8 15

8_14

8_16

Visit 8 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 15's page at Knotilus! Visit 8 15's page at the original Knot Atlas!

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]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-13][3] Hyperbolic Volume 9.93065 A-Polynomial See Data:8 15/A-polynomial

[edit Notes for 8 15's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $2$ Rasmussen s-Invariant -4

[edit Notes for 8 15's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 8_15.

A1 Invariants.

Weight	Invariant
1	$q^{21}-2q^{19}+q^{17}-2q^{15}+q^{11}+3q^{7}-q^{5}+q^{3}$
2	$q^{58}-2q^{56}-2q^{54}+6q^{52}-2q^{50}-6q^{48}+9q^{46}+q^{44}-9q^{42}+6q^{40}+3q^{38}-6q^{36}-q^{34}+2q^{32}-7q^{28}+2q^{26}+7q^{24}-8q^{22}+10q^{18}-5q^{16}-2q^{14}+6q^{12}-q^{8}+q^{6}$
3	$q^{111}-2q^{109}-2q^{107}+3q^{105}+6q^{103}-2q^{101}-13q^{99}+2q^{97}+18q^{95}+4q^{93}-25q^{91}-13q^{89}+27q^{87}+22q^{85}-27q^{83}-29q^{81}+20q^{79}+35q^{77}-12q^{75}-32q^{73}+7q^{71}+27q^{69}+4q^{67}-21q^{65}-9q^{63}+10q^{61}+17q^{59}-5q^{57}-21q^{55}-7q^{53}+27q^{51}+13q^{49}-30q^{47}-23q^{45}+27q^{43}+27q^{41}-22q^{39}-32q^{37}+12q^{35}+32q^{33}-6q^{31}-23q^{29}-2q^{27}+19q^{25}+7q^{23}-8q^{21}-4q^{19}+4q^{17}+3q^{15}-q^{11}+q^{9}$
4	$q^{180}-2q^{178}-2q^{176}+3q^{174}+3q^{172}+6q^{170}-9q^{168}-13q^{166}+2q^{164}+11q^{162}+30q^{160}-11q^{158}-41q^{156}-22q^{154}+14q^{152}+80q^{150}+22q^{148}-62q^{146}-83q^{144}-27q^{142}+123q^{140}+97q^{138}-31q^{136}-135q^{134}-108q^{132}+102q^{130}+151q^{128}+41q^{126}-124q^{124}-159q^{122}+38q^{120}+138q^{118}+88q^{116}-63q^{114}-137q^{112}-19q^{110}+80q^{108}+92q^{106}-2q^{104}-81q^{102}-57q^{100}+17q^{98}+75q^{96}+49q^{94}-23q^{92}-93q^{90}-46q^{88}+62q^{86}+102q^{84}+34q^{82}-121q^{80}-107q^{78}+32q^{76}+141q^{74}+101q^{72}-112q^{70}-148q^{68}-28q^{66}+124q^{64}+148q^{62}-49q^{60}-133q^{58}-86q^{56}+53q^{54}+136q^{52}+19q^{50}-62q^{48}-83q^{46}-12q^{44}+70q^{42}+38q^{40}-38q^{36}-24q^{34}+16q^{32}+14q^{30}+11q^{28}-5q^{26}-7q^{24}+2q^{22}+q^{20}+3q^{18}-q^{14}+q^{12}$
5	$q^{265}-2q^{263}-2q^{261}+3q^{259}+3q^{257}+3q^{255}-q^{253}-9q^{251}-13q^{249}+2q^{247}+20q^{245}+23q^{243}+6q^{241}-27q^{239}-49q^{237}-33q^{235}+37q^{233}+92q^{231}+71q^{229}-21q^{227}-131q^{225}-155q^{223}-29q^{221}+172q^{219}+255q^{217}+121q^{215}-157q^{213}-369q^{211}-275q^{209}+93q^{207}+447q^{205}+449q^{203}+42q^{201}-464q^{199}-612q^{197}-218q^{195}+403q^{193}+725q^{191}+409q^{189}-288q^{187}-747q^{185}-557q^{183}+129q^{181}+696q^{179}+645q^{177}+23q^{175}-589q^{173}-647q^{171}-147q^{169}+441q^{167}+591q^{165}+229q^{163}-299q^{161}-511q^{159}-260q^{157}+162q^{155}+395q^{153}+289q^{151}-49q^{149}-310q^{147}-287q^{145}-48q^{143}+215q^{141}+319q^{139}+152q^{137}-156q^{135}-340q^{133}-254q^{131}+77q^{129}+394q^{127}+373q^{125}-7q^{123}-426q^{121}-499q^{119}-98q^{117}+448q^{115}+620q^{113}+214q^{111}-413q^{109}-712q^{107}-360q^{105}+339q^{103}+747q^{101}+494q^{99}-202q^{97}-709q^{95}-595q^{93}+27q^{91}+607q^{89}+634q^{87}+132q^{85}-435q^{83}-592q^{81}-272q^{79}+239q^{77}+496q^{75}+313q^{73}-70q^{71}-337q^{69}-309q^{67}-55q^{65}+194q^{63}+240q^{61}+101q^{59}-72q^{57}-154q^{55}-101q^{53}+6q^{51}+77q^{49}+74q^{47}+18q^{45}-27q^{43}-36q^{41}-14q^{39}+4q^{37}+17q^{35}+12q^{33}-q^{31}-4q^{29}-q^{27}-q^{25}+q^{23}+3q^{21}-q^{17}+q^{15}$

