9 18

9_17

9_19

Visit 9 18's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 18's page at Knotilus! Visit 9 18's page at the original Knot Atlas!

9 18 Quick Notes

9 18 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,14,6,15 X9,18,10,1 X17,6,18,7 X7,16,8,17 X15,8,16,9 X13,10,14,11 X11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4
Dowker-Thistlethwaite code	4 12 14 16 18 2 10 8 6
Conway Notation	[3222]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-14][3] Hyperbolic Volume 10.0577 A-Polynomial See Data:9 18/A-polynomial

[edit Notes for 9 18'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 9 18's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{2}-10t+13-10t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, -4 }
Jones polynomial	$q^{-2}-2q^{-3}+5q^{-4}-6q^{-5}+7q^{-6}-7q^{-7}+6q^{-8}-4q^{-9}+2q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}-a^{10}+z^{4}a^{8}+z^{2}a^{8}+2z^{4}a^{6}+4z^{2}a^{6}+a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-3z^{3}a^{13}+2za^{13}+2z^{6}a^{12}-5z^{4}a^{12}+3z^{2}a^{12}+2z^{7}a^{11}-3z^{5}a^{11}+z^{8}a^{10}+z^{6}a^{10}-2z^{4}a^{10}-2z^{2}a^{10}+a^{10}+4z^{7}a^{9}-5z^{5}a^{9}+z^{3}a^{9}+z^{8}a^{8}+2z^{6}a^{8}-2z^{4}a^{8}+2z^{7}a^{7}+z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+3z^{6}a^{6}-4z^{4}a^{6}+3z^{2}a^{6}-a^{6}+2z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-2z^{2}a^{4}+a^{4}$
The A2 invariant	$-q^{34}-2q^{28}+q^{26}+q^{20}-q^{18}+2q^{16}+q^{12}+2q^{10}-q^{8}+q^{6}$
The G2 invariant	$q^{176}-q^{174}+3q^{172}-4q^{170}+3q^{168}-2q^{166}-2q^{164}+10q^{162}-15q^{160}+18q^{158}-15q^{156}+5q^{154}+9q^{152}-26q^{150}+36q^{148}-37q^{146}+23q^{144}-2q^{142}-24q^{140}+37q^{138}-38q^{136}+27q^{134}-6q^{132}-16q^{130}+24q^{128}-23q^{126}+5q^{124}+16q^{122}-29q^{120}+31q^{118}-16q^{116}-7q^{114}+34q^{112}-51q^{110}+54q^{108}-40q^{106}+9q^{104}+23q^{102}-48q^{100}+58q^{98}-47q^{96}+23q^{94}+4q^{92}-26q^{90}+32q^{88}-25q^{86}+4q^{84}+16q^{82}-23q^{80}+20q^{78}-2q^{76}-18q^{74}+35q^{72}-36q^{70}+29q^{68}-12q^{66}-11q^{64}+29q^{62}-34q^{60}+33q^{58}-18q^{56}+6q^{54}+6q^{52}-14q^{50}+16q^{48}-13q^{46}+9q^{44}-2q^{42}-q^{40}+3q^{38}-3q^{36}+3q^{34}-q^{32}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 9_18.

A1 Invariants.

