10 139

10_138

10_140

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10 139 Quick Notes

10 139 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X11,19,12,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8 X13,1,14,20 X19,13,20,12 X2,10,3,9
Gauss code	1, -10, 2, -1, -4, 6, -5, 7, 10, -2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8
Dowker-Thistlethwaite code	4 10 -14 -16 2 -18 -20 -6 -8 -12
Conway Notation	[4,3,3-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 4 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [7][-16] Hyperbolic Volume 4.85117 A-Polynomial See Data:10 139/A-polynomial

[edit Notes for 10 139's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $4$ Topological 4 genus $[3,4]$ Concordance genus $4$ Rasmussen s-Invariant -8

[edit Notes for 10 139's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-t^{3}+2t-3+2t^{-1}-t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+7z^{6}+14z^{4}+9z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 3, 6 }
Jones polynomial	$-q^{12}+q^{11}-q^{10}+q^{9}-q^{8}+q^{6}+q^{4}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-7z^{4}a^{-10}+21z^{2}a^{-8}-13z^{2}a^{-10}+z^{2}a^{-12}+6a^{-8}-6a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-8z^{6}a^{-10}-7z^{5}a^{-9}-7z^{5}a^{-11}+21z^{4}a^{-8}+20z^{4}a^{-10}+z^{4}a^{-14}+13z^{3}a^{-9}+13z^{3}a^{-11}+z^{3}a^{-13}+z^{3}a^{-15}-21z^{2}a^{-8}-19z^{2}a^{-10}-2z^{2}a^{-14}-6za^{-9}-5za^{-11}-za^{-13}-2za^{-15}+6a^{-8}+6a^{-10}+a^{-12}$
The A2 invariant	$q^{-14}+q^{-16}+2q^{-18}+2q^{-20}+q^{-22}-q^{-28}-q^{-32}-q^{-34}-q^{-36}-q^{-38}+q^{-40}$
The G2 invariant	$q^{-70}+q^{-72}+q^{-74}+q^{-76}+2q^{-80}+3q^{-82}+q^{-84}+q^{-86}+q^{-88}+3q^{-90}+3q^{-92}+2q^{-94}-2q^{-96}+2q^{-98}+3q^{-100}+q^{-102}-3q^{-106}+q^{-108}+2q^{-110}-2q^{-112}-3q^{-114}-3q^{-116}-q^{-118}+4q^{-120}-3q^{-122}-3q^{-124}-q^{-126}-q^{-128}+2q^{-130}-3q^{-132}-q^{-134}+2q^{-140}-q^{-152}-q^{-156}-q^{-158}+2q^{-160}-2q^{-162}-q^{-164}-3q^{-168}+q^{-170}-2q^{-172}+q^{-176}-q^{-178}+2q^{-180}+2q^{-186}-q^{-188}-q^{-190}+q^{-192}+q^{-196}$

Further Quantum Invariants

Further quantum knot invariants for 10_139.

A1 Invariants.