A2 Invariants.

Weight	Invariant
1,0	$q^{32}+q^{30}-2q^{28}-q^{26}-2q^{24}-2q^{22}+q^{20}+3q^{16}+q^{14}+q^{12}+2q^{10}-q^{8}+q^{6}$
1,1	$q^{84}-4q^{82}+10q^{80}-20q^{78}+34q^{76}-54q^{74}+74q^{72}-92q^{70}+105q^{68}-104q^{66}+90q^{64}-60q^{62}+18q^{60}+36q^{58}-90q^{56}+142q^{54}-176q^{52}+200q^{50}-202q^{48}+182q^{46}-153q^{44}+96q^{42}-52q^{40}-10q^{38}+48q^{36}-86q^{34}+104q^{32}-104q^{30}+99q^{28}-72q^{26}+62q^{24}-34q^{22}+25q^{20}-10q^{18}+6q^{16}-2q^{14}+q^{12}$
2,0	$q^{80}+q^{78}-q^{76}-4q^{74}-3q^{72}+2q^{70}+q^{68}+3q^{64}+8q^{62}+4q^{60}-3q^{58}+2q^{54}-4q^{52}-6q^{50}-3q^{48}-3q^{46}-6q^{44}-2q^{42}+q^{40}-3q^{38}+6q^{34}+3q^{32}-3q^{30}+4q^{28}+6q^{26}+q^{24}-2q^{22}+3q^{20}+4q^{18}-q^{16}-q^{14}+q^{12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{68}-2q^{66}+3q^{62}-6q^{60}+q^{58}+6q^{56}-5q^{54}+4q^{52}+10q^{50}-3q^{48}-q^{46}-q^{44}-7q^{42}-9q^{40}-7q^{38}+q^{36}-q^{34}-2q^{32}+10q^{30}+5q^{28}-4q^{26}+9q^{24}+2q^{22}-3q^{20}+4q^{18}+q^{16}-q^{14}+q^{12}$
1,0,0	$q^{43}+q^{41}+q^{39}-2q^{37}-q^{35}-4q^{33}-2q^{31}-2q^{29}+q^{27}+q^{25}+2q^{23}+3q^{21}+q^{19}+2q^{17}+2q^{13}-q^{11}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{90}+q^{88}-q^{86}-3q^{84}-q^{82}-5q^{78}-4q^{76}+6q^{74}+8q^{72}+3q^{70}+9q^{68}+14q^{66}+5q^{64}-5q^{62}-4q^{60}-9q^{58}-20q^{56}-17q^{54}-7q^{52}-10q^{50}-6q^{48}+9q^{46}+9q^{44}+2q^{42}+7q^{40}+12q^{38}+4q^{36}-q^{34}+4q^{32}+6q^{30}-q^{28}+3q^{24}+q^{22}-q^{20}+q^{18}$
1,0,0,0	$q^{54}+q^{52}+q^{50}+q^{48}-2q^{46}-q^{44}-4q^{42}-4q^{40}-2q^{38}-2q^{36}+q^{34}+q^{32}+3q^{30}+2q^{28}+3q^{26}+q^{24}+2q^{22}+q^{20}+2q^{16}-q^{14}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$q^{68}-2q^{66}+4q^{64}-5q^{62}+6q^{60}-7q^{58}+6q^{56}-5q^{54}+2q^{52}-5q^{48}+7q^{46}-11q^{44}+11q^{42}-13q^{40}+11q^{38}-9q^{36}+7q^{34}-2q^{32}+5q^{28}-4q^{26}+7q^{24}-6q^{22}+7q^{20}-4q^{18}+3q^{16}-q^{14}+q^{12}$
