9 21

9_20

9_22

Visit 9 21's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 21's page at Knotilus! Visit 9 21's page at the original Knot Atlas!

9 21 Quick Notes

9 21 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8
Gauss code	-1, 4, -3, 1, -6, 7, -8, 9, -2, 3, -4, 2, -5, 6, -9, 8, -7, 5
Dowker-Thistlethwaite code	4 10 14 16 12 2 18 8 6
Conway Notation	[31122]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 10.1833 A-Polynomial See Data:9 21/A-polynomial

[edit Notes for 9 21'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 9 21's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+11t-17+11t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, 2 }
Jones polynomial	$-q^{8}+2q^{7}-4q^{6}+6q^{5}-7q^{4}+8q^{3}-6q^{2}+5q-3+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-z^{4}a^{-4}+2z^{2}a^{-6}+z^{2}+a^{-2}+a^{-6}-a^{-8}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-4}+z^{8}a^{-6}+3z^{7}a^{-3}+5z^{7}a^{-5}+2z^{7}a^{-7}+4z^{6}a^{-2}+4z^{6}a^{-4}+2z^{6}a^{-6}+2z^{6}a^{-8}+3z^{5}a^{-1}-3z^{5}a^{-3}-10z^{5}a^{-5}-3z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-2}-9z^{4}a^{-4}-7z^{4}a^{-6}-5z^{4}a^{-8}+z^{4}-4z^{3}a^{-1}+2z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-9}+3z^{2}a^{-2}+6z^{2}a^{-4}+5z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-za^{-3}-3za^{-5}+2za^{-9}-a^{-2}-a^{-6}-a^{-8}$
The A2 invariant	$q^{4}-q^{2}-1+2q^{-2}-q^{-4}+2q^{-6}+q^{-8}+q^{-12}-q^{-14}+2q^{-16}-q^{-20}+q^{-22}-q^{-24}-q^{-26}$
The G2 invariant	$q^{18}-2q^{16}+4q^{14}-6q^{12}+4q^{10}-q^{8}-4q^{6}+13q^{4}-17q^{2}+22-19q^{-2}+5q^{-4}+10q^{-6}-27q^{-8}+38q^{-10}-39q^{-12}+28q^{-14}-7q^{-16}-17q^{-18}+36q^{-20}-40q^{-22}+30q^{-24}-9q^{-26}-13q^{-28}+24q^{-30}-21q^{-32}+8q^{-34}+18q^{-36}-33q^{-38}+40q^{-40}-24q^{-42}-4q^{-44}+36q^{-46}-58q^{-48}+63q^{-50}-46q^{-52}+17q^{-54}+18q^{-56}-45q^{-58}+57q^{-60}-50q^{-62}+27q^{-64}-25q^{-68}+32q^{-70}-23q^{-72}+7q^{-74}+16q^{-76}-28q^{-78}+26q^{-80}-10q^{-82}-14q^{-84}+36q^{-86}-44q^{-88}+36q^{-90}-17q^{-92}-8q^{-94}+27q^{-96}-37q^{-98}+35q^{-100}-23q^{-102}+6q^{-104}+6q^{-106}-16q^{-108}+16q^{-110}-14q^{-112}+9q^{-114}-3q^{-116}-2q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 9_21.

A1 Invariants.

