10 8

10_7

10_9

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10 8 Quick Notes

10 8 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X7,16,8,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X9,18,10,19 X11,20,12,1 X17,8,18,9 X19,10,20,11
Gauss code	-1, 6, -4, 5, -3, 1, -2, 9, -7, 10, -8, 3, -5, 4, -6, 2, -9, 7, -10, 8
Dowker-Thistlethwaite code	6 14 12 16 18 20 4 2 8 10
Conway Notation	[514]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-11][-1] Hyperbolic Volume 6.08323 A-Polynomial See Data:10 8/A-polynomial

[edit Notes for 10 8's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for 10 8's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+5t^{2}-5t+5-5t^{-1}+5t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-7z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 29, -4 }
Jones polynomial	$q^{2}-q+2-3q^{-1}+4q^{-2}-4q^{-3}+4q^{-4}-4q^{-5}+3q^{-6}-2q^{-7}+q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{6}+3z^{2}a^{6}+a^{6}-z^{6}a^{4}-4z^{4}a^{4}-3z^{2}a^{4}-z^{6}a^{2}-5z^{4}a^{2}-7z^{2}a^{2}-3a^{2}+z^{4}+4z^{2}+3$
Kauffman polynomial (db, data sources)	$z^{2}a^{10}+2z^{3}a^{9}+3z^{4}a^{8}-2z^{2}a^{8}+4z^{5}a^{7}-7z^{3}a^{7}+2za^{7}+4z^{6}a^{6}-10z^{4}a^{6}+5z^{2}a^{6}-a^{6}+3z^{7}a^{5}-8z^{5}a^{5}+2z^{3}a^{5}+za^{5}+2z^{8}a^{4}-6z^{6}a^{4}+z^{4}a^{4}+3z^{2}a^{4}+z^{9}a^{3}-3z^{7}a^{3}-z^{5}a^{3}+5z^{3}a^{3}-za^{3}+3z^{8}a^{2}-17z^{6}a^{2}+30z^{4}a^{2}-18z^{2}a^{2}+3a^{2}+z^{9}a-6z^{7}a+11z^{5}a-6z^{3}a+z^{8}-7z^{6}+16z^{4}-13z^{2}+3$
The A2 invariant	$q^{24}+q^{14}-q^{12}-q^{8}-q^{6}+1+q^{-2}+q^{-4}+q^{-6}$
The G2 invariant	$q^{134}-q^{132}+q^{130}-q^{128}-q^{122}+2q^{120}-2q^{118}+2q^{116}-2q^{114}+q^{110}-q^{108}+3q^{106}-3q^{104}+2q^{102}-q^{100}-q^{98}+2q^{96}-3q^{94}+3q^{92}-2q^{90}+q^{88}+q^{86}-q^{84}+2q^{82}+q^{78}+q^{74}+2q^{70}+q^{68}+q^{66}+q^{62}-q^{60}-3q^{54}+3q^{52}-5q^{50}+3q^{48}-q^{46}-5q^{44}+5q^{42}-7q^{40}+2q^{38}-2q^{36}-2q^{34}+2q^{32}-3q^{30}+3q^{28}-q^{26}-2q^{20}+q^{18}+q^{16}-q^{14}+2q^{12}-3q^{10}+3q^{8}+q^{6}-3q^{4}+6q^{2}-6+4q^{-2}+q^{-4}-3q^{-6}+6q^{-8}-4q^{-10}+4q^{-12}+2q^{-18}-2q^{-20}+2q^{-22}+q^{-26}$

Further Quantum Invariants

Further quantum knot invariants for 10_8.

A1 Invariants.

