8 15

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 15 at Knotilus! Two trefoil knots along a closed loop, mutually interlinked. (See also 10 120.)
 Symmetrical depiction.

Knot presentations

 Planar diagram presentation X1425 X3849 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X7283 Gauss code -1, 8, -2, 1, -3, 7, -8, 2, -5, 6, -7, 3, -4, 5, -6, 4 Dowker-Thistlethwaite code 4 8 12 2 14 6 16 10 Conway Notation [21,21,2]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 9, width is 4,

Braid index is 4

[{11, 3}, {2, 9}, {7, 10}, {9, 11}, {8, 4}, {3, 7}, {4, 1}, {5, 8}, {6, 2}, {10, 5}, {1, 6}]
 Knot 8_15. A graph, knot 8_15.

Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index ${\displaystyle \{4,6\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-13][3] Hyperbolic Volume 9.93065 A-Polynomial See Data:8 15/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 ${\displaystyle 3t^{2}-8t+11-8t^{-1}+3t^{-2}}$ Conway polynomial ${\displaystyle 3z^{4}+4z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 33, -4 } Jones polynomial ${\displaystyle q^{-2}-2q^{-3}+5q^{-4}-5q^{-5}+6q^{-6}-6q^{-7}+4q^{-8}-3q^{-9}+q^{-10}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle a^{10}-3z^{2}a^{8}-4a^{8}+2z^{4}a^{6}+5z^{2}a^{6}+3a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-5z^{3}a^{11}+2za^{11}+3z^{6}a^{10}-3z^{4}a^{10}-a^{10}+z^{7}a^{9}+6z^{5}a^{9}-14z^{3}a^{9}+8za^{9}+6z^{6}a^{8}-10z^{4}a^{8}+8z^{2}a^{8}-4a^{8}+z^{7}a^{7}+5z^{5}a^{7}-11z^{3}a^{7}+6za^{7}+3z^{6}a^{6}-5z^{4}a^{6}+5z^{2}a^{6}-3a^{6}+2z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-2z^{2}a^{4}+a^{4}}$ The A2 invariant ${\displaystyle q^{32}+q^{30}-2q^{28}-q^{26}-2q^{24}-2q^{22}+q^{20}+3q^{16}+q^{14}+q^{12}+2q^{10}-q^{8}+q^{6}}$ The G2 invariant ${\displaystyle q^{162}-2q^{160}+4q^{158}-6q^{156}+3q^{154}-q^{152}-6q^{150}+14q^{148}-18q^{146}+20q^{144}-12q^{142}-q^{140}+17q^{138}-27q^{136}+34q^{134}-24q^{132}+7q^{130}+10q^{128}-21q^{126}+25q^{124}-16q^{122}+q^{120}+14q^{118}-20q^{116}+12q^{114}-24q^{110}+34q^{108}-36q^{106}+18q^{104}-2q^{102}-24q^{100}+40q^{98}-47q^{96}+34q^{94}-18q^{92}-7q^{90}+25q^{88}-33q^{86}+26q^{84}-9q^{82}-2q^{80}+16q^{78}-17q^{76}+9q^{74}+10q^{72}-21q^{70}+29q^{68}-21q^{66}+6q^{64}+17q^{62}-27q^{60}+33q^{58}-23q^{56}+12q^{54}+2q^{52}-14q^{50}+17q^{48}-14q^{46}+10q^{44}-2q^{42}-q^{40}+3q^{38}-3q^{36}+3q^{34}-q^{32}+q^{30}}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n65,}

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

Vassiliev invariants

 V2 and V3: (4, -7)
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 ${\displaystyle 16}$ ${\displaystyle -56}$ ${\displaystyle 128}$ ${\displaystyle {\frac {776}{3}}}$ ${\displaystyle {\frac {112}{3}}}$ ${\displaystyle -896}$ ${\displaystyle -{\frac {4016}{3}}}$ ${\displaystyle -{\frac {704}{3}}}$ ${\displaystyle -152}$ ${\displaystyle {\frac {2048}{3}}}$ ${\displaystyle 1568}$ ${\displaystyle {\frac {12416}{3}}}$ ${\displaystyle {\frac {1792}{3}}}$ ${\displaystyle {\frac {107222}{15}}}$ ${\displaystyle {\frac {6512}{15}}}$ ${\displaystyle {\frac {107648}{45}}}$ ${\displaystyle {\frac {250}{9}}}$ ${\displaystyle {\frac {4502}{15}}}$

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 ${\displaystyle s=}$-4 is the signature of 8 15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-8-7-6-5-4-3-2-10χ
-3        11
-5       21-1
-7      3  3
-9     22  0
-11    43   1
-13   22    0
-15  24     -2
-17 12      1
-19 2       -2
-211        1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$