9 49

9_48

10_1

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9 49 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X12,8,13,7 X5,15,6,14 X3,11,4,10 X11,3,12,2 X15,5,16,4 X17,9,18,8 X9,17,10,16 X18,14,1,13
Gauss code	1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9
Dowker-Thistlethwaite code	6 -10 -14 12 -16 -2 18 -4 -8
Conway Notation	[-20:-20:-20]

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 2 Bridge index 3 Super bridge index $\{4,5\}$ Nakanishi index 2 Maximal Thurston-Bennequin number [3][-12] Hyperbolic Volume 9.42707 A-Polynomial See Data:9 49/A-polynomial

[edit Notes for 9 49's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 49's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-6t+7-6t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{5,t+1\}$
Determinant and Signature	{ 25, 4 }
Jones polynomial	$-2q^{9}+3q^{8}-4q^{7}+5q^{6}-4q^{5}+4q^{4}-2q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+2z^{4}a^{-6}+2z^{2}a^{-4}+6z^{2}a^{-6}-2z^{2}a^{-8}+4a^{-6}-3a^{-8}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-7}+z^{7}a^{-9}+3z^{6}a^{-6}+4z^{6}a^{-8}+z^{6}a^{-10}+2z^{5}a^{-5}+z^{5}a^{-7}-z^{5}a^{-9}+z^{4}a^{-4}-8z^{4}a^{-6}-9z^{4}a^{-8}-3z^{3}a^{-5}-3z^{3}a^{-7}+3z^{3}a^{-9}+3z^{3}a^{-11}-2z^{2}a^{-4}+9z^{2}a^{-6}+10z^{2}a^{-8}-z^{2}a^{-10}+2za^{-7}-2za^{-9}-4za^{-11}-4a^{-6}-3a^{-8}$
The A2 invariant	$q^{-6}-q^{-8}+q^{-10}+q^{-14}+3q^{-16}+q^{-18}+2q^{-20}-q^{-22}-q^{-24}-q^{-26}-2q^{-28}$
The G2 invariant	$q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+8q^{-44}-10q^{-46}+12q^{-48}-7q^{-50}-q^{-52}+10q^{-54}-16q^{-56}+19q^{-58}-11q^{-60}+q^{-62}+10q^{-64}-14q^{-66}+13q^{-68}-2q^{-70}-6q^{-72}+14q^{-74}-12q^{-76}+4q^{-78}+9q^{-80}-15q^{-82}+21q^{-84}-16q^{-86}+9q^{-88}+5q^{-90}-13q^{-92}+22q^{-94}-22q^{-96}+16q^{-98}-4q^{-100}-7q^{-102}+13q^{-104}-16q^{-106}+9q^{-108}-q^{-110}-10q^{-112}+10q^{-114}-11q^{-116}-2q^{-118}+10q^{-120}-20q^{-122}+16q^{-124}-9q^{-126}-4q^{-128}+11q^{-130}-16q^{-132}+15q^{-134}-7q^{-136}+q^{-138}+3q^{-140}-7q^{-142}+7q^{-144}-2q^{-146}+q^{-148}+q^{-150}$

Further Quantum Invariants

Further quantum knot invariants for 9_49.

A1 Invariants.