Weight	Invariant
1	$q^{17}-q^{15}+q^{13}-q^{11}+q^{3}-q+q^{-1}+q^{-5}$
2	$q^{46}-q^{44}+q^{40}-2q^{38}+q^{36}+q^{34}-q^{32}-q^{26}+q^{22}-q^{20}+2q^{16}+q^{14}-q^{12}+q^{8}-q^{6}-q^{4}+2q^{2}-q^{-2}+2q^{-4}-2q^{-8}+q^{-10}+q^{-12}-q^{-14}+q^{-18}$
3	$q^{87}-q^{85}+q^{73}-q^{71}-q^{69}-q^{67}+q^{65}+q^{63}-2q^{59}+4q^{55}+2q^{53}-3q^{51}-2q^{49}+q^{47}+2q^{45}+q^{43}-q^{41}-2q^{39}-q^{37}+3q^{35}+q^{33}-3q^{31}-3q^{29}+q^{27}+q^{25}-q^{23}-q^{21}+q^{19}+q^{17}+q^{15}+q^{11}+2q^{9}+q^{7}-q^{5}-q^{3}+q+2q^{-1}-3q^{-5}-2q^{-7}+2q^{-9}+3q^{-11}-3q^{-15}-q^{-17}+3q^{-19}+2q^{-21}-q^{-23}-3q^{-25}+2q^{-29}+q^{-31}-q^{-33}-q^{-35}+q^{-39}$
4	$q^{140}-q^{138}-q^{132}+2q^{130}-q^{128}+q^{126}-q^{124}-2q^{122}+q^{120}-2q^{118}+3q^{116}+2q^{114}-2q^{112}-2q^{110}-2q^{108}+6q^{106}+5q^{104}-2q^{102}-6q^{100}-7q^{98}+6q^{96}+9q^{94}+2q^{92}-6q^{90}-11q^{88}+q^{86}+8q^{84}+6q^{82}-q^{80}-10q^{78}-5q^{76}+2q^{74}+6q^{72}+8q^{70}-q^{68}-7q^{66}-7q^{64}+9q^{60}+7q^{58}-q^{56}-8q^{54}-6q^{52}+4q^{50}+8q^{48}+2q^{46}-4q^{44}-4q^{42}+2q^{40}+3q^{38}-2q^{36}-4q^{34}-2q^{32}+3q^{30}+5q^{28}-2q^{26}-5q^{24}-3q^{22}+2q^{20}+7q^{18}+2q^{16}-q^{14}-4q^{12}-5q^{10}+3q^{8}+4q^{6}+5q^{4}+q^{2}-7-3q^{-2}+6q^{-6}+7q^{-8}-2q^{-10}-4q^{-12}-6q^{-14}+7q^{-18}+4q^{-20}+q^{-22}-6q^{-24}-6q^{-26}+3q^{-30}+7q^{-32}+q^{-34}-4q^{-36}-4q^{-38}-3q^{-40}+4q^{-42}+4q^{-44}+2q^{-46}-q^{-48}-5q^{-50}-q^{-52}+q^{-54}+2q^{-56}+2q^{-58}-q^{-60}-q^{-62}-q^{-64}+q^{-68}$
5	$q^{205}-q^{203}-q^{197}+q^{195}+q^{193}-q^{189}-q^{187}-2q^{185}+2q^{181}+3q^{179}+q^{177}-2q^{175}-4q^{173}-2q^{171}+5q^{169}+8q^{167}+2q^{165}-8q^{163}-11q^{161}-3q^{159}+10q^{157}+16q^{155}+3q^{153}-15q^{151}-16q^{149}-4q^{147}+13q^{145}+19q^{143}+5q^{141}-15q^{139}-20q^{137}-5q^{135}+14q^{133}+21q^{131}+8q^{129}-12q^{127}-24q^{125}-12q^{123}+11q^{121}+22q^{119}+15q^{117}-3q^{115}-16q^{113}-18q^{111}-6q^{109}+10q^{107}+16q^{105}+13q^{103}+3q^{101}-11q^{99}-19q^{97}-15q^{95}+4q^{93}+17q^{91}+20q^{89}+8q^{87}-11q^{85}-20q^{83}-11q^{81}+5q^{79}+14q^{77}+10q^{75}-8q^{71}-7q^{69}+q^{67}+3q^{65}-5q^{61}-7q^{59}+q^{57}+11q^{55}+10q^{53}-13q^{49}-16q^{47}-3q^{45}+15q^{43}+20q^{41}+10q^{39}-9q^{37}-20q^{35}-14q^{33}+4q^{31}+17q^{29}+16q^{27}+3q^{25}-14q^{23}-17q^{21}-7q^{19}+7q^{17}+14q^{15}+10q^{13}+q^{11}-9q^{9}-11q^{7}-4q^{5}+3q^{3}+8q+10q^{-1}+5q^{-3}-4q^{-5}-9q^{-7}-9q^{-9}-4q^{-11}+4q^{-13}+12q^{-15}+11q^{-17}+q^{-19}-8q^{-21}-13q^{-23}-9q^{-25}+q^{-27}+12q^{-29}+14q^{-31}+6q^{-33}-4q^{-35}-12q^{-37}-12q^{-39}-4q^{-41}+7q^{-43}+12q^{-45}+9q^{-47}+2q^{-49}-7q^{-51}-11q^{-53}-8q^{-55}+7q^{-59}+9q^{-61}+6q^{-63}-q^{-65}-6q^{-67}-8q^{-69}-4q^{-71}+2q^{-73}+5q^{-75}+6q^{-77}+3q^{-79}-2q^{-81}-5q^{-83}-3q^{-85}-q^{-87}+q^{-89}+3q^{-91}+2q^{-93}-q^{-97}-q^{-99}-q^{-101}+q^{-105}$

