10 11

10_10

10_12

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10 11 Quick Notes

10 11 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X11,16,12,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X7,18,8,19 X9,20,10,1 X19,8,20,9 X17,10,18,11
Gauss code	-1, 6, -4, 5, -3, 1, -7, 9, -8, 10, -2, 3, -5, 4, -6, 2, -10, 7, -9, 8
Dowker-Thistlethwaite code	6 14 12 18 20 16 4 2 10 8
Conway Notation	[433]

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-3] Hyperbolic Volume 9.37044 A-Polynomial See Data:10 11/A-polynomial

[edit Notes for 10 11's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant -2

[edit Notes for 10 11's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-4t^{2}+11t-13+11t^{-1}-4t^{-2}$
Conway polynomial	$-4z^{4}-5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, -2 }
Jones polynomial	$q^{3}-q^{2}+3q-5+6q^{-1}-7q^{-2}+7q^{-3}-6q^{-4}+4q^{-5}-2q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-z^{4}a^{4}-z^{2}a^{4}-2z^{4}a^{2}-4z^{2}a^{2}-a^{2}-z^{4}-2z^{2}-1+z^{2}a^{-2}+2a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{9}+az^{9}+2a^{4}z^{8}+3a^{2}z^{8}+z^{8}+3a^{5}z^{7}-2a^{3}z^{7}-4az^{7}+z^{7}a^{-1}+3a^{6}z^{6}-4a^{4}z^{6}-10a^{2}z^{6}+z^{6}a^{-2}-2z^{6}+2a^{7}z^{5}-7a^{5}z^{5}+5a^{3}z^{5}+11az^{5}-3z^{5}a^{-1}+a^{8}z^{4}-6a^{6}z^{4}+5a^{4}z^{4}+16a^{2}z^{4}-5z^{4}a^{-2}-z^{4}-3a^{7}z^{3}+9a^{5}z^{3}-5a^{3}z^{3}-16az^{3}+z^{3}a^{-1}-2a^{8}z^{2}+5a^{6}z^{2}-12a^{2}z^{2}+7z^{2}a^{-2}+2z^{2}-2a^{5}z+2a^{3}z+5az+za^{-1}-a^{6}+a^{2}-2a^{-2}-1$
The A2 invariant	$q^{22}+2q^{16}-q^{14}-q^{8}+q^{6}-2q^{4}-1-q^{-2}+2q^{-4}+q^{-6}+q^{-8}+q^{-10}$
The G2 invariant	$q^{114}-q^{112}+2q^{110}-3q^{108}+2q^{106}-q^{104}-2q^{102}+6q^{100}-8q^{98}+10q^{96}-9q^{94}+5q^{92}+2q^{90}-11q^{88}+19q^{86}-21q^{84}+18q^{82}-12q^{80}-q^{78}+15q^{76}-23q^{74}+29q^{72}-21q^{70}+9q^{68}+5q^{66}-16q^{64}+19q^{62}-13q^{60}+2q^{58}+14q^{56}-19q^{54}+16q^{52}+q^{50}-19q^{48}+33q^{46}-37q^{44}+25q^{42}-7q^{40}-18q^{38}+36q^{36}-42q^{34}+36q^{32}-20q^{30}-5q^{28}+19q^{26}-29q^{24}+25q^{22}-17q^{20}+q^{18}+13q^{16}-18q^{14}+15q^{12}-q^{10}-15q^{8}+24q^{6}-24q^{4}+10q^{2}+5-21q^{-2}+31q^{-4}-28q^{-6}+19q^{-8}-3q^{-10}-14q^{-12}+20q^{-14}-19q^{-16}+16q^{-18}-8q^{-20}+q^{-22}+5q^{-24}-6q^{-26}+9q^{-28}-6q^{-30}+4q^{-32}+2q^{-38}-2q^{-40}+2q^{-42}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 10_11.

A1 Invariants.