Weight	Invariant
1	$q^{3}-2q+2q^{-1}-q^{-3}+2q^{-5}+q^{-7}-q^{-9}+2q^{-11}-2q^{-13}+q^{-15}-q^{-17}$
2	$q^{10}-2q^{8}-q^{6}+6q^{4}-4q^{2}-5+11q^{-2}-3q^{-4}-9q^{-6}+11q^{-8}+q^{-10}-8q^{-12}+5q^{-14}+4q^{-16}-2q^{-18}-5q^{-20}+6q^{-22}+5q^{-24}-11q^{-26}+3q^{-28}+8q^{-30}-11q^{-32}-q^{-34}+8q^{-36}-5q^{-38}-2q^{-40}+4q^{-42}-q^{-44}-q^{-46}+q^{-48}$
3	$q^{21}-2q^{19}-q^{17}+3q^{15}+3q^{13}-4q^{11}-8q^{9}+8q^{7}+13q^{5}-9q^{3}-21q+7q^{-1}+32q^{-3}-3q^{-5}-40q^{-7}-5q^{-9}+46q^{-11}+14q^{-13}-41q^{-15}-22q^{-17}+38q^{-19}+26q^{-21}-25q^{-23}-28q^{-25}+13q^{-27}+25q^{-29}+2q^{-31}-21q^{-33}-16q^{-35}+20q^{-37}+26q^{-39}-12q^{-41}-38q^{-43}+6q^{-45}+42q^{-47}+q^{-49}-46q^{-51}-10q^{-53}+41q^{-55}+19q^{-57}-36q^{-59}-24q^{-61}+24q^{-63}+27q^{-65}-13q^{-67}-23q^{-69}+5q^{-71}+17q^{-73}+q^{-75}-11q^{-77}-2q^{-79}+6q^{-81}+2q^{-83}-3q^{-85}-q^{-87}+q^{-89}+q^{-91}-q^{-93}$
4	$q^{36}-2q^{34}-q^{32}+3q^{30}+3q^{26}-7q^{24}-3q^{22}+9q^{20}+q^{18}+9q^{16}-22q^{14}-16q^{12}+22q^{10}+21q^{8}+29q^{6}-50q^{4}-59q^{2}+16+62q^{-2}+97q^{-4}-54q^{-6}-135q^{-8}-51q^{-10}+79q^{-12}+191q^{-14}+6q^{-16}-168q^{-18}-144q^{-20}+28q^{-22}+231q^{-24}+92q^{-26}-118q^{-28}-179q^{-30}-51q^{-32}+173q^{-34}+128q^{-36}-27q^{-38}-136q^{-40}-97q^{-42}+68q^{-44}+109q^{-46}+52q^{-48}-59q^{-50}-104q^{-52}-38q^{-54}+77q^{-56}+114q^{-58}+5q^{-60}-107q^{-62}-126q^{-64}+47q^{-66}+162q^{-68}+69q^{-70}-94q^{-72}-198q^{-74}-2q^{-76}+178q^{-78}+135q^{-80}-42q^{-82}-224q^{-84}-75q^{-86}+123q^{-88}+168q^{-90}+50q^{-92}-170q^{-94}-123q^{-96}+20q^{-98}+125q^{-100}+110q^{-102}-65q^{-104}-96q^{-106}-50q^{-108}+42q^{-110}+91q^{-112}+4q^{-114}-33q^{-116}-47q^{-118}-8q^{-120}+39q^{-122}+13q^{-124}+q^{-126}-19q^{-128}-11q^{-130}+11q^{-132}+3q^{-134}+4q^{-136}-4q^{-138}-4q^{-140}+3q^{-142}+q^{-146}-q^{-148}-q^{-150}+q^{-152}$
5	$q^{55}-2q^{53}-q^{51}+3q^{49}-2q^{41}-2q^{39}+5q^{37}+4q^{35}-6q^{33}-8q^{31}-5q^{29}+6q^{27}+18q^{25}+24q^{23}-3q^{21}-46q^{19}-51q^{17}-10q^{15}+62q^{13}+109q^{11}+65q^{9}-75q^{7}-195q^{5}-154q^{3}+44q+271q^{-1}+307q^{-3}+54q^{-5}-326q^{-7}-489q^{-9}-214q^{-11}+313q^{-13}+652q^{-15}+448q^{-17}-214q^{-19}-771q^{-21}-680q^{-23}+38q^{-25}+782q^{-27}+884q^{-29}+188q^{-31}-703q^{-33}-984q^{-35}-407q^{-37}+529q^{-39}+991q^{-41}+568q^{-43}-320q^{-45}-875q^{-47}-655q^{-49}+101q^{-51}+710q^{-53}+657q^{-55}+70q^{-57}-492q^{-59}-598q^{-61}-214q^{-63}+299q^{-65}+509q^{-67}+303q^{-69}-122q^{-71}-417q^{-73}-373q^{-75}-36q^{-77}+353q^{-79}+446q^{-81}+147q^{-83}-302q^{-85}-520q^{-87}-281q^{-89}+266q^{-91}+622q^{-93}+398q^{-95}-228q^{-97}-701q^{-99}-552q^{-101}+156q^{-103}+774q^{-105}+703q^{-107}-39q^{-109}-782q^{-111}-844q^{-113}-126q^{-115}+722q^{-117}+932q^{-119}+326q^{-121}-568q^{-123}-957q^{-125}-506q^{-127}+348q^{-129}+865q^{-131}+645q^{-133}-88q^{-135}-699q^{-137}-689q^{-139}-133q^{-141}+458q^{-143}+632q^{-145}+303q^{-147}-218q^{-149}-501q^{-151}-364q^{-153}+17q^{-155}+324q^{-157}+340q^{-159}+111q^{-161}-160q^{-163}-263q^{-165}-154q^{-167}+40q^{-169}+161q^{-171}+142q^{-173}+28q^{-175}-80q^{-177}-102q^{-179}-46q^{-181}+26q^{-183}+60q^{-185}+38q^{-187}-q^{-189}-27q^{-191}-27q^{-193}-5q^{-195}+13q^{-197}+12q^{-199}+3q^{-201}-q^{-203}-6q^{-205}-4q^{-207}+3q^{-209}+2q^{-211}-q^{-213}-q^{-219}+q^{-221}+q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}-q^{2}-1+2q^{-2}-q^{-4}+2q^{-6}+q^{-8}+q^{-12}-q^{-14}+2q^{-16}-q^{-20}+q^{-22}-q^{-24}-q^{-26}$
1,1	$q^{12}-4q^{10}+10q^{8}-20q^{6}+34q^{4}-52q^{2}+78-104q^{-2}+124q^{-4}-142q^{-6}+152q^{-8}-140q^{-10}+113q^{-12}-60q^{-14}+2q^{-16}+74q^{-18}-150q^{-20}+220q^{-22}-268q^{-24}+298q^{-26}-297q^{-28}+272q^{-30}-228q^{-32}+162q^{-34}-89q^{-36}+14q^{-38}+50q^{-40}-100q^{-42}+136q^{-44}-152q^{-46}+144q^{-48}-130q^{-50}+106q^{-52}-82q^{-54}+56q^{-56}-36q^{-58}+23q^{-60}-12q^{-62}+6q^{-64}-2q^{-66}+q^{-68}$
2,0	$q^{12}-q^{10}-2q^{8}+q^{6}+4q^{4}+q^{2}-7-q^{-2}+8q^{-4}-7q^{-8}+2q^{-10}+8q^{-12}-4q^{-16}+2q^{-18}+4q^{-20}-3q^{-22}+q^{-24}+2q^{-26}-2q^{-28}+2q^{-30}+7q^{-32}-q^{-34}-6q^{-36}+q^{-38}+5q^{-40}-4q^{-42}-9q^{-44}+q^{-46}+5q^{-48}-q^{-50}-4q^{-52}+3q^{-56}-2q^{-60}+q^{-64}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{8}-2q^{6}+4q^{2}-5+q^{-2}+8q^{-4}-8q^{-6}-q^{-8}+9q^{-10}-6q^{-12}-2q^{-14}+8q^{-16}+2q^{-22}+5q^{-24}-q^{-26}-6q^{-28}+6q^{-30}+q^{-32}-10q^{-34}+6q^{-36}+2q^{-38}-9q^{-40}+3q^{-42}+q^{-44}-4q^{-46}+2q^{-48}+q^{-50}-q^{-52}+q^{-54}$
1,0,0	$q^{5}-q^{3}-q^{-1}+2q^{-3}-q^{-5}+2q^{-7}+q^{-9}+q^{-11}+q^{-17}-q^{-19}+2q^{-21}+q^{-25}-q^{-27}+q^{-29}-q^{-31}-q^{-33}-q^{-35}$