A2 Invariants.

Weight	Invariant
1,0	$q^{24}+q^{14}-q^{12}-q^{8}-q^{6}+1+q^{-2}+q^{-4}+q^{-6}$
1,1	$q^{68}-2q^{66}+2q^{64}-2q^{62}+3q^{60}-4q^{58}+4q^{56}-2q^{54}+5q^{52}-6q^{50}+6q^{48}-6q^{46}+5q^{44}-8q^{42}+2q^{40}-2q^{38}+2q^{34}-2q^{32}+4q^{30}-2q^{28}+8q^{26}-6q^{24}+10q^{22}-14q^{20}+16q^{18}-18q^{16}+16q^{14}-15q^{12}+16q^{10}-10q^{8}+6q^{6}-3q^{4}-2q^{2}+6-12q^{-2}+13q^{-4}-14q^{-6}+14q^{-8}-10q^{-10}+8q^{-12}-4q^{-14}+4q^{-16}+q^{-20}$
2,0	$q^{60}+q^{50}-q^{46}-q^{44}-2q^{38}-q^{36}+q^{34}-2q^{30}+2q^{26}+q^{24}+2q^{22}+2q^{20}+2q^{18}+q^{16}+q^{14}-2q^{10}-q^{8}-q^{4}-2q^{2}+q^{-2}-q^{-6}+q^{-10}+q^{-12}+q^{-16}+q^{-18}+q^{-20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{56}-q^{54}-q^{52}+2q^{50}-2q^{46}+q^{44}-2q^{40}+q^{38}+q^{36}+q^{32}+2q^{30}+q^{28}+q^{24}+q^{22}-q^{20}-2q^{18}-2q^{14}-2q^{12}-2q^{8}+q^{4}+1+2q^{-2}+q^{-4}+q^{-6}+2q^{-8}+q^{-12}$
1,0,0	$q^{31}+q^{27}-q^{25}+q^{23}+q^{19}-q^{13}-2q^{11}-q^{9}-2q^{7}+2q+q^{-1}+2q^{-3}+q^{-5}+q^{-7}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{70}-q^{66}+q^{62}-q^{58}+q^{54}-2q^{50}-q^{44}+q^{40}+2q^{38}+2q^{36}+3q^{34}+2q^{32}+q^{30}+q^{26}-q^{24}-3q^{22}-q^{16}-2q^{14}-q^{12}-2q^{10}-3q^{8}-3q^{6}-2q^{4}+1+3q^{-2}+3q^{-4}+4q^{-6}+3q^{-8}+2q^{-10}+q^{-12}+q^{-14}$
1,0,0,0	$q^{38}+q^{34}+q^{28}+q^{24}+q^{20}-q^{18}-q^{16}-2q^{14}-2q^{12}-2q^{10}-2q^{8}+2q^{2}+2+2q^{-2}+2q^{-4}+q^{-6}+q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$q^{56}-q^{54}+q^{52}-2q^{50}+2q^{48}-2q^{46}+3q^{44}-2q^{42}+2q^{40}-q^{38}+q^{36}-q^{32}+2q^{30}-3q^{28}+4q^{26}-5q^{24}+5q^{22}-5q^{20}+4q^{18}-4q^{16}+2q^{14}-2q^{12}-2q^{6}+3q^{4}-2q^{2}+3-2q^{-2}+3q^{-4}-q^{-6}+2q^{-8}+q^{-12}$
1,0	$q^{90}-q^{86}-q^{84}+2q^{80}+q^{78}-q^{76}-2q^{74}-q^{72}+2q^{70}+q^{68}-q^{66}-2q^{64}+2q^{60}+q^{58}-q^{56}-q^{54}+q^{52}+2q^{50}+q^{48}-q^{46}-q^{44}+q^{42}+2q^{40}-q^{36}+q^{32}-q^{30}-2q^{28}+2q^{24}-3q^{20}-2q^{18}+q^{16}+2q^{14}-3q^{10}-q^{8}+2q^{6}+2q^{4}-2+2q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-12}+2q^{-14}+q^{-22}$