Weight	Invariant
1	$q^{-7}+q^{-9}+q^{-11}+q^{-13}-q^{-15}-q^{-25}$
2	$q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}+q^{-24}+q^{-26}-q^{-28}-q^{-30}-q^{-36}+2q^{-40}-q^{-44}-q^{-48}-q^{-50}-q^{-52}-q^{-56}+2q^{-60}-q^{-64}+q^{-68}$
3	$q^{-21}+q^{-23}+q^{-25}+q^{-27}+q^{-29}+q^{-31}+q^{-33}+q^{-35}+q^{-37}+q^{-39}-q^{-41}-q^{-43}-q^{-45}-q^{-53}-q^{-55}+2q^{-59}+2q^{-61}-2q^{-65}-q^{-67}+2q^{-69}-2q^{-73}-4q^{-75}+q^{-79}-q^{-81}-q^{-83}+q^{-87}-q^{-101}+3q^{-105}+q^{-107}-2q^{-109}+3q^{-113}+2q^{-115}-2q^{-117}-3q^{-119}+q^{-123}+q^{-125}-q^{-129}$
5	$q^{-35}+q^{-37}+q^{-39}+q^{-41}+q^{-43}+q^{-45}+q^{-47}+q^{-49}+q^{-51}+q^{-53}+q^{-55}+q^{-57}+q^{-59}+q^{-61}+q^{-63}+q^{-65}-q^{-67}-q^{-69}-q^{-71}-q^{-73}-q^{-75}-q^{-87}-q^{-89}-q^{-91}-q^{-93}+2q^{-97}+2q^{-99}+2q^{-101}+2q^{-103}-2q^{-107}-2q^{-109}-2q^{-111}-q^{-113}+2q^{-115}+2q^{-117}+2q^{-119}-2q^{-123}-5q^{-125}-5q^{-127}-4q^{-129}+3q^{-133}+5q^{-135}+3q^{-137}-2q^{-139}-6q^{-141}-7q^{-143}-3q^{-145}+2q^{-147}+7q^{-149}+9q^{-151}+4q^{-153}-4q^{-155}-9q^{-157}-7q^{-159}+7q^{-163}+11q^{-165}+4q^{-167}-6q^{-169}-11q^{-171}-7q^{-173}-q^{-175}+9q^{-177}+10q^{-179}-6q^{-183}-7q^{-185}+q^{-187}+7q^{-189}+7q^{-191}+2q^{-193}-4q^{-195}-4q^{-197}+q^{-199}+3q^{-201}+2q^{-203}-q^{-207}-q^{-209}+q^{-229}+q^{-231}-q^{-233}-3q^{-235}-2q^{-237}+q^{-239}+4q^{-241}+2q^{-243}-3q^{-245}-9q^{-247}-8q^{-249}+q^{-251}+10q^{-253}+12q^{-255}+2q^{-257}-9q^{-259}-12q^{-261}-7q^{-263}+6q^{-265}+13q^{-267}+10q^{-269}+q^{-271}-7q^{-273}-9q^{-275}-7q^{-277}+6q^{-281}+7q^{-283}+3q^{-285}-3q^{-289}-4q^{-291}-q^{-293}+q^{-297}+q^{-299}+q^{-301}-q^{-305}$
6	$q^{-42}+q^{-44}+q^{-46}+q^{-48}+q^{-50}+q^{-52}+q^{-54}+q^{-56}+q^{-58}+q^{-60}+q^{-62}+q^{-64}+q^{-66}+q^{-68}+q^{-70}+q^{-72}+q^{-74}+q^{-76}+q^{-78}-q^{-80}-q^{-82}-q^{-84}-q^{-86}-q^{-88}-q^{-90}-q^{-104}-q^{-106}-q^{-108}-q^{-110}-q^{-112}+2q^{-116}+2q^{-118}+2q^{-120}+2q^{-122}+2q^{-124}-2q^{-128}-2q^{-130}-2q^{-132}-2q^{-134}-q^{-136}+2q^{-138}+2q^{-140}+2q^{-142}+2q^{-144}-2q^{-148}-5q^{-150}-5q^{-152}-5q^{-154}-4q^{-156}+3q^{-160}+5q^{-162}+5q^{-164}+3q^{-166}-2q^{-168}-6q^{-170}-8q^{-172}-7q^{-174}-3q^{-176}+2q^{-178}+7q^{-180}+11q^{-182}+9q^{-184}+4q^{-186}-4q^{-188}-10q^{-190}-12q^{-192}-9q^{-194}+9q^{-198}+15q^{-200}+14q^{-202}+6q^{-204}-7q^{-206}-16q^{-208}-18q^{-210}-13q^{-212}+q^{-214}+15q^{-216}+22q^{-218}+15q^{-220}-13q^{-224}-22q^{-226}-17q^{-228}+19q^{-232}+25q^{-234}+17q^{-236}+q^{-238}-14q^{-240}-23q^{-242}-14q^{-244}+6q^{-246}+17q^{-248}+17q^{-250}+7q^{-252}-8q^{-254}-18q^{-256}-13q^{-258}+q^{-260}+11q^{-262}+11q^{-264}+6q^{-266}-3q^{-268}-10q^{-270}-6q^{-272}+q^{-274}+5q^{-276}+4q^{-278}+q^{-280}-q^{-282}-2q^{-284}-q^{-286}-q^{-294}-q^{-296}-q^{-298}-q^{-300}-q^{-302}-q^{-304}-q^{-306}-q^{-314}-2q^{-316}-q^{-318}+q^{-320}+4q^{-322}+5q^{-324}+q^{-326}-5q^{-328}-11q^{-330}-6q^{-332}+4q^{-334}+13q^{-336}+17q^{-338}+6q^{-340}-10q^{-342}-23q^{-344}-19q^{-346}-2q^{-348}+18q^{-350}+29q^{-352}+23q^{-354}+q^{-356}-22q^{-358}-28q^{-360}-21q^{-362}-3q^{-364}+18q^{-366}+27q^{-368}+21q^{-370}+2q^{-372}-12q^{-374}-19q^{-376}-19q^{-378}-8q^{-380}+5q^{-382}+12q^{-384}+12q^{-386}+9q^{-388}+2q^{-390}-5q^{-392}-8q^{-394}-6q^{-396}-3q^{-398}+3q^{-402}+4q^{-404}+2q^{-406}-q^{-412}-q^{-414}-q^{-416}+q^{-420}$