B2 Invariants.

Weight	Invariant
0,1	$q^{8}-2q^{6}+4q^{4}-6q^{2}+7-9q^{-2}+10q^{-4}-8q^{-6}+7q^{-8}-3q^{-10}+6q^{-14}-10q^{-16}+14q^{-18}-16q^{-20}+18q^{-22}-17q^{-24}+15q^{-26}-10q^{-28}+6q^{-30}-q^{-32}-2q^{-34}+6q^{-36}-8q^{-38}+9q^{-40}-9q^{-42}+7q^{-44}-6q^{-46}+4q^{-48}-3q^{-50}+q^{-52}-q^{-54}$
1,0	$q^{14}-2q^{10}-2q^{8}+2q^{6}+5q^{4}-6-4q^{-2}+6q^{-4}+9q^{-6}-2q^{-8}-10q^{-10}-4q^{-12}+8q^{-14}+7q^{-16}-4q^{-18}-8q^{-20}+2q^{-22}+8q^{-24}+2q^{-26}-6q^{-28}-q^{-30}+7q^{-32}+5q^{-34}-4q^{-36}-4q^{-38}+4q^{-40}+6q^{-42}-3q^{-44}-7q^{-46}+q^{-48}+8q^{-50}+q^{-52}-9q^{-54}-7q^{-56}+6q^{-58}+9q^{-60}-2q^{-62}-10q^{-64}-4q^{-66}+6q^{-68}+5q^{-70}-2q^{-72}-5q^{-74}-q^{-76}+3q^{-78}+2q^{-80}-q^{-82}-q^{-84}+q^{-88}$

G2 Invariants.

