10 97

10_96

10_98

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10 97 Quick Notes

10 97 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X12,6,13,5 X8394 X2,9,3,10 X16,12,17,11 X10,18,11,17 X18,8,19,7 X20,14,1,13 X14,20,15,19 X6,16,7,15
Gauss code	1, -4, 3, -1, 2, -10, 7, -3, 4, -6, 5, -2, 8, -9, 10, -5, 6, -7, 9, -8
Dowker-Thistlethwaite code	4 8 12 18 2 16 20 6 10 14
Conway Notation	[.2.210.2]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-11] Hyperbolic Volume 14.8527 A-Polynomial See Data:10 97/A-polynomial

[edit Notes for 10 97's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 2 }[/math] Rasmussen s-Invariant -2

[edit Notes for 10 97's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -5 t^2+22 t-33+22 t^{-1} -5 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -5 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 87, 2 }
Jones polynomial	[math]\displaystyle{ q^9-4 q^8+7 q^7-11 q^6+14 q^5-14 q^4+14 q^3-11 q^2+7 q-3+ q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^4 a^{-2} -3 z^4 a^{-4} -z^4 a^{-6} +2 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +z^2 a^{-8} +z^2+2 a^{-2} -2 a^{-4} +2 a^{-6} - a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^9 a^{-5} +2 z^9 a^{-7} +6 z^8 a^{-4} +11 z^8 a^{-6} +5 z^8 a^{-8} +8 z^7 a^{-3} +11 z^7 a^{-5} +7 z^7 a^{-7} +4 z^7 a^{-9} +6 z^6 a^{-2} -3 z^6 a^{-4} -21 z^6 a^{-6} -11 z^6 a^{-8} +z^6 a^{-10} +3 z^5 a^{-1} -12 z^5 a^{-3} -32 z^5 a^{-5} -28 z^5 a^{-7} -11 z^5 a^{-9} -7 z^4 a^{-2} -9 z^4 a^{-4} +5 z^4 a^{-6} +4 z^4 a^{-8} -2 z^4 a^{-10} +z^4-2 z^3 a^{-1} +10 z^3 a^{-3} +24 z^3 a^{-5} +20 z^3 a^{-7} +8 z^3 a^{-9} +6 z^2 a^{-2} +10 z^2 a^{-4} +3 z^2 a^{-6} +z^2 a^{-8} +z^2 a^{-10} -z^2-2 z a^{-3} -6 z a^{-5} -4 z a^{-7} -2 a^{-2} -2 a^{-4} -2 a^{-6} - a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ q^4-q^2-1+4 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} +2 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} -2 q^{-20} +3 q^{-22} -2 q^{-24} -2 q^{-26} + q^{-28} }[/math]
The G2 invariant	[math]\displaystyle{ q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+6 q^{10}-5 q^8+10 q^4-19 q^2+31-39 q^{-2} +37 q^{-4} -22 q^{-6} -8 q^{-8} +53 q^{-10} -94 q^{-12} +123 q^{-14} -126 q^{-16} +82 q^{-18} - q^{-20} -105 q^{-22} +205 q^{-24} -241 q^{-26} +205 q^{-28} -91 q^{-30} -61 q^{-32} +196 q^{-34} -257 q^{-36} +214 q^{-38} -82 q^{-40} -83 q^{-42} +199 q^{-44} -208 q^{-46} +106 q^{-48} +71 q^{-50} -229 q^{-52} +296 q^{-54} -241 q^{-56} +68 q^{-58} +148 q^{-60} -329 q^{-62} +404 q^{-64} -336 q^{-66} +158 q^{-68} +73 q^{-70} -266 q^{-72} +359 q^{-74} -328 q^{-76} +186 q^{-78} +3 q^{-80} -172 q^{-82} +252 q^{-84} -210 q^{-86} +77 q^{-88} +98 q^{-90} -221 q^{-92} +231 q^{-94} -136 q^{-96} -38 q^{-98} +207 q^{-100} -299 q^{-102} +277 q^{-104} -150 q^{-106} -24 q^{-108} +177 q^{-110} -251 q^{-112} +231 q^{-114} -142 q^{-116} +28 q^{-118} +61 q^{-120} -111 q^{-122} +108 q^{-124} -71 q^{-126} +33 q^{-128} +2 q^{-130} -19 q^{-132} +20 q^{-134} -16 q^{-136} +8 q^{-138} -3 q^{-140} + q^{-142} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_97.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^3-2 q+4 q^{-1} -4 q^{-3} +3 q^{-5} +3 q^{-11} -4 q^{-13} +3 q^{-15} -3 q^{-17} + q^{-19} }[/math]
2	[math]\displaystyle{ q^{10}-2 q^8+q^6+5 q^4-11 q^2+4+19 q^{-2} -27 q^{-4} -2 q^{-6} +38 q^{-8} -27 q^{-10} -17 q^{-12} +36 q^{-14} -6 q^{-16} -23 q^{-18} +12 q^{-20} +19 q^{-22} -16 q^{-24} -16 q^{-26} +30 q^{-28} - q^{-30} -37 q^{-32} +25 q^{-34} +18 q^{-36} -37 q^{-38} +8 q^{-40} +24 q^{-42} -18 q^{-44} -5 q^{-46} +12 q^{-48} -2 q^{-50} -3 q^{-52} + q^{-54} }[/math]