Weight	Invariant
1	$-q^{23}+q^{21}-2q^{19}+2q^{17}-q^{15}+q^{11}-q^{9}+3q^{7}-q^{5}+q^{3}$
2	$q^{64}-q^{62}-q^{60}+4q^{58}-2q^{56}-6q^{54}+8q^{52}+q^{50}-10q^{48}+8q^{46}+4q^{44}-11q^{42}+2q^{40}+5q^{38}-4q^{36}-3q^{34}+3q^{32}+5q^{30}-8q^{28}-q^{26}+11q^{24}-8q^{22}-4q^{20}+11q^{18}-3q^{16}-3q^{14}+5q^{12}-q^{8}+q^{6}$
3	$-q^{123}+q^{121}+q^{119}-q^{117}-3q^{115}+2q^{113}+7q^{111}-q^{109}-13q^{107}-3q^{105}+18q^{103}+11q^{101}-22q^{99}-21q^{97}+21q^{95}+31q^{93}-14q^{91}-39q^{89}+9q^{87}+42q^{85}+q^{83}-42q^{81}-10q^{79}+38q^{77}+15q^{75}-30q^{73}-18q^{71}+19q^{69}+21q^{67}-8q^{65}-23q^{63}-7q^{61}+22q^{59}+18q^{57}-20q^{55}-33q^{53}+17q^{51}+41q^{49}-9q^{47}-45q^{45}+2q^{43}+41q^{41}+6q^{39}-35q^{37}-11q^{35}+27q^{33}+10q^{31}-14q^{29}-9q^{27}+10q^{25}+7q^{23}-4q^{21}-3q^{19}+3q^{17}+2q^{15}-q^{11}+q^{9}$
4	$q^{200}-q^{198}-q^{196}+q^{194}+3q^{190}-4q^{188}-5q^{186}+3q^{184}+4q^{182}+15q^{180}-7q^{178}-23q^{176}-10q^{174}+7q^{172}+50q^{170}+14q^{168}-39q^{166}-55q^{164}-30q^{162}+83q^{160}+77q^{158}-4q^{156}-94q^{154}-117q^{152}+55q^{150}+132q^{148}+88q^{146}-68q^{144}-192q^{142}-29q^{140}+122q^{138}+164q^{136}+6q^{134}-197q^{132}-104q^{130}+64q^{128}+181q^{126}+66q^{124}-146q^{122}-124q^{120}+9q^{118}+142q^{116}+87q^{114}-72q^{112}-111q^{110}-38q^{108}+83q^{106}+93q^{104}+10q^{102}-86q^{100}-88q^{98}+4q^{96}+96q^{94}+110q^{92}-43q^{90}-137q^{88}-92q^{86}+73q^{84}+195q^{82}+30q^{80}-137q^{78}-171q^{76}+6q^{74}+209q^{72}+94q^{70}-69q^{68}-172q^{66}-64q^{64}+138q^{62}+99q^{60}+6q^{58}-106q^{56}-76q^{54}+56q^{52}+52q^{50}+30q^{48}-37q^{46}-43q^{44}+15q^{42}+14q^{40}+19q^{38}-9q^{36}-15q^{34}+7q^{32}+2q^{30}+7q^{28}-2q^{26}-4q^{24}+3q^{22}+2q^{18}-q^{14}+q^{12}$
5	$-q^{295}+q^{293}+q^{291}-q^{289}-q^{283}+2q^{281}+4q^{279}-3q^{277}-7q^{275}-4q^{273}-q^{271}+11q^{269}+20q^{267}+8q^{265}-20q^{263}-39q^{261}-29q^{259}+14q^{257}+67q^{255}+74q^{253}+13q^{251}-86q^{249}-137q^{247}-76q^{245}+73q^{243}+199q^{241}+182q^{239}-5q^{237}-239q^{235}-308q^{233}-121q^{231}+209q^{229}+422q^{227}+305q^{225}-100q^{223}-487q^{221}-502q^{219}-79q^{217}+462q^{215}+665q^{213}+314q^{211}-353q^{209}-771q^{207}-532q^{205}+180q^{203}