Weight

Invariant

1,0

$q^{18}-2q^{16}+4q^{14}-6q^{12}+4q^{10}-q^{8}-4q^{6}+13q^{4}-17q^{2}+22-19q^{-2}+5q^{-4}+10q^{-6}-27q^{-8}+38q^{-10}-39q^{-12}+28q^{-14}-7q^{-16}-17q^{-18}+36q^{-20}-40q^{-22}+30q^{-24}-9q^{-26}-13q^{-28}+24q^{-30}-21q^{-32}+8q^{-34}+18q^{-36}-33q^{-38}+40q^{-40}-24q^{-42}-4q^{-44}+36q^{-46}-58q^{-48}+63q^{-50}-46q^{-52}+17q^{-54}+18q^{-56}-45q^{-58}+57q^{-60}-50q^{-62}+27q^{-64}-25q^{-68}+32q^{-70}-23q^{-72}+7q^{-74}+16q^{-76}-28q^{-78}+26q^{-80}-10q^{-82}-14q^{-84}+36q^{-86}-44q^{-88}+36q^{-90}-17q^{-92}-8q^{-94}+27q^{-96}-37q^{-98}+35q^{-100}-23q^{-102}+6q^{-104}+6q^{-106}-16q^{-108}+16q^{-110}-14q^{-112}+9q^{-114}-3q^{-116}-2q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

.

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 21"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+11t-17+11t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$-q^{8}+2q^{7}-4q^{6}+6q^{5}-7q^{4}+8q^{3}-6q^{2}+5q-3+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(3, 6)

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 $12$ $48$ $72$ $238$ $50$ $576$ $1248$ $192$ $272$ $288$ $1152$ $2856$ $600$ ${\frac {65311}{10}}$ $-{\frac {3182}{5}}$ ${\frac {18154}{5}}$ ${\frac {235}{2}}$ ${\frac {5631}{10}}$

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 9 21. 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	χ
17										1	-1
15									1		1
13								3	1		-2
11							3	1			2
9						4	3				-1
7					4	3					1
5				2	4						2
3			3	4							-1
1		1	3								2
-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.

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 21]]
Out[2]=	9
In[3]:=	PD[Knot[9, 21]]
Out[3]=	PD[X[1, 4, 2, 5], X[9, 12, 10, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[13, 1, 14, 18], X[5, 15, 6, 14], X[17, 7, 18, 6], X[7, 17, 8, 16], X[15, 9, 16, 8]]
In[4]:=	GaussCode[Knot[9, 21]]
Out[4]=	GaussCode[-1, 4, -3, 1, -6, 7, -8, 9, -2, 3, -4, 2, -5, 6, -9, 8, -7, 5]
In[5]:=	BR[Knot[9, 21]]
Out[5]=	BR[5, {1, 1, 2, -1, 2, -3, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[9, 21]][t]
Out[6]=	2 11 2 -17 - -- + -- + 11 t - 2 t 2 t t
In[7]:=	Conway[Knot[9, 21]][z]
Out[7]=	2 4 1 + 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 21]}
In[9]:=	{KnotDet[Knot[9, 21]], KnotSignature[Knot[9, 21]]}
Out[9]=	{43, 2}
In[10]:=	J=Jones[Knot[9, 21]][q]
Out[10]=	1 2 3 4 5 6 7 8 -3 + - + 5 q - 6 q + 8 q - 7 q + 6 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 21], Knot[11, NonAlternating, 129]}
In[12]:=	A2Invariant[Knot[9, 21]][q]
Out[12]=	-4 -2 2 4 6 8 12 14 16 20 -1 + q - q + 2 q - q + 2 q + q + q - q + 2 q - q + 22 24 26 q - q - q
In[13]:=	Kauffman[Knot[9, 21]][a, z]
Out[13]=	2 2 2 2 -8 -6 -2 2 z 3 z z 2 3 z 5 z 6 z 3 z -a - a - a + --- - --- - -- - z + ---- + ---- + ---- + ---- - 9 5 3 8 6 4 2 a a a a a a a 3 3 3 3 4 4 4 4 5 3 z 9 z 2 z 4 z 4 5 z 7 z 9 z 6 z z ---- + ---- + ---- - ---- + z - ---- - ---- - ---- - ---- + -- - 9 5 3 a 8 6 4 2 9 a a a a a a a a 5 5 5 5 6 6 6 6 7 3 z 10 z 3 z 3 z 2 z 2 z 4 z 4 z 2 z ---- - ----- - ---- + ---- + ---- + ---- + ---- + ---- + ---- + 7 5 3 a 8 6 4 2 7 a a a a a a a a 7 7 8 8 5 z 3 z z z ---- + ---- + -- + -- 5 3 6 4 a a a a
In[14]:=	{Vassiliev[2][Knot[9, 21]], Vassiliev[3][Knot[9, 21]]}
Out[14]=	{0, 6}
In[15]:=	Kh[Knot[9, 21]][q, t]
Out[15]=	3 1 2 q 3 5 5 2 7 2 3 q + 3 q + ----- + --- + - + 4 q t + 2 q t + 4 q t + 4 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 3 q t + 4 q t + 3 q t + 3 q t + q t + 3 q t + q t + 15 6 17 7 q t + q t