3	[math]\displaystyle{ q^{21}-2 q^{19}+q^{17}+2 q^{15}-2 q^{13}-5 q^{11}+7 q^9+13 q^7-20 q^5-29 q^3+39 q+61 q^{-1} -51 q^{-3} -119 q^{-5} +54 q^{-7} +189 q^{-9} -27 q^{-11} -248 q^{-13} -37 q^{-15} +284 q^{-17} +116 q^{-19} -271 q^{-21} -184 q^{-23} +206 q^{-25} +236 q^{-27} -116 q^{-29} -245 q^{-31} +13 q^{-33} +227 q^{-35} +78 q^{-37} -186 q^{-39} -153 q^{-41} +137 q^{-43} +215 q^{-45} -89 q^{-47} -257 q^{-49} +27 q^{-51} +285 q^{-53} +37 q^{-55} -287 q^{-57} -119 q^{-59} +266 q^{-61} +183 q^{-63} -202 q^{-65} -235 q^{-67} +117 q^{-69} +250 q^{-71} -26 q^{-73} -215 q^{-75} -48 q^{-77} +155 q^{-79} +86 q^{-81} -85 q^{-83} -86 q^{-85} +28 q^{-87} +60 q^{-89} +4 q^{-91} -34 q^{-93} -9 q^{-95} +11 q^{-97} +7 q^{-99} -2 q^{-101} -3 q^{-103} + q^{-105} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^4-q^2-1+4 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} +2 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} -2 q^{-20} +3 q^{-22} -2 q^{-24} -2 q^{-26} + q^{-28} }[/math]
2,0	[math]\displaystyle{ q^{12}-q^{10}-2 q^8+3 q^6+5 q^4-4 q^2-11+7 q^{-2} +20 q^{-4} -11 q^{-6} -22 q^{-8} +12 q^{-10} +24 q^{-12} -10 q^{-14} -22 q^{-16} +11 q^{-18} +16 q^{-20} -8 q^{-22} -8 q^{-24} +8 q^{-26} +2 q^{-30} +10 q^{-32} -6 q^{-34} -10 q^{-36} +6 q^{-38} +14 q^{-40} -17 q^{-42} -19 q^{-44} +14 q^{-46} +18 q^{-48} -13 q^{-50} -17 q^{-52} +12 q^{-54} +15 q^{-56} -6 q^{-58} -15 q^{-60} +3 q^{-62} +9 q^{-64} +2 q^{-66} -3 q^{-68} -2 q^{-70} + q^{-72} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^8-2 q^6+6 q^2-7-4 q^{-2} +19 q^{-4} -11 q^{-6} -13 q^{-8} +30 q^{-10} -11 q^{-12} -18 q^{-14} +29 q^{-16} -7 q^{-18} -15 q^{-20} +13 q^{-22} +3 q^{-24} -6 q^{-26} -7 q^{-28} +11 q^{-30} +9 q^{-32} -23 q^{-34} +9 q^{-36} +18 q^{-38} -29 q^{-40} +5 q^{-42} +18 q^{-44} -22 q^{-46} +7 q^{-48} +11 q^{-50} -12 q^{-52} +4 q^{-54} +2 q^{-56} -3 q^{-58} + q^{-60} }[/math]
1,0,0	[math]\displaystyle{ q^5-q^3- q^{-1} +4 q^{-3} -2 q^{-5} +3 q^{-7} +2 q^{-11} -2 q^{-13} -2 q^{-19} +2 q^{-21} +3 q^{-25} -2 q^{-27} +3 q^{-29} -2 q^{-31} - q^{-33} -2 q^{-35} + q^{-37} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^8-2 q^6+4 q^4-8 q^2+13-18 q^{-2} +27 q^{-4} -31 q^{-6} +35 q^{-8} -34 q^{-10} +29 q^{-12} -18 q^{-14} +3 q^{-16} +13 q^{-18} -31 q^{-20} +47 q^{-22} -61 q^{-24} +68 q^{-26} -67 q^{-28} +63 q^{-30} -49 q^{-32} +35 q^{-34} -15 q^{-36} -2 q^{-38} +17 q^{-40} -29 q^{-42} +34 q^{-44} -36 q^{-46} +33 q^{-48} -29 q^{-50} +20 q^{-52} -14 q^{-54} +8 q^{-56} -3 q^{-58} + q^{-60} }[/math]
1,0	[math]\displaystyle{ q^{14}-2 q^{10}-2 q^8+2 q^6+7 q^4+2 q^2-10-11 q^{-2} +5 q^{-4} +23 q^{-6} +8 q^{-8} -23 q^{-10} -24 q^{-12} +12 q^{-14} +36 q^{-16} +6 q^{-18} -34 q^{-20} -20 q^{-22} +25 q^{-24} +30 q^{-26} -12 q^{-28} -31 q^{-30} + q^{-32} +29 q^{-34} +7 q^{-36} -25 q^{-38} -12 q^{-40} +20 q^{-42} +16 q^{-44} -15 q^{-46} -19 q^{-48} +14 q^{-50} +24 q^{-52} -8 q^{-54} -32 q^{-56} -2 q^{-58} +34 q^{-60} +17 q^{-62} -30 q^{-64} -34 q^{-66} +14 q^{-68} +37 q^{-70} +4 q^{-72} -31 q^{-74} -18 q^{-76} +19 q^{-78} +23 q^{-80} -5 q^{-82} -16 q^{-84} -4 q^{-86} +9 q^{-88} +5 q^{-90} -3 q^{-92} -3 q^{-94} + q^{-98} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+6 q^{10}-5 q^8+10 q^4-19 q^2+31-39 q^{-2} +37 q^{-4} -22 q^{-6} -8 q^{-8} +53 q^{-10} -94 q^{-12} +123 q^{-14} -126 q^{-16} +82 q^{-18} - q^{-20} -105 q^{-22} +205 q^{-24} -241 q^{-26} +205 q^{-28} -91 q^{-30} -61 q^{-32} +196 q^{-34} -257 q^{-36} +214 q^{-38} -82 q^{-40} -83 q^{-42} +199 q^{-44} -208 q^{-46} +106 q^{-48} +71 q^{-50} -229 q^{-52} +296 q^{-54} -241 q^{-56} +68 q^{-58} +148 q^{-60} -329 q^{-62} +404 q^{-64} -336 q^{-66} +158 q^{-68} +73 q^{-70} -266 q^{-72} +359 q^{-74} -328 q^{-76} +186 q^{-78} +3 q^{-80} -172 q^{-82} +252 q^{-84} -210 q^{-86} +77 q^{-88} +98 q^{-90} -221 q^{-92} +231 q^{-94} -136 q^{-96} -38 q^{-98} +207 q^{-100} -299 q^{-102} +277 q^{-104} -150 q^{-106} -24 q^{-108} +177 q^{-110} -251 q^{-112} +231 q^{-114} -142 q^{-116} +28 q^{-118} +61 q^{-120} -111 q^{-122} +108 q^{-124} -71 q^{-126} +33 q^{-128} +2 q^{-130} -19 q^{-132} +20 q^{-134} -16 q^{-136} +8 q^{-138} -3 q^{-140} + q^{-142} }[/math]