Weight	Invariant
1	$q^{15}-q^{13}+2q^{11}-2q^{9}+q^{7}-q^{3}+q-2q^{-1}+2q^{-3}+q^{-7}$
2	$q^{42}-q^{40}+3q^{36}-3q^{34}-2q^{32}+6q^{30}-5q^{28}-5q^{26}+10q^{24}-3q^{22}-6q^{20}+6q^{18}+q^{16}-3q^{14}-q^{12}+5q^{10}-6q^{6}+6q^{4}+4q^{2}-8+3q^{-2}+6q^{-4}-7q^{-6}-q^{-8}+4q^{-10}-3q^{-12}-q^{-14}+2q^{-16}+q^{-22}$
3	$q^{81}-q^{79}+q^{75}+q^{73}-2q^{71}-2q^{69}+2q^{67}+2q^{65}-4q^{63}-3q^{61}+6q^{59}+6q^{57}-10q^{55}-11q^{53}+14q^{51}+17q^{49}-12q^{47}-25q^{45}+12q^{43}+28q^{41}-6q^{39}-26q^{37}-q^{35}+22q^{33}+7q^{31}-14q^{29}-12q^{27}+5q^{25}+15q^{23}+q^{21}-18q^{19}-9q^{17}+20q^{15}+13q^{13}-19q^{11}-16q^{9}+16q^{7}+22q^{5}-13q^{3}-24q+6q^{-1}+26q^{-3}+2q^{-5}-21q^{-7}-9q^{-9}+19q^{-11}+12q^{-13}-10q^{-15}-12q^{-17}+4q^{-19}+10q^{-21}-q^{-23}-6q^{-25}-2q^{-27}+3q^{-29}-q^{-33}-q^{-35}+q^{-37}+q^{-45}$
4	$q^{132}-q^{130}+q^{126}-q^{124}+2q^{122}-3q^{120}-q^{118}+2q^{116}-3q^{114}+6q^{112}-3q^{110}-q^{108}+3q^{106}-9q^{104}+6q^{102}-5q^{100}+6q^{98}+15q^{96}-9q^{94}-4q^{92}-29q^{90}+4q^{88}+46q^{86}+19q^{84}-6q^{82}-76q^{80}-33q^{78}+62q^{76}+71q^{74}+33q^{72}-100q^{70}-89q^{68}+31q^{66}+92q^{64}+79q^{62}-64q^{60}-100q^{58}-24q^{56}+53q^{54}+88q^{52}+2q^{50}-57q^{48}-52q^{46}-3q^{44}+53q^{42}+45q^{40}-4q^{38}-56q^{36}-41q^{34}+20q^{32}+69q^{30}+29q^{28}-58q^{26}-61q^{24}+q^{22}+88q^{20}+51q^{18}-61q^{16}-80q^{14}-25q^{12}+93q^{10}+79q^{8}-31q^{6}-84q^{4}-72q^{2}+60+94q^{-2}+25q^{-4}-48q^{-6}-98q^{-8}-5q^{-10}+60q^{-12}+61q^{-14}+16q^{-16}-72q^{-18}-44q^{-20}+5q^{-22}+44q^{-24}+47q^{-26}-18q^{-28}-30q^{-30}-24q^{-32}+6q^{-34}+31q^{-36}+7q^{-38}-3q^{-40}-15q^{-42}-8q^{-44}+8q^{-46}+3q^{-48}+5q^{-50}-3q^{-52}-4q^{-54}+q^{-56}-2q^{-58}+2q^{-60}-q^{-64}+q^{-66}-q^{-68}+q^{-76}$
5	$q^{195}-q^{193}+q^{189}-q^{187}+q^{183}-2q^{181}-2q^{179}+2q^{177}+q^{175}+4q^{171}-q^{169}-6q^{167}-4q^{165}-q^{163}+3q^{161}+10q^{159}+10q^{157}-q^{155}-11q^{153}-18q^{151}-10q^{149}+5q^{147}+26q^{145}+34q^{143}+12q^{141}-29q^{139}-61q^{137}-51q^{135}+11q^{133}+91q^{131}+118q^{129}+34q^{127}-109q^{125}-196q^{123}-119q^{121}+90q^{119}+268q^{117}+239q^{115}-30q^{113}-320q^{111}-357q^{109}-72q^{107}+303q^{105}+458q^{103}+210q^{101}-249q^{99}-495q^{97}-320q^{95}+131q^{93}+464q^{91}+396q^{89}+3q^{87}-374q^{85}-408q^{83}-111q^{81}+237q^{79}+361q^{77}+192q^{75}-102q^{73}-277q^{71}-221q^{69}-11q^{67}+177q^{65}+217q^{63}+96q^{61}-92q^{59}-202q^{57}-145q^{55}+31q^{53}+185q^{51}+182q^{49}+3q^{47}-194q^{45}-210q^{43}-16q^{41}+212q^{39}+244q^{37}+31q^{35}-243q^{33}-297q^{31}-53q^{29}+266q^{27}+353q^{25}+100q^{23}-261q^{21}-402q^{19}-180q^{17}+227q^{15}+432q^{13}+260q^{11}-136q^{9}-420q^{7}-339q^{5}+22q^{3}+355q+388q^{-1}+111q^{-3}-247q^{-5}-380q^{-7}-213q^{-9}+97q^{-11}+317q^{-13}+283q^{-15}+37q^{-17}-208q^{-19}-275q^{-21}-146q^{-23}+73q^{-25}+227q^{-27}+194q^{-29}+29q^{-31}-132q^{-33}-183q^{-35}-102q^{-37}+38q^{-39}+135q^{-41}+118q^{-43}+25q^{-45}-66q^{-47}-97q^{-49}-60q^{-51}+16q^{-53}+63q^{-55}+55q^{-57}+16q^{-59}-24q^{-61}-41q^{-63}-22q^{-65}+4q^{-67}+19q^{-69}+20q^{-71}+7q^{-73}-9q^{-75}-10q^{-77}-6q^{-79}-2q^{-81}+6q^{-83}+6q^{-85}-q^{-89}-q^{-91}-4q^{-93}+2q^{-97}+q^{-103}-q^{-105}-q^{-107}+q^{-115}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}+2q^{16}-q^{14}-q^{8}+q^{6}-2q^{4}-1-q^{-2}+2q^{-4}+q^{-6}+q^{-8}+q^{-10}$