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{134}-q^{132}+q^{130}-q^{128}-q^{122}+2q^{120}-2q^{118}+2q^{116}-2q^{114}+q^{110}-q^{108}+3q^{106}-3q^{104}+2q^{102}-q^{100}-q^{98}+2q^{96}-3q^{94}+3q^{92}-2q^{90}+q^{88}+q^{86}-q^{84}+2q^{82}+q^{78}+q^{74}+2q^{70}+q^{68}+q^{66}+q^{62}-q^{60}-3q^{54}+3q^{52}-5q^{50}+3q^{48}-q^{46}-5q^{44}+5q^{42}-7q^{40}+2q^{38}-2q^{36}-2q^{34}+2q^{32}-3q^{30}+3q^{28}-q^{26}-2q^{20}+q^{18}+q^{16}-q^{14}+2q^{12}-3q^{10}+3q^{8}+q^{6}-3q^{4}+6q^{2}-6+4q^{-2}+q^{-4}-3q^{-6}+6q^{-8}-4q^{-10}+4q^{-12}+2q^{-18}-2q^{-20}+2q^{-22}+q^{-26}$

.

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 8"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+5t^{2}-5t+5-5t^{-1}+5t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 29, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{6}+3z^{2}a^{6}+a^{6}-z^{6}a^{4}-4z^{4}a^{4}-3z^{2}a^{4}-z^{6}a^{2}-5z^{4}a^{2}-7z^{2}a^{2}-3a^{2}+z^{4}+4z^{2}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{2}a^{10}+2z^{3}a^{9}+3z^{4}a^{8}-2z^{2}a^{8}+4z^{5}a^{7}-7z^{3}a^{7}+2za^{7}+4z^{6}a^{6}-10z^{4}a^{6}+5z^{2}a^{6}-a^{6}+3z^{7}a^{5}-8z^{5}a^{5}+2z^{3}a^{5}+za^{5}+2z^{8}a^{4}-6z^{6}a^{4}+z^{4}a^{4}+3z^{2}a^{4}+z^{9}a^{3}-3z^{7}a^{3}-z^{5}a^{3}+5z^{3}a^{3}-za^{3}+3z^{8}a^{2}-17z^{6}a^{2}+30z^{4}a^{2}-18z^{2}a^{2}+3a^{2}+z^{9}a-6z^{7}a+11z^{5}a-6z^{3}a+z^{8}-7z^{6}+16z^{4}-13z^{2}+3$