+772q^{201}+718q^{199}+25q^{197}-709q^{195}-827q^{193}-204q^{191}+583q^{189}+846q^{187}+349q^{185}-443q^{183}-803q^{181}-430q^{179}+309q^{177}+714q^{175}+452q^{173}-186q^{171}-605q^{169}-448q^{167}+91q^{165}+500q^{163}+427q^{161}-7q^{159}-389q^{157}-415q^{155}-89q^{153}+291q^{151}+419q^{149}+199q^{147}-177q^{145}-426q^{143}-345q^{141}+44q^{139}+437q^{137}+497q^{135}+130q^{133}-414q^{131}-656q^{129}-324q^{127}+342q^{125}+771q^{123}+538q^{121}-213q^{119}-828q^{117}-719q^{115}+35q^{113}+775q^{111}+850q^{109}+160q^{107}-656q^{105}-871q^{103}-332q^{101}+455q^{99}+800q^{97}+446q^{95}-249q^{93}-656q^{91}-466q^{89}+70q^{87}+461q^{85}+418q^{83}+59q^{81}-290q^{79}-326q^{77}-96q^{75}+139q^{73}+214q^{71}+107q^{69}-55q^{67}-129q^{65}-76q^{63}+13q^{61}+62q^{59}+49q^{57}-27q^{53}-23q^{51}-3q^{49}+15q^{47}+11q^{45}-q^{43}-6q^{41}-q^{37}+4q^{35}+5q^{33}-2q^{31}-2q^{29}+2q^{27}+2q^{21}-q^{17}+q^{15}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{34}-2q^{28}+q^{26}+q^{20}-q^{18}+2q^{16}+q^{12}+2q^{10}-q^{8}+q^{6}$
1,1	$q^{92}-2q^{90}+6q^{88}-12q^{86}+23q^{84}-40q^{82}+60q^{80}-84q^{78}+110q^{76}-132q^{74}+142q^{72}-136q^{70}+114q^{68}-76q^{66}+18q^{64}+52q^{62}-119q^{60}+184q^{58}-236q^{56}+268q^{54}-280q^{52}+260q^{50}-224q^{48}+162q^{46}-97q^{44}+24q^{42}+38q^{40}-92q^{38}+123q^{36}-134q^{34}+136q^{32}-120q^{30}+103q^{28}-72q^{26}+56q^{24}-34q^{22}+23q^{20}-10q^{18}+6q^{16}-2q^{14}+q^{12}$
2,0	$q^{86}+2q^{78}-4q^{74}-q^{72}+4q^{70}+2q^{68}-4q^{66}-q^{64}+5q^{62}+q^{60}-7q^{58}-2q^{56}+2q^{54}-3q^{52}-3q^{50}+q^{48}+q^{46}-3q^{44}+q^{42}+3q^{40}-4q^{38}-q^{36}+7q^{34}+2q^{32}-5q^{30}+3q^{28}+7q^{26}+q^{24}-4q^{22}+2q^{20}+4q^{18}-q^{16}-q^{14}+q^{12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{74}-q^{72}+q^{70}+2q^{68}-4q^{66}+2q^{64}+4q^{62}-8q^{60}+3q^{58}+5q^{56}-10q^{54}+q^{52}+5q^{50}-6q^{48}-2q^{46}+2q^{44}-3q^{40}-2q^{38}+7q^{36}-2q^{34}-5q^{32}+10q^{30}-6q^{26}+9q^{24}+q^{22}-3q^{20}+4q^{18}+q^{16}-q^{14}+q^{12}$