1,1	$q^{60}-2q^{58}+4q^{56}-8q^{54}+15q^{52}-20q^{50}+28q^{48}-40q^{46}+54q^{44}-60q^{42}+68q^{40}-80q^{38}+80q^{36}-70q^{34}+52q^{32}-34q^{30}-3q^{28}+44q^{26}-86q^{24}+120q^{22}-150q^{20}+170q^{18}-168q^{16}+166q^{14}-137q^{12}+110q^{10}-66q^{8}+30q^{6}+7q^{4}-48q^{2}+72-84q^{-2}+84q^{-4}-86q^{-6}+72q^{-8}-56q^{-10}+40q^{-12}-32q^{-14}+22q^{-16}-12q^{-18}+10q^{-20}-4q^{-22}+4q^{-24}+q^{-28}$
2,0	$q^{56}+q^{50}+2q^{48}-3q^{44}+3q^{40}-3q^{38}-5q^{36}+q^{34}+4q^{32}-2q^{30}-5q^{28}+3q^{26}+2q^{24}-3q^{22}+q^{20}+4q^{18}+3q^{12}-2q^{8}+3q^{6}+6q^{4}-q^{2}-3+4q^{-2}+q^{-4}-5q^{-6}-5q^{-8}-q^{-10}-3q^{-14}-q^{-16}+2q^{-18}+3q^{-20}+q^{-22}+q^{-24}+q^{-26}+q^{-28}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-q^{46}+2q^{42}-3q^{40}+6q^{36}-5q^{34}-2q^{32}+8q^{30}-4q^{28}-3q^{26}+6q^{24}-2q^{22}-3q^{20}+q^{18}+2q^{16}-2q^{12}+6q^{10}+2q^{8}-7q^{6}+2q^{4}+2q^{2}-9+q^{-2}+2q^{-4}-4q^{-6}+3q^{-8}+2q^{-10}+2q^{-14}+2q^{-16}+q^{-20}$
1,0,0	$q^{29}+q^{25}+2q^{21}-q^{19}+q^{17}-q^{15}-q^{11}-2q^{5}-2q-q^{-3}+2q^{-5}+q^{-7}+2q^{-9}+q^{-11}+q^{-13}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-q^{46}+2q^{44}-4q^{42}+5q^{40}-6q^{38}+8q^{36}-7q^{34}+8q^{32}-6q^{30}+4q^{28}+q^{26}-4q^{24}+8q^{22}-11q^{20}+13q^{18}-16q^{16}+14q^{14}-14q^{12}+10q^{10}-8q^{8}+3q^{6}-4q^{2}+5-7q^{-2}+8q^{-4}-6q^{-6}+7q^{-8}-4q^{-10}+4q^{-12}-2q^{-14}+2q^{-16}+q^{-20}$
1,0	$q^{78}-q^{74}-q^{72}+q^{70}+3q^{68}-4q^{64}-3q^{62}+3q^{60}+7q^{58}-7q^{54}-5q^{52}+5q^{50}+8q^{48}-q^{46}-8q^{44}-2q^{42}+6q^{40}+4q^{38}-5q^{36}-5q^{34}+3q^{32}+5q^{30}-q^{28}-5q^{26}+q^{24}+5q^{22}+2q^{20}-4q^{18}-q^{16}+5q^{14}+4q^{12}-5q^{10}-7q^{8}+2q^{6}+8q^{4}+q^{2}-8-7q^{-2}+4q^{-4}+7q^{-6}-q^{-8}-6q^{-10}-3q^{-12}+4q^{-14}+3q^{-16}-2q^{-20}+2q^{-24}+2q^{-26}+q^{-34}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-q^{112}+2q^{110}-3q^{108}+2q^{106}-q^{104}-2q^{102}+6q^{100}-8q^{98}+10q^{96}-9q^{94}+5q^{92}+2q^{90}-11q^{88}+19q^{86}-21q^{84}+18q^{82}-12q^{80}-q^{78}+15q^{76}-23q^{74}+29q^{72}-21q^{70}+9q^{68}+5q^{66}-16q^{64}+19q^{62}-13q^{60}+2q^{58}+14q^{56}-19q^{54}+16q^{52}+q^{50}-19q^{48}+33q^{46}-37q^{44}+25q^{42}-7q^{40}-18q^{38}+36q^{36}-42q^{34}+36q^{32}-20q^{30}-5q^{28}+19q^{26}-29q^{24}+25q^{22}-17q^{20}+q^{18}+13q^{16}-18q^{14}+15q^{12}-q^{10}-15q^{8}+24q^{6}-24q^{4}+10q^{2}+5-21q^{-2}+31q^{-4}-28q^{-6}+19q^{-8}-3q^{-10}-14q^{-12}+20q^{-14}-19q^{-16}+16q^{-18}-8q^{-20}+q^{-22}+5q^{-24}-6q^{-26}+9q^{-28}-6q^{-30}+4q^{-32}+2q^{-38}-2q^{-40}+2q^{-42}+q^{-46}$