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-28}+q^{-30}+3q^{-32}+4q^{-34}+5q^{-36}+4q^{-38}+5q^{-40}+q^{-42}+q^{-44}-2q^{-46}-4q^{-48}-4q^{-50}-6q^{-52}-4q^{-54}-3q^{-56}-q^{-58}+q^{-60}+2q^{-62}+q^{-64}+q^{-66}-q^{-70}-q^{-74}+q^{-78}+q^{-82}$
1,0,0	$q^{-21}+q^{-23}+2q^{-25}+3q^{-27}+2q^{-29}+2q^{-31}+q^{-33}-q^{-37}-q^{-39}-2q^{-41}-2q^{-43}-2q^{-45}-q^{-47}+q^{-53}$
1,0,1	$q^{-42}+2q^{-44}+5q^{-46}+9q^{-48}+12q^{-50}+15q^{-52}+17q^{-54}+14q^{-56}+14q^{-58}+7q^{-60}+2q^{-62}-7q^{-64}-15q^{-66}-19q^{-68}-26q^{-70}-22q^{-72}-21q^{-74}-12q^{-76}-7q^{-78}+2q^{-80}+8q^{-82}+8q^{-84}+14q^{-86}+7q^{-88}+8q^{-90}+5q^{-92}+3q^{-94}-4q^{-98}-q^{-100}-6q^{-102}-2q^{-104}-q^{-106}+q^{-108}+2q^{-110}+q^{-112}+4q^{-114}-2q^{-116}-q^{-120}-2q^{-122}-q^{-124}+2q^{-128}+q^{-132}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-42}+q^{-44}+3q^{-46}+5q^{-48}+7q^{-50}+8q^{-52}+11q^{-54}+9q^{-56}+9q^{-58}+5q^{-60}+q^{-62}-4q^{-64}-8q^{-66}-13q^{-68}-14q^{-70}-12q^{-72}-10q^{-74}-6q^{-76}-2q^{-78}+4q^{-80}+4q^{-82}+4q^{-84}+4q^{-86}+q^{-88}-2q^{-90}-q^{-92}-q^{-94}-q^{-96}+q^{-98}+3q^{-100}+2q^{-102}+q^{-104}+q^{-106}+q^{-108}-q^{-110}-q^{-112}$
1,0,0,0	$q^{-28}+q^{-30}+2q^{-32}+3q^{-34}+3q^{-36}+3q^{-38}+3q^{-40}+q^{-42}+q^{-44}-q^{-46}-2q^{-48}-3q^{-50}-3q^{-52}-3q^{-54}-2q^{-56}-q^{-58}+q^{-62}+q^{-66}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-28}+q^{-30}+q^{-32}+2q^{-34}+q^{-36}+2q^{-38}+q^{-40}+q^{-42}+q^{-44}-2q^{-50}-2q^{-54}-q^{-56}-q^{-58}-q^{-60}-q^{-64}+q^{-66}+q^{-70}+q^{-74}-q^{-78}-q^{-82}$
1,0	$q^{-42}+q^{-46}+q^{-48}+2q^{-50}+q^{-52}+3q^{-54}+2q^{-56}+2q^{-58}+q^{-60}+2q^{-62}+q^{-64}+q^{-66}-q^{-68}-q^{-74}-2q^{-76}-q^{-78}-2q^{-82}-2q^{-84}-2q^{-86}-q^{-88}-2q^{-90}-q^{-92}-q^{-94}+q^{-100}+q^{-106}+q^{-108}-q^{-112}-q^{-120}+q^{-124}+q^{-132}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-42}+q^{-44}+2q^{-46}+4q^{-48}+4q^{-50}+6q^{-52}+6q^{-54}+6q^{-56}+5q^{-58}+4q^{-60}+q^{-62}-2q^{-64}-4q^{-66}-7q^{-68}-6q^{-70}-8q^{-72}-6q^{-74}-5q^{-76}-2q^{-78}+q^{-82}+2q^{-84}+2q^{-86}+2q^{-88}+q^{-90}+q^{-92}-q^{-94}-q^{-98}-q^{-102}+q^{-110}+q^{-114}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-70}+q^{-72}+q^{-74}+q^{-76}+2q^{-80}+3q^{-82}+q^{-84}+q^{-86}+q^{-88}+3q^{-90}+3q^{-92}+2q^{-94}-2q^{-96}+2q^{-98}+3q^{-100}+q^{-102}-3q^{-106}+q^{-108}+2q^{-110}-2q^{-112}-3q^{-114}-3q^{-116}-q^{-118}+4q^{-120}-3q^{-122}-3q^{-124}-q^{-126}-q^{-128}+2q^{-130}-3q^{-132}-q^{-134}+2q^{-140}-q^{-152}-q^{-156}-q^{-158}+2q^{-160}-2q^{-162}-q^{-164}-3q^{-168}+q^{-170}-2q^{-172}+q^{-176}-q^{-178}+2q^{-180}+2q^{-186}-q^{-188}-q^{-190}+q^{-192}+q^{-196}$

.