1,0,0	$-q^{45}-q^{41}-2q^{37}+q^{35}-q^{33}+q^{31}+q^{27}+2q^{21}+2q^{17}+2q^{13}-q^{11}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{96}-q^{92}+2q^{90}+3q^{88}-2q^{86}-q^{84}+5q^{82}+3q^{80}-6q^{78}-4q^{76}+4q^{74}-3q^{72}-10q^{70}+4q^{66}-5q^{64}-3q^{62}+4q^{60}-q^{58}-6q^{56}+q^{54}+4q^{52}-5q^{50}-2q^{48}+9q^{46}+2q^{44}-5q^{42}+4q^{40}+8q^{38}-2q^{34}+4q^{32}+5q^{30}-q^{28}+3q^{24}+q^{22}-q^{20}+q^{18}$
1,0,0,0	$-q^{56}-q^{52}-q^{50}-2q^{46}+q^{44}-q^{42}+q^{38}+q^{34}+q^{30}+2q^{26}+2q^{22}+q^{20}+2q^{16}-q^{14}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{74}+q^{72}-3q^{70}+4q^{68}-6q^{66}+8q^{64}-8q^{62}+8q^{60}-7q^{58}+5q^{56}-2q^{54}-3q^{52}+7q^{50}-12q^{48}+14q^{46}-16q^{44}+16q^{42}-15q^{40}+12q^{38}-7q^{36}+4q^{34}+q^{32}-4q^{30}+8q^{28}-8q^{26}+9q^{24}-7q^{22}+7q^{20}-4q^{18}+3q^{16}-q^{14}+q^{12}$
1,0	$q^{120}-q^{116}-q^{114}+2q^{112}+3q^{110}-q^{108}-5q^{106}-2q^{104}+6q^{102}+6q^{100}-4q^{98}-9q^{96}-q^{94}+9q^{92}+5q^{90}-7q^{88}-8q^{86}+2q^{84}+7q^{82}-7q^{78}-2q^{76}+5q^{74}+2q^{72}-6q^{70}-4q^{68}+4q^{66}+5q^{64}-3q^{62}-6q^{60}+2q^{58}+7q^{56}+q^{54}-8q^{52}-3q^{50}+8q^{48}+8q^{46}-4q^{44}-8q^{42}+9q^{38}+5q^{36}-3q^{34}-5q^{32}+q^{30}+4q^{28}+2q^{26}-q^{24}-q^{22}+q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{102}-q^{100}+2q^{98}-2q^{96}+4q^{94}-5q^{92}+5q^{90}-6q^{88}+7q^{86}-7q^{84}+5q^{82}-5q^{80}+4q^{78}-3q^{76}-3q^{74}+2q^{72}-5q^{70}+7q^{68}-12q^{66}+10q^{64}-12q^{62}+13q^{60}-13q^{58}+9q^{56}-10q^{54}+9q^{52}-4q^{50}+3q^{48}-q^{46}+7q^{42}-4q^{40}+6q^{38}-6q^{36}+9q^{34}-4q^{32}+6q^{30}-4q^{28}+5q^{26}-q^{24}+2q^{22}-q^{20}+q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{176}-q^{174}+3q^{172}-4q^{170}+3q^{168}-2q^{166}-2q^{164}+10q^{162}-15q^{160}+18q^{158}-15q^{156}+5q^{154}+9q^{152}-26q^{150}+36q^{148}-37q^{146}+23q^{144}-2q^{142}-24q^{140}+37q^{138}-38q^{136}+27q^{134}-6q^{132}-16q^{130}+24q^{128}-23q^{126}+5q^{124}+16q^{122}-29q^{120}+31q^{118}-16q^{116}-7q^{114}+34q^{112}-51q^{110}+54q^{108}-40q^{106}+9q^{104}+23q^{102}-48q^{100}+58q^{98}-47q^{96}+23q^{94}+4q^{92}-26q^{90}+32q^{88}-25q^{86}+4q^{84}+16q^{82}-23q^{80}+20q^{78}-2q^{76}-18q^{74}+35q^{72}-36q^{70}+29q^{68}-12q^{66}-11q^{64}+29q^{62}-34q^{60}+33q^{58}-18q^{56}+6q^{54}+6q^{52}-14q^{50}+16q^{48}-13q^{46}+9q^{44}-2q^{42}-q^{40}+3q^{38}-3q^{36}+3q^{34}-q^{32}+q^{30}$