.

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["10 11"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-4t^{2}+11t-13+11t^{-1}-4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-4z^{4}-5z^{2}+1$

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]=

{ 43, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-q^{2}+3q-5+6q^{-1}-7q^{-2}+7q^{-3}-6q^{-4}+4q^{-5}-2q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{9}+az^{9}+2a^{4}z^{8}+3a^{2}z^{8}+z^{8}+3a^{5}z^{7}-2a^{3}z^{7}-4az^{7}+z^{7}a^{-1}+3a^{6}z^{6}-4a^{4}z^{6}-10a^{2}z^{6}+z^{6}a^{-2}-2z^{6}+2a^{7}z^{5}-7a^{5}z^{5}+5a^{3}z^{5}+11az^{5}-3z^{5}a^{-1}+a^{8}z^{4}-6a^{6}z^{4}+5a^{4}z^{4}+16a^{2}z^{4}-5z^{4}a^{-2}-z^{4}-3a^{7}z^{3}+9a^{5}z^{3}-5a^{3}z^{3}-16az^{3}+z^{3}a^{-1}-2a^{8}z^{2}+5a^{6}z^{2}-12a^{2}z^{2}+7z^{2}a^{-2}+2z^{2}-2a^{5}z+2a^{3}z+5az+za^{-1}-a^{6}+a^{2}-2a^{-2}-1$