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["10 139"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+7z^{6}+14z^{4}+9z^{2}+1$

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]=

{ 3, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{12}+q^{11}-q^{10}+q^{9}-q^{8}+q^{6}+q^{4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-7z^{4}a^{-10}+21z^{2}a^{-8}-13z^{2}a^{-10}+z^{2}a^{-12}+6a^{-8}-6a^{-10}+a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-8z^{6}a^{-10}-7z^{5}a^{-9}-7z^{5}a^{-11}+21z^{4}a^{-8}+20z^{4}a^{-10}+z^{4}a^{-14}+13z^{3}a^{-9}+13z^{3}a^{-11}+z^{3}a^{-13}+z^{3}a^{-15}-21z^{2}a^{-8}-19z^{2}a^{-10}-2z^{2}a^{-14}-6za^{-9}-5za^{-11}-za^{-13}-2za^{-15}+6a^{-8}+6a^{-10}+a^{-12}$

Vassiliev invariants

V2 and V3:

(9, 25)

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 $36$ $200$ $648$ $1466$ $206$ $7200$ ${\frac {35888}{3}}$ ${\frac {6272}{3}}$ $1384$ $7776$ $20000$ $52776$ $7416$ ${\frac {1001773}{10}}$ ${\frac {77402}{15}}$ ${\frac {523826}{15}}$ ${\frac {3059}{6}}$ ${\frac {43373}{10}}$

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=$6 is the signature of 10 139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	0	1	2	3	4	5	6	7	8	9	χ
25										1	-1
23											0
21								1	1		0
19						1	1				0
17						1	1				0
15				1	1	1					-1
13					1						1
11			1								1
9	1										1
7	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, 139]]
Out[2]=	10
In[3]:=	PD[Knot[10, 139]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[11, 19, 12, 18], X[5, 15, 6, 14], X[7, 17, 8, 16], X[15, 7, 16, 6], X[17, 9, 18, 8], X[13, 1, 14, 20], X[19, 13, 20, 12], X[2, 10, 3, 9]]
In[4]:=	GaussCode[Knot[10, 139]]
Out[4]=	GaussCode[1, -10, 2, -1, -4, 6, -5, 7, 10, -2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8]
In[5]:=	BR[Knot[10, 139]]
Out[5]=	BR[3, {1, 1, 1, 1, 2, 1, 1, 1, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 139]][t]
Out[6]=	-4 -3 2 3 4 -3 + t - t + - + 2 t - t + t t
In[7]:=	Conway[Knot[10, 139]][z]
Out[7]=	2 4 6 8 1 + 9 z + 14 z + 7 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 139]}
In[9]:=	{KnotDet[Knot[10, 139]], KnotSignature[Knot[10, 139]]}
Out[9]=	{3, 6}
In[10]:=	J=Jones[Knot[10, 139]][q]
Out[10]=	4 6 8 9 10 11 12 q + q - q + q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 139]}
In[12]:=	A2Invariant[Knot[10, 139]][q]
Out[12]=	14 16 18 20 22 28 32 34 36 38 40 q + q + 2 q + 2 q + q - q - q - q - q - q + q
In[13]:=	Kauffman[Knot[10, 139]][a, z]
Out[13]=	2 2 2 3 -12 6 6 2 z z 5 z 6 z 2 z 19 z 21 z z a + --- + -- - --- - --- - --- - --- - ---- - ----- - ----- + --- + 10 8 15 13 11 9 14 10 8 15 a a a a a a a a a a 3 3 3 4 4 4 5 5 6 z 13 z 13 z z 20 z 21 z 7 z 7 z 8 z --- + ----- + ----- + --- + ----- + ----- - ---- - ---- - ---- - 13 11 9 14 10 8 11 9 10 a a a a a a a a a 6 7 7 8 8 8 z z z z z ---- + --- + -- + --- + -- 8 11 9 10 8 a a a a a
In[14]:=	{Vassiliev[2][Knot[10, 139]], Vassiliev[3][Knot[10, 139]]}
Out[14]=	{0, 25}
In[15]:=	Kh[Knot[10, 139]][q, t]
Out[15]=	7 9 11 2 15 3 13 4 15 4 15 5 17 5 q + q + q t + q t + q t + q t + q t + q t + 19 5 17 6 19 6 21 7 21 8 25 9 q t + q t + q t + q t + q t + q t