1,0	$q^{110}-2q^{106}-2q^{104}+2q^{102}+4q^{100}-q^{98}-6q^{96}-3q^{94}+6q^{92}+6q^{90}-2q^{88}-6q^{86}+q^{84}+8q^{82}+5q^{80}-4q^{78}-3q^{76}+3q^{74}+3q^{72}-5q^{70}-7q^{68}-2q^{66}+2q^{64}-3q^{62}-7q^{60}-3q^{58}+3q^{56}+3q^{54}-4q^{52}-3q^{50}+5q^{48}+9q^{46}-5q^{42}-q^{40}+8q^{38}+6q^{36}-2q^{34}-5q^{32}+q^{30}+4q^{28}+2q^{26}-q^{24}-q^{22}+q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{94}-2q^{92}+2q^{90}-3q^{88}+4q^{86}-6q^{84}+4q^{82}-5q^{80}+6q^{78}-3q^{76}+4q^{74}+2q^{72}+5q^{70}+6q^{68}-4q^{66}+5q^{64}-10q^{62}+4q^{60}-17q^{58}-16q^{54}+4q^{52}-8q^{50}+5q^{48}-2q^{46}+6q^{44}+7q^{42}+2q^{40}+6q^{38}-2q^{36}+8q^{34}-3q^{32}+6q^{30}-4q^{28}+5q^{26}-q^{24}+2q^{22}-q^{20}+q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{162}-2q^{160}+4q^{158}-6q^{156}+3q^{154}-q^{152}-6q^{150}+14q^{148}-18q^{146}+20q^{144}-12q^{142}-q^{140}+17q^{138}-27q^{136}+34q^{134}-24q^{132}+7q^{130}+10q^{128}-21q^{126}+25q^{124}-16q^{122}+q^{120}+14q^{118}-20q^{116}+12q^{114}-24q^{110}+34q^{108}-36q^{106}+18q^{104}-2q^{102}-24q^{100}+40q^{98}-47q^{96}+34q^{94}-18q^{92}-7q^{90}+25q^{88}-33q^{86}+26q^{84}-9q^{82}-2q^{80}+16q^{78}-17q^{76}+9q^{74}+10q^{72}-21q^{70}+29q^{68}-21q^{66}+6q^{64}+17q^{62}-27q^{60}+33q^{58}-23q^{56}+12q^{54}+2q^{52}-14q^{50}+17q^{48}-14q^{46}+10q^{44}-2q^{42}-q^{40}+3q^{38}-3q^{36}+3q^{34}-q^{32}+q^{30}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 15"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$3t^{2}-8t+11-8t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}+4z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 33, -4 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{-2}-2q^{-3}+5q^{-4}-5q^{-5}+6q^{-6}-6q^{-7}+4q^{-8}-3q^{-9}+q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$a^{10}-3z^{2}a^{8}-4a^{8}+2z^{4}a^{6}+5z^{2}a^{6}+3a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-5z^{3}a^{11}+2za^{11}+3z^{6}a^{10}-3z^{4}a^{10}-a^{10}+z^{7}a^{9}+6z^{5}a^{9}-14z^{3}a^{9}+8za^{9}+6z^{6}a^{8}-10z^{4}a^{8}+8z^{2}a^{8}-4a^{8}+z^{7}a^{7}+5z^{5}a^{7}-11z^{3}a^{7}+6za^{7}+3z^{6}a^{6}-5z^{4}a^{6}+5z^{2}a^{6}-3a^{6}+2z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-2z^{2}a^{4}+a^{4}$