Vassiliev invariants

V2 and V3:

(-3, 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 $-12$ $32$ $72$ $114$ $94$ $-384$ $-{\frac {2368}{3}}$ $-{\frac {736}{3}}$ $-224$ $-288$ $512$ $-1368$ $-1128$ ${\frac {3969}{10}}$ ${\frac {16346}{15}}$ $-{\frac {8314}{5}}$ ${\frac {949}{2}}$ $-{\frac {6111}{10}}$

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 $s=$-4 is the signature of 10 8. 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
5											1	1
3												0
1									2	1		1
-1								1				-1
-3							3	2				1
-5						2	2					0
-7					2	2						0
-9				2	2							0
-11			1	2								-1
-13		1	2									1
-15		1										-1
-17	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, 8]]
Out[2]=	10
In[3]:=	PD[Knot[10, 8]]
Out[3]=	PD[X[1, 6, 2, 7], X[7, 16, 8, 17], X[5, 13, 6, 12], X[3, 15, 4, 14], X[13, 5, 14, 4], X[15, 3, 16, 2], X[9, 18, 10, 19], X[11, 20, 12, 1], X[17, 8, 18, 9], X[19, 10, 20, 11]]
In[4]:=	GaussCode[Knot[10, 8]]
Out[4]=	GaussCode[-1, 6, -4, 5, -3, 1, -2, 9, -7, 10, -8, 3, -5, 4, -6, 2, -9, 7, -10, 8]
In[5]:=	BR[Knot[10, 8]]
Out[5]=	BR[4, {-1, -1, -1, -1, -1, 2, -1, 2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[10, 8]][t]
Out[6]=	2 5 5 2 3 5 - -- + -- - - - 5 t + 5 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 8]][z]
Out[7]=	2 4 6 1 - 3 z - 7 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 8]}
In[9]:=	{KnotDet[Knot[10, 8]], KnotSignature[Knot[10, 8]]}
Out[9]=	{29, -4}
In[10]:=	J=Jones[Knot[10, 8]][q]
Out[10]=	-8 2 3 4 4 4 4 3 2 2 + q - -- + -- - -- + -- - -- + -- - - - q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 8]}
In[12]:=	A2Invariant[Knot[10, 8]][q]
Out[12]=	-24 -14 -12 -8 -6 2 4 6 1 + q + q - q - q - q + q + q + q
In[13]:=	Kauffman[Knot[10, 8]][a, z]
Out[13]=	2 6 3 5 7 2 2 2 4 2 3 + 3 a - a - a z + a z + 2 a z - 13 z - 18 a z + 3 a z + 6 2 8 2 10 2 3 3 3 5 3 7 3 5 a z - 2 a z + a z - 6 a z + 5 a z + 2 a z - 7 a z + 9 3 4 2 4 4 4 6 4 8 4 5 2 a z + 16 z + 30 a z + a z - 10 a z + 3 a z + 11 a z - 3 5 5 5 7 5 6 2 6 4 6 6 6 a z - 8 a z + 4 a z - 7 z - 17 a z - 6 a z + 4 a z - 7 3 7 5 7 8 2 8 4 8 9 3 9 6 a z - 3 a z + 3 a z + z + 3 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 8]], Vassiliev[3][Knot[10, 8]]}
Out[14]=	{0, 4}
In[15]:=	Kh[Knot[10, 8]][q, t]
Out[15]=	2 3 1 1 1 2 1 2 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 5 3 17 6 15 5 13 5 13 4 11 4 11 3 9 3 q q q t q t q t q t q t q t q t 2 2 2 2 2 t t 2 3 5 4 ----- + ----- + ---- + ---- + --- + - + 2 q t + q t + q t 9 2 7 2 7 5 3 q q t q t q t q t q