{4}-5z^{4}a^{4}-7z^{2}a^{4}-3a^{4}+z^{4}a^{2}+4z^{2}a^{2}+3a^{2}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{2}a^{10}+2z^{3}a^{9}-za^{9}+2z^{4}a^{8}-z^{2}a^{8}+2z^{5}a^{7}-2z^{3}a^{7}-za^{7}+2z^{6}a^{6}-5z^{4}a^{6}+3z^{2}a^{6}-a^{6}+z^{7}a^{5}-2z^{5}a^{5}-z^{3}a^{5}+za^{5}+3z^{6}a^{4}-12z^{4}a^{4}+12z^{2}a^{4}-3a^{4}+z^{7}a^{3}-4z^{5}a^{3}+3z^{3}a^{3}+za^{3}+z^{6}a^{2}-5z^{4}a^{2}+7z^{2}a^{2}-3a^{2}}$ The A2 invariant ${\displaystyle q^{24}-q^{18}-q^{16}-q^{12}+q^{10}+q^{6}+q^{4}+q^{2}+1}$ The G2 invariant ${\displaystyle q^{134}-q^{132}+q^{130}-q^{128}-q^{126}-q^{122}+2q^{120}-2q^{118}+q^{116}+q^{110}+q^{106}-q^{104}+q^{102}+q^{96}+2q^{92}-q^{84}+q^{82}-2q^{78}+q^{76}-q^{74}+q^{70}-4q^{68}+2q^{66}-3q^{64}-3q^{58}+3q^{56}-2q^{54}+q^{52}-q^{50}-q^{48}+q^{46}-2q^{44}+q^{42}-q^{38}+q^{36}+q^{32}+2q^{30}-2q^{28}+3q^{26}-q^{24}+q^{22}+3q^{20}-3q^{18}+4q^{16}+q^{12}+q^{10}-q^{8}+2q^{6}+q^{2}}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n6,}

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

Vassiliev invariants

 V2 and V3: (0, 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 ${\displaystyle 0}$ ${\displaystyle 8}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 24}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {304}{3}}}$ ${\displaystyle -{\frac {160}{3}}}$ ${\displaystyle -88}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 432}$ ${\displaystyle -104}$ ${\displaystyle 472}$ ${\displaystyle 80}$ ${\displaystyle 32}$

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 8 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-1012χ
1        11
-1         0
-3      21 1
-5     11  0
-7    21   1
-9   11    0
-11  12     -1
-13 11      0
-15 1       -1
-171        1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$