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 $16$ $-56$ $128$ ${\frac {776}{3}}$ ${\frac {112}{3}}$ $-896$ $-{\frac {4016}{3}}$ $-{\frac {704}{3}}$ $-152$ ${\frac {2048}{3}}$ $1568$ ${\frac {12416}{3}}$ ${\frac {1792}{3}}$ ${\frac {107222}{15}}$ ${\frac {6512}{15}}$ ${\frac {107648}{45}}$ ${\frac {250}{9}}$ ${\frac {4502}{15}}$

V_2,1 through V_6,9 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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s+1$, where $s=$-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	-1	0	χ
-3									1	1
-5								2	1	-1
-7							3			3
-9						2	2			0
-11					4	3				1
-13				2	2					0
-15			2	4						-2
-17		1	2							1
-19		2								-2
-21	1									1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[8, 15]]
Out[2]=	8
In[3]:=	PD[Knot[8, 15]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 12, 6, 13], X[13, 16, 14, 1], X[9, 14, 10, 15], X[15, 10, 16, 11], X[11, 6, 12, 7], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[8, 15]]
Out[4]=	GaussCode[-1, 8, -2, 1, -3, 7, -8, 2, -5, 6, -7, 3, -4, 5, -6, 4]
In[5]:=	BR[Knot[8, 15]]
Out[5]=	BR[4, {-1, -1, 2, -1, -3, -2, -2, -2, -3}]
In[6]:=	alex = Alexander[Knot[8, 15]][t]
Out[6]=	3 8 2 11 + -- - - - 8 t + 3 t 2 t t
In[7]:=	Conway[Knot[8, 15]][z]
Out[7]=	2 4 1 + 4 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 15], Knot[11, NonAlternating, 65]}
In[9]:=	{KnotDet[Knot[8, 15]], KnotSignature[Knot[8, 15]]}
Out[9]=	{33, -4}
In[10]:=	J=Jones[Knot[8, 15]][q]
Out[10]=	-10 3 4 6 6 5 5 2 -2 q - -- + -- - -- + -- - -- + -- - -- + q 9 8 7 6 5 4 3 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 15]}
In[12]:=	A2Invariant[Knot[8, 15]][q]
Out[12]=	-32 -30 2 -26 2 2 -20 3 -14 -12 2 q + q - --- - q - --- - --- + q + --- + q + q + --- - 28 24 22 16 10 q q q q q -8 -6 q + q
In[13]:=	Kauffman[Knot[8, 15]][a, z]
Out[13]=	4 6 8 10 7 9 11 4 2 a - 3 a - 4 a - a + 6 a z + 8 a z + 2 a z - 2 a z + 6 2 8 2 12 2 5 3 7 3 9 3 5 a z + 8 a z - a z - 2 a z - 11 a z - 14 a z - 11 3 4 4 6 4 8 4 10 4 12 4 5 5 5 a z + a z - 5 a z - 10 a z - 3 a z + a z + 2 a z + 7 5 9 5 11 5 6 6 8 6 10 6 7 7 5 a z + 6 a z + 3 a z + 3 a z + 6 a z + 3 a z + a z + 9 7 a z
In[14]:=	{Vassiliev[2][Knot[8, 15]], Vassiliev[3][Knot[8, 15]]}
Out[14]=	{0, -7}
In[15]:=	Kh[Knot[8, 15]][q, t]
Out[15]=	-5 -3 1 2 1 2 2 4 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 21 8 19 7 17 7 17 6 15 6 15 5 q t q t q t q t q t q t 2 2 4 3 2 2 3 2 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 13 5 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t q t