.

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 97"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -5 t^2+22 t-33+22 t^{-1} -5 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -5 z^4+2 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 87, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^9-4 q^8+7 q^7-11 q^6+14 q^5-14 q^4+14 q^3-11 q^2+7 q-3+ q^{-1} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^4 a^{-2} -3 z^4 a^{-4} -z^4 a^{-6} +2 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +z^2 a^{-8} +z^2+2 a^{-2} -2 a^{-4} +2 a^{-6} - a^{-8} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^9 a^{-5} +2 z^9 a^{-7} +6 z^8 a^{-4} +11 z^8 a^{-6} +5 z^8 a^{-8} +8 z^7 a^{-3} +11 z^7 a^{-5} +7 z^7 a^{-7} +4 z^7 a^{-9} +6 z^6 a^{-2} -3 z^6 a^{-4} -21 z^6 a^{-6} -11 z^6 a^{-8} +z^6 a^{-10} +3 z^5 a^{-1} -12 z^5 a^{-3} -32 z^5 a^{-5} -28 z^5 a^{-7} -11 z^5 a^{-9} -7 z^4 a^{-2} -9 z^4 a^{-4} +5 z^4 a^{-6} +4 z^4 a^{-8} -2 z^4 a^{-10} +z^4-2 z^3 a^{-1} +10 z^3 a^{-3} +24 z^3 a^{-5} +20 z^3 a^{-7} +8 z^3 a^{-9} +6 z^2 a^{-2} +10 z^2 a^{-4} +3 z^2 a^{-6} +z^2 a^{-8} +z^2 a^{-10} -z^2-2 z a^{-3} -6 z a^{-5} -4 z a^{-7} -2 a^{-2} -2 a^{-4} -2 a^{-6} - a^{-8} }[/math]