Weight	Invariant
1	$q^{-3}-q^{-5}+2q^{-7}+q^{-11}+q^{-13}-q^{-15}+q^{-17}-2q^{-19}$
2	$q^{-6}-q^{-8}+5q^{-12}-q^{-14}-5q^{-16}+5q^{-18}+2q^{-20}-5q^{-22}+4q^{-24}+4q^{-26}-3q^{-28}-q^{-30}+q^{-32}+q^{-34}-5q^{-36}+q^{-38}+5q^{-40}-6q^{-42}-2q^{-44}+5q^{-46}-3q^{-48}-3q^{-50}+3q^{-52}+q^{-54}$
3	$q^{-9}-q^{-11}+3q^{-15}+3q^{-17}-3q^{-19}-7q^{-21}+3q^{-23}+14q^{-25}+4q^{-27}-14q^{-29}-13q^{-31}+14q^{-33}+18q^{-35}-9q^{-37}-21q^{-39}+4q^{-41}+23q^{-43}+q^{-45}-19q^{-47}-3q^{-49}+14q^{-51}+6q^{-53}-9q^{-55}-9q^{-57}+2q^{-59}+8q^{-61}+q^{-63}-14q^{-65}-7q^{-67}+14q^{-69}+14q^{-71}-16q^{-73}-17q^{-75}+12q^{-77}+21q^{-79}-6q^{-81}-21q^{-83}-q^{-85}+18q^{-87}+5q^{-89}-10q^{-91}-7q^{-93}+5q^{-95}+8q^{-97}-q^{-99}-2q^{-101}-2q^{-103}$
4	$q^{-12}-q^{-14}+3q^{-18}+q^{-20}+q^{-22}-6q^{-24}-4q^{-26}+8q^{-28}+10q^{-30}+14q^{-32}-12q^{-34}-28q^{-36}-13q^{-38}+12q^{-40}+51q^{-42}+23q^{-44}-29q^{-46}-58q^{-48}-37q^{-50}+56q^{-52}+76q^{-54}+19q^{-56}-66q^{-58}-93q^{-60}+10q^{-62}+84q^{-64}+69q^{-66}-28q^{-68}-97q^{-70}-28q^{-72}+51q^{-74}+71q^{-76}+4q^{-78}-62q^{-80}-38q^{-82}+14q^{-84}+45q^{-86}+17q^{-88}-25q^{-90}-34q^{-92}-9q^{-94}+25q^{-96}+32q^{-98}+7q^{-100}-43q^{-102}-38q^{-104}+13q^{-106}+56q^{-108}+43q^{-110}-49q^{-112}-73q^{-114}-13q^{-116}+71q^{-118}+87q^{-120}-25q^{-122}-88q^{-124}-57q^{-126}+39q^{-128}+99q^{-130}+24q^{-132}-50q^{-134}-75q^{-136}-17q^{-138}+58q^{-140}+45q^{-142}+7q^{-144}-37q^{-146}-35q^{-148}+5q^{-150}+19q^{-152}+20q^{-154}-q^{-156}-13q^{-158}-7q^{-160}-3q^{-162}+3q^{-164}+3q^{-166}+q^{-168}$
5	$q^{-15}-q^{-17}+3q^{-21}+q^{-23}-q^{-25}-2q^{-27}-3q^{-29}+9q^{-33}+14q^{-35}+3q^{-37}-10q^{-39}-24q^{-41}-23q^{-43}+40q^{-47}+59q^{-49}+34q^{-51}-28q^{-53}-92q^{-55}-97q^{-57}-27q^{-59}+96q^{-61}+171q^{-63}+117q^{-65}-44q^{-67}-206q^{-69}-231q^{-71}-64q^{-73}+197q^{-75}+325q^{-77}+194q^{-79}-114q^{-81}-363q^{-83}-326q^{-85}-8q^{-87}+339q^{-89}+402q^{-91}+127q^{-93}-257q^{-95}-424q^{-97}-219q^{-99}+159q^{-101}+387q^{-103}+263q^{-105}-72q^{-107}-310q^{-109}-263q^{-111}+5q^{-113}+228q^{-115}+229q^{-117}+33q^{-119}-156q^{-121}-183q^{-123}-56q^{-125}+92q^{-127}+141q^{-129}+66q^{-131}-52q^{-133}-117q^{-135}-81q^{-137}+20q^{-139}+108q^{-141}+112q^{-143}+16q^{-145}-116q^{-147}-156q^{-149}-50q^{-151}+121q^{-153}+216q^{-155}+116q^{-157}-128q^{-159}-281q^{-161}-184q^{-163}+100q^{-165}+334q^{-167}+279q^{-169}-42q^{-171}-353q^{-173}-359q^{-175}-46q^{-177}+317q^{-179}+417q^{-181}+151q^{-183}-229q^{-185}-414q^{-187}-254q^{-189}+106q^{-191}+349q^{-193}+299q^{-195}+25q^{-197}-232q^{-199}-292q^{-201}-124q^{-203}+107q^{-205}+219q^{-207}+160q^{-209}+3q^{-211}-120q^{-213}-141q^{-215}-61q^{-217}+42q^{-219}+82q^{-221}+63q^{-223}+12q^{-225}-31q^{-227}-46q^{-229}-21q^{-231}+3q^{-233}+12q^{-235}+17q^{-237}+8q^{-239}-q^{-241}-4q^{-243}-2q^{-245}-2q^{-247}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-6}-q^{-8}+q^{-10}+q^{-14}+3q^{-16}+q^{-18}+2q^{-20}-q^{-22}-q^{-24}-q^{-26}-2q^{-28}$