Vassiliev invariants

V2 and V3:

(-5, 4)

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 $-20$ $32$ $200$ ${\frac {842}{3}}$ ${\frac {406}{3}}$ $-640$ $-{\frac {3232}{3}}$ $-{\frac {832}{3}}$ $-224$ $-{\frac {4000}{3}}$ $512$ $-{\frac {16840}{3}}$ $-{\frac {8120}{3}}$ $-{\frac {22639}{6}}$ ${\frac {4990}{3}}$ $-{\frac {45566}{9}}$ ${\frac {13547}{18}}$ $-{\frac {7855}{6}}$

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 $s=$-2 is the signature of 10 11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-6	-5	-4	-3	-2	-1	0	1	2	3	4	χ
7											1	1
5												0
3									3	1		2
1								2				-2
-1							4	3				1
-3						4	3					-1
-5					3	3						0
-7				3	4							1
-9			1	3								-2
-11		1	3									2
-13		1										-1
-15	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[10, 11]]
Out[2]=	10
In[3]:=	PD[Knot[10, 11]]
Out[3]=	PD[X[1, 6, 2, 7], X[11, 16, 12, 17], X[5, 13, 6, 12], X[3, 15, 4, 14], X[13, 5, 14, 4], X[15, 3, 16, 2], X[7, 18, 8, 19], X[9, 20, 10, 1], X[19, 8, 20, 9], X[17, 10, 18, 11]]
In[4]:=	GaussCode[Knot[10, 11]]
Out[4]=	GaussCode[-1, 6, -4, 5, -3, 1, -7, 9, -8, 10, -2, 3, -5, 4, -6, 2, -10, 7, -9, 8]
In[5]:=	BR[Knot[10, 11]]
Out[5]=	BR[5, {-1, -1, -1, -1, -2, 1, 3, -2, 3, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[10, 11]][t]
Out[6]=	4 11 2 -13 - -- + -- + 11 t - 4 t 2 t t
In[7]:=	Conway[Knot[10, 11]][z]
Out[7]=	2 4 1 - 5 z - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 11]}
In[9]:=	{KnotDet[Knot[10, 11]], KnotSignature[Knot[10, 11]]}
Out[9]=	{43, -2}
In[10]:=	J=Jones[Knot[10, 11]][q]
Out[10]=	-7 2 4 6 7 7 6 2 3 -5 + q - -- + -- - -- + -- - -- + - + 3 q - q + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 11]}
In[12]:=	A2Invariant[Knot[10, 11]][q]
Out[12]=	-22 2 -14 -8 -6 2 2 4 6 8 10 -1 + q + --- - q - q + q - -- - q + 2 q + q + q + q 16 4 q q
In[13]:=	Kauffman[Knot[10, 11]][a, z]
Out[13]=	2 2 2 6 z 3 5 2 7 z -1 - -- + a - a + - + 5 a z + 2 a z - 2 a z + 2 z + ---- - 2 a 2 a a 3 2 2 6 2 8 2 z 3 3 3 5 3 12 a z + 5 a z - 2 a z + -- - 16 a z - 5 a z + 9 a z - a 4 5 7 3 4 5 z 2 4 4 4 6 4 8 4 3 z 3 a z - z - ---- + 16 a z + 5 a z - 6 a z + a z - ---- + 2 a a 6 5 3 5 5 5 7 5 6 z 2 6 11 a z + 5 a z - 7 a z + 2 a z - 2 z + -- - 10 a z - 2 a 7 4 6 6 6 z 7 3 7 5 7 8 2 8 4 a z + 3 a z + -- - 4 a z - 2 a z + 3 a z + z + 3 a z + a 4 8 9 3 9 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 11]], Vassiliev[3][Knot[10, 11]]}
Out[14]=	{0, 4}
In[15]:=	Kh[Knot[10, 11]][q, t]
Out[15]=	3 4 1 1 1 3 1 3 3 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 4 3 3 4 3 t 3 2 3 3 7 4 ----- + ----- + ---- + ---- + --- + 2 q t + 3 q t + q t + q t 7 2 5 2 5 3 q q t q t q t q t