Vassiliev invariants

V2 and V3:

(2, 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 [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{508}{3} }[/math] [math]\displaystyle{ \frac{164}{3} }[/math] [math]\displaystyle{ 256 }[/math] [math]\displaystyle{ \frac{2432}{3} }[/math] [math]\displaystyle{ \frac{512}{3} }[/math] [math]\displaystyle{ 224 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{4064}{3} }[/math] [math]\displaystyle{ \frac{1312}{3} }[/math] [math]\displaystyle{ \frac{55471}{15} }[/math] [math]\displaystyle{ -\frac{2868}{5} }[/math] [math]\displaystyle{ \frac{117724}{45} }[/math] [math]\displaystyle{ \frac{593}{9} }[/math] [math]\displaystyle{ \frac{6511}{15} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]2 is the signature of 10 97. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-2	-1	0	1	2	3	4	5	6	7	8	χ
19											1	1
17										3		-3
15									4	1		3
13								7	3			-4
11							7	4				3
9						7	7					0
7					7	7						0
5				4	7							3
3			3	7								-4
1		1	5									4
-1		2										-2
-3	1											1

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 97]]
Out[2]=	10
In[3]:=	PD[Knot[10, 97]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 6, 13, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[16, 12, 17, 11], X[10, 18, 11, 17], X[18, 8, 19, 7], X[20, 14, 1, 13], X[14, 20, 15, 19], X[6, 16, 7, 15]]
In[4]:=	GaussCode[Knot[10, 97]]
Out[4]=	GaussCode[1, -4, 3, -1, 2, -10, 7, -3, 4, -6, 5, -2, 8, -9, 10, -5, 6, -7, 9, -8]
In[5]:=	BR[Knot[10, 97]]
Out[5]=	BR[5, {1, 1, 2, -1, 2, 1, -3, 2, -1, 2, 3, -4, 3, -4}]
In[6]:=	alex = Alexander[Knot[10, 97]][t]
Out[6]=	5 22 2 -33 - -- + -- + 22 t - 5 t 2 t t
In[7]:=	Conway[Knot[10, 97]][z]
Out[7]=	2 4 1 + 2 z - 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 97]}
In[9]:=	{KnotDet[Knot[10, 97]], KnotSignature[Knot[10, 97]]}
Out[9]=	{87, 2}
In[10]:=	J=Jones[Knot[10, 97]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 -3 + - + 7 q - 11 q + 14 q - 14 q + 14 q - 11 q + 7 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 97]}
In[12]:=	A2Invariant[Knot[10, 97]][q]
Out[12]=	-4 -2 2 4 6 8 10 12 14 -1 + q - q + 4 q - 2 q + q + 2 q - 2 q + 2 q - 2 q + 16 18 20 22 24 26 28 2 q + q - 2 q + 3 q - 2 q - 2 q + q
In[13]:=	Kauffman[Knot[10, 97]][a, z]
Out[13]=	2 2 2 2 -8 2 2 2 4 z 6 z 2 z 2 z z 3 z 10 z -a - -- - -- - -- - --- - --- - --- - z + --- + -- + ---- + ----- + 6 4 2 7 5 3 10 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 4 4 6 z 8 z 20 z 24 z 10 z 2 z 4 2 z 4 z ---- + ---- + ----- + ----- + ----- - ---- + z - ---- + ---- + 2 9 7 5 3 a 10 8 a a a a a a a 4 4 4 5 5 5 5 5 6 5 z 9 z 7 z 11 z 28 z 32 z 12 z 3 z z ---- - ---- - ---- - ----- - ----- - ----- - ----- + ---- + --- - 6 4 2 9 7 5 3 a 10 a a a a a a a a 6 6 6 6 7 7 7 7 8 11 z 21 z 3 z 6 z 4 z 7 z 11 z 8 z 5 z ----- - ----- - ---- + ---- + ---- + ---- + ----- + ---- + ---- + 8 6 4 2 9 7 5 3 8 a a a a a a a a a 8 8 9 9 11 z 6 z 2 z 2 z ----- + ---- + ---- + ---- 6 4 7 5 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 97]], Vassiliev[3][Knot[10, 97]]}
Out[14]=	{0, 4}
In[15]:=	Kh[Knot[10, 97]][q, t]
Out[15]=	3 1 2 q 3 5 5 2 7 2 5 q + 3 q + ----- + --- + - + 7 q t + 4 q t + 7 q t + 7 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 7 q t + 7 q t + 7 q t + 7 q t + 4 q t + 7 q t + 13 6 15 6 15 7 17 7 19 8 3 q t + 4 q t + q t + 3 q t + q t