.

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["9 18"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t^{2}-10t+13-10t^{-1}+4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{4}+6z^{2}+1$

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]=

{ 41, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-2}-2q^{-3}+5q^{-4}-6q^{-5}+7q^{-6}-7q^{-7}+6q^{-8}-4q^{-9}+2q^{-10}-q^{-11}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{13}-3z^{3}a^{13}+2za^{13}+2z^{6}a^{12}-5z^{4}a^{12}+3z^{2}a^{12}+2z^{7}a^{11}-3z^{5}a^{11}+z^{8}a^{10}+z^{6}a^{10}-2z^{4}a^{10}-2z^{2}a^{10}+a^{10}+4z^{7}a^{9}-5z^{5}a^{9}+z^{3}a^{9}+z^{8}a^{8}+2z^{6}a^{8}-2z^{4}a^{8}+2z^{7}a^{7}+z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+3z^{6}a^{6}-4z^{4}a^{6}+3z^{2}a^{6}-a^{6}+2z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-2z^{2}a^{4}+a^{4}$

Vassiliev invariants

V2 and V3:

(6, -15)

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 $24$ $-120$ $288$ $748$ $108$ $-2880$ $-5200$ $-896$ $-664$ $2304$ $7200$ $17952$ $2592$ ${\frac {184471}{5}}$ ${\frac {17828}{15}}$ ${\frac {207524}{15}}$ ${\frac {889}{3}}$ ${\frac {8791}{5}}$

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 9 18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-9	-8	-7	-6	-5	-4	-3	-2	-1	0	χ
-3										1	1
-5									2	1	-1
-7								3			3
-9							3	2			-1
-11						4	3				1
-13					3	3					0
-15				3	4						-1
-17			1	3							2
-19		1	3								-2
-21		1									1
-23	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[9, 18]]
Out[2]=	9
In[3]:=	PD[Knot[9, 18]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 14, 6, 15], X[9, 18, 10, 1], X[17, 6, 18, 7], X[7, 16, 8, 17], X[15, 8, 16, 9], X[13, 10, 14, 11], X[11, 2, 12, 3]]
In[4]:=	GaussCode[Knot[9, 18]]
Out[4]=	GaussCode[-1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4]
In[5]:=	BR[Knot[9, 18]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, -2, -2, -2, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 18]][t]
Out[6]=	4 10 2 13 + -- - -- - 10 t + 4 t 2 t t
In[7]:=	Conway[Knot[9, 18]][z]
Out[7]=	2 4 1 + 6 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 18], Knot[11, Alternating, 246]}
In[9]:=	{KnotDet[Knot[9, 18]], KnotSignature[Knot[9, 18]]}
Out[9]=	{41, -4}
In[10]:=	J=Jones[Knot[9, 18]][q]
Out[10]=	-11 2 4 6 7 7 6 5 2 -2 -q + --- - -- + -- - -- + -- - -- + -- - -- + q 10 9 8 7 6 5 4 3 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 18]}
In[12]:=	A2Invariant[Knot[9, 18]][q]
Out[12]=	-34 2 -26 -20 -18 2 -12 2 -8 -6 -q - --- + q + q - q + --- + q + --- - q + q 28 16 10 q q q
In[13]:=	Kauffman[Knot[9, 18]][a, z]
Out[13]=	4 6 10 7 13 4 2 6 2 10 2 a - a + a + 2 a z + 2 a z - 2 a z + 3 a z - 2 a z + 12 2 5 3 7 3 9 3 13 3 4 4 6 4 3 a z - 2 a z - 4 a z + a z - 3 a z + a z - 4 a z - 8 4 10 4 12 4 5 5 7 5 9 5 2 a z - 2 a z - 5 a z + 2 a z + a z - 5 a z - 11 5 13 5 6 6 8 6 10 6 12 6 7 7 3 a z + a z + 3 a z + 2 a z + a z + 2 a z + 2 a z + 9 7 11 7 8 8 10 8 4 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 18]], Vassiliev[3][Knot[9, 18]]}
Out[14]=	{0, -15}
In[15]:=	Kh[Knot[9, 18]][q, t]
Out[15]=	-5 -3 1 1 1 3 1 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 23 9 21 8 19 8 19 7 17 7 17 6 q t q t q t q t q t q t 3 4 3 3 4 3 3 2 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 15 6 15 5 13 5 13 4 11 4 11 3 9 3 9 2 q t q t q t q t q t q t q t q t 3 2 ----- + ---- 7 2 5 q t q t