1,1	$q^{-12}-2q^{-14}+4q^{-16}-8q^{-18}+19q^{-20}-22q^{-22}+36q^{-24}-42q^{-26}+48q^{-28}-40q^{-30}+34q^{-32}-10q^{-34}-6q^{-36}+38q^{-38}-56q^{-40}+70q^{-42}-87q^{-44}+78q^{-46}-84q^{-48}+58q^{-50}-45q^{-52}+18q^{-54}+10q^{-56}-22q^{-58}+45q^{-60}-48q^{-62}+48q^{-64}-42q^{-66}+25q^{-68}-22q^{-70}+10q^{-72}-2q^{-74}+2q^{-76}+2q^{-78}$
2,0	$q^{-12}-q^{-14}-q^{-16}+3q^{-18}+3q^{-20}-q^{-22}-q^{-24}+2q^{-26}+2q^{-28}-2q^{-30}+q^{-32}+4q^{-34}+2q^{-36}+3q^{-38}+5q^{-40}+2q^{-42}-q^{-44}-q^{-46}-2q^{-48}-6q^{-50}-4q^{-52}-q^{-54}-q^{-56}-5q^{-58}-q^{-60}+q^{-62}+3q^{-70}+q^{-72}+q^{-74}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-12}-q^{-14}+2q^{-18}-3q^{-20}+2q^{-22}+6q^{-24}-2q^{-26}+5q^{-28}+6q^{-30}+q^{-34}+2q^{-36}-q^{-38}-2q^{-40}-3q^{-42}-3q^{-46}-6q^{-48}+2q^{-50}-2q^{-52}-5q^{-54}+4q^{-56}+q^{-58}-q^{-60}+3q^{-62}$
1,0,0	$q^{-9}-q^{-11}+q^{-13}-q^{-15}+q^{-17}+q^{-19}+3q^{-21}+3q^{-23}+2q^{-25}+2q^{-27}-q^{-29}-q^{-31}-3q^{-33}-q^{-35}-2q^{-37}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-18}-q^{-20}+q^{-24}-q^{-26}-q^{-28}+4q^{-30}+2q^{-32}-q^{-34}+4q^{-36}+7q^{-38}+3q^{-40}+2q^{-42}+8q^{-44}+8q^{-46}+3q^{-50}+5q^{-52}-3q^{-54}-6q^{-56}-2q^{-58}-8q^{-60}-13q^{-62}-8q^{-64}-5q^{-66}-6q^{-68}-3q^{-70}+6q^{-72}+6q^{-74}+2q^{-76}+3q^{-78}+4q^{-80}-q^{-84}$
1,0,0,0	$q^{-12}-q^{-14}+q^{-16}-q^{-18}+q^{-22}+q^{-24}+3q^{-26}+3q^{-28}+4q^{-30}+2q^{-32}+2q^{-34}-q^{-36}-q^{-38}-3q^{-40}-3q^{-42}-q^{-44}-2q^{-46}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-12}-q^{-14}+2q^{-16}-4q^{-18}+5q^{-20}-4q^{-22}+4q^{-24}+3q^{-28}+2q^{-30}-2q^{-32}+7q^{-34}-8q^{-36}+7q^{-38}-8q^{-40}+5q^{-42}-6q^{-44}+3q^{-46}-2q^{-50}+4q^{-52}-3q^{-54}+4q^{-56}-5q^{-58}+3q^{-60}-3q^{-62}$
1,0	$q^{-18}-q^{-22}-q^{-24}+q^{-26}+3q^{-28}-4q^{-32}-q^{-34}+5q^{-36}+5q^{-38}-2q^{-40}-2q^{-42}+q^{-44}+6q^{-46}+2q^{-48}-2q^{-50}-q^{-52}+4q^{-54}+2q^{-56}-2q^{-58}-2q^{-60}+q^{-62}+3q^{-64}-q^{-66}-4q^{-68}-2q^{-70}+2q^{-72}-5q^{-76}-4q^{-78}+2q^{-80}+3q^{-82}-3q^{-84}-5q^{-86}-q^{-88}+4q^{-90}+3q^{-92}-2q^{-94}-2q^{-96}+q^{-98}+3q^{-100}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-18}-q^{-20}+q^{-22}-2q^{-24}+3q^{-26}-4q^{-28}+4q^{-30}-2q^{-32}+5q^{-34}+q^{-36}+4q^{-38}+4q^{-40}+5q^{-42}+6q^{-44}-2q^{-46}+6q^{-48}-5q^{-50}+5q^{-52}-8q^{-54}+2q^{-56}-8q^{-58}+3q^{-60}-5q^{-62}-4q^{-66}-2q^{-68}+q^{-70}-4q^{-72}+q^{-74}-4q^{-76}+4q^{-78}-2q^{-80}+4q^{-82}-2q^{-84}+3q^{-86}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+8q^{-44}-10q^{-46}+12q^{-48}-7q^{-50}-q^{-52}+10q^{-54}-16q^{-56}+19q^{-58}-11q^{-60}+q^{-62}+10q^{-64}-14q^{-66}+13q^{-68}-2q^{-70}-6q^{-72}+14q^{-74}-12q^{-76}+4q^{-78}+9q^{-80}-15q^{-82}+21q^{-84}-16q^{-86}+9q^{-88}+5q^{-90}-13q^{-92}+22q^{-94}-22q^{-96}+16q^{-98}-4q^{-100}-7q^{-102}+13q^{-104}-16q^{-106}+9q^{-108}-q^{-110}-10q^{-112}+10q^{-114}-11q^{-116}-2q^{-118}+10q^{-120}-20q^{-122}+16q^{-124}-9q^{-126}-4q^{-128}+11q^{-130}-16q^{-132}+15q^{-134}-7q^{-136}+q^{-138}+3q^{-140}-7q^{-142}+7q^{-144}-2q^{-146}+q^{-148}+q^{-150}$

.

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 49"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-6t+7-6t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{5,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 25, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-2q^{9}+3q^{8}-4q^{7}+5q^{6}-4q^{5}+4q^{4}-2q^{3}+q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{-4}+2z^{4}a^{-6}+2z^{2}a^{-4}+6z^{2}a^{-6}-2z^{2}a^{-8}+4a^{-6}-3a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(6, 14)

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 $24$ $112$ $288$ $700$ $116$ $2688$ ${\frac {14368}{3}}$ ${\frac {2560}{3}}$ $688$ $2304$ $6272$ $16800$ $2784$ ${\frac {166191}{5}}$ ${\frac {1876}{5}}$ ${\frac {207244}{15}}$ ${\frac {689}{3}}$ ${\frac {9391}{5}}$

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 9 49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$

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, 49]]
Out[2]=	9
In[3]:=	PD[Knot[9, 49]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 8, 13, 7], X[5, 15, 6, 14], X[3, 11, 4, 10], X[11, 3, 12, 2], X[15, 5, 16, 4], X[17, 9, 18, 8], X[9, 17, 10, 16], X[18, 14, 1, 13]]
In[4]:=	GaussCode[Knot[9, 49]]
Out[4]=	GaussCode[1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9]
In[5]:=	BR[Knot[9, 49]]
Out[5]=	BR[4, {1, 1, 2, 1, 1, -3, 2, -1, 2, 3, 3}]
In[6]:=	alex = Alexander[Knot[9, 49]][t]
Out[6]=	3 6 2 7 + -- - - - 6 t + 3 t 2 t t
In[7]:=	Conway[Knot[9, 49]][z]
Out[7]=	2 4 1 + 6 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 49]}
In[9]:=	{KnotDet[Knot[9, 49]], KnotSignature[Knot[9, 49]]}
Out[9]=	{25, 4}
In[10]:=	J=Jones[Knot[9, 49]][q]
Out[10]=	2 3 4 5 6 7 8 9 q - 2 q + 4 q - 4 q + 5 q - 4 q + 3 q - 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 49]}
In[12]:=	A2Invariant[Knot[9, 49]][q]
Out[12]=	6 8 10 14 16 18 20 22 24 26 28 q - q + q + q + 3 q + q + 2 q - q - q - q - 2 q
In[13]:=	Kauffman[Knot[9, 49]][a, z]
Out[13]=	2 2 2 2 3 3 -3 4 4 z 2 z 2 z z 10 z 9 z 2 z 3 z 3 z -- - -- - --- - --- + --- - --- + ----- + ---- - ---- + ---- + ---- - 8 6 11 9 7 10 8 6 4 11 9 a a a a a a a a a a a 3 3 4 4 4 5 5 5 6 6 6 3 z 3 z 9 z 8 z z z z 2 z z 4 z 3 z ---- - ---- - ---- - ---- + -- - -- + -- + ---- + --- + ---- + ---- + 7 5 8 6 4 9 7 5 10 8 6 a a a a a a a a a a a 7 7 z z -- + -- 9 7 a a
In[14]:=	{Vassiliev[2][Knot[9, 49]], Vassiliev[3][Knot[9, 49]]}
Out[14]=	{0, 14}
In[15]:=	Kh[Knot[9, 49]][q, t]
Out[15]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 2 q t + 2 q t + 2 q t + 2 q t + 2 q t + 3 q t + 13 4 13 5 15 5 15 6 17 6 19 7 2 q t + q t + 3 q t + 2 q t + q t + 2 q t