6 3

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6 2.gif

6_2

7 1.gif

7_1

6 3.gif Visit 6 3's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 6 3's page at Knotilus!

Visit 6 3's page at the original Knot Atlas!

The Eskimo bowline knot of practical knot tying deforms to 6_3. The standard bowline is at 6_2.



3D depiction
Irish knot, sum of four 6.3

Knot presentations

Planar diagram presentation X4251 X8493 X12,9,1,10 X10,5,11,6 X6,11,7,12 X2837
Gauss code 1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3
Dowker-Thistlethwaite code 4 8 10 2 12 6
Conway Notation [2112]

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index [math]\displaystyle{ \{3,4\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-4][-4]
Hyperbolic Volume 5.69302
A-Polynomial See Data:6 3/A-polynomial

[edit Notes for 6 3'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 0

[edit Notes for 6 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^2-3 t+5-3 t^{-1} + t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 13, 0 }
Jones polynomial [math]\displaystyle{ -q^3+2 q^2-2 q+3-2 q^{-1} +2 q^{-2} - q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4-a^2 z^2-z^2 a^{-2} +3 z^2-a^2- a^{-2} +3 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^5+z^5 a^{-1} +2 a^2 z^4+2 z^4 a^{-2} +4 z^4+a^3 z^3+a z^3+z^3 a^{-1} +z^3 a^{-3} -3 a^2 z^2-3 z^2 a^{-2} -6 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3 }[/math]
The A2 invariant [math]\displaystyle{ -q^{10}+2 q^2+1+2 q^{-2} - q^{-10} }[/math]
The G2 invariant [math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-q^{44}+q^{42}-3 q^{40}+4 q^{38}-4 q^{36}+q^{34}-3 q^{30}+3 q^{28}-3 q^{26}+q^{24}+q^{22}-2 q^{20}+q^{18}+q^{16}-q^{14}+4 q^{12}-3 q^{10}+3 q^8+q^6-q^4+6 q^2-5+6 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +4 q^{-12} - q^{-14} + q^{-16} + q^{-18} -2 q^{-20} + q^{-22} + q^{-24} -3 q^{-26} +3 q^{-28} -3 q^{-30} + q^{-34} -4 q^{-36} +4 q^{-38} -3 q^{-40} + q^{-42} - q^{-44} -2 q^{-46} +2 q^{-48} - q^{-50} + q^{-52} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 6_3.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^7+q^5+q+ q^{-1} + q^{-5} - q^{-7} }[/math]
2 [math]\displaystyle{ q^{20}-q^{18}-2 q^{16}+2 q^{14}-2 q^{10}+2 q^8+q^6-q^4+q^2+1+ q^{-2} - q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} +2 q^{-14} -2 q^{-16} - q^{-18} + q^{-20} }[/math]
3 [math]\displaystyle{ -q^{39}+q^{37}+2 q^{35}-3 q^{31}-2 q^{29}+4 q^{27}+2 q^{25}-4 q^{23}-4 q^{21}+3 q^{19}+4 q^{17}-2 q^{15}-4 q^{13}+3 q^{11}+4 q^9-2 q^5+q+ q^{-1} -2 q^{-5} +4 q^{-9} +3 q^{-11} -4 q^{-13} -2 q^{-15} +4 q^{-17} +3 q^{-19} -4 q^{-21} -4 q^{-23} +2 q^{-25} +4 q^{-27} -2 q^{-29} -3 q^{-31} +2 q^{-35} + q^{-37} - q^{-39} }[/math]
4 [math]\displaystyle{ q^{64}-q^{62}-2 q^{60}+q^{56}+5 q^{54}-4 q^{50}-4 q^{48}-2 q^{46}+10 q^{44}+5 q^{42}-4 q^{40}-9 q^{38}-8 q^{36}+9 q^{34}+9 q^{32}+q^{30}-9 q^{28}-11 q^{26}+7 q^{24}+10 q^{22}+4 q^{20}-6 q^{18}-9 q^{16}+3 q^{14}+6 q^{12}+4 q^{10}-2 q^8-5 q^6+2 q^2+3+2 q^{-2} -5 q^{-6} -2 q^{-8} +4 q^{-10} +6 q^{-12} +3 q^{-14} -9 q^{-16} -6 q^{-18} +4 q^{-20} +10 q^{-22} +7 q^{-24} -11 q^{-26} -9 q^{-28} + q^{-30} +9 q^{-32} +9 q^{-34} -8 q^{-36} -9 q^{-38} -4 q^{-40} +5 q^{-42} +10 q^{-44} -2 q^{-46} -4 q^{-48} -4 q^{-50} +5 q^{-54} + q^{-56} -2 q^{-60} - q^{-62} + q^{-64} }[/math]
5 [math]\displaystyle{ -q^{95}+q^{93}+2 q^{91}-q^{87}-3 q^{85}-3 q^{83}+6 q^{79}+6 q^{77}+q^{75}-6 q^{73}-10 q^{71}-6 q^{69}+5 q^{67}+17 q^{65}+13 q^{63}-3 q^{61}-17 q^{59}-20 q^{57}-5 q^{55}+17 q^{53}+25 q^{51}+9 q^{49}-15 q^{47}-27 q^{45}-17 q^{43}+12 q^{41}+27 q^{39}+19 q^{37}-6 q^{35}-25 q^{33}-19 q^{31}+4 q^{29}+21 q^{27}+17 q^{25}-16 q^{21}-16 q^{19}+11 q^{15}+11 q^{13}+3 q^{11}-6 q^9-8 q^7-3 q^5+3 q^3+6 q+6 q^{-1} +3 q^{-3} -3 q^{-5} -8 q^{-7} -6 q^{-9} +3 q^{-11} +11 q^{-13} +11 q^{-15} -16 q^{-19} -16 q^{-21} +17 q^{-25} +21 q^{-27} +4 q^{-29} -19 q^{-31} -25 q^{-33} -6 q^{-35} +19 q^{-37} +27 q^{-39} +12 q^{-41} -17 q^{-43} -27 q^{-45} -15 q^{-47} +9 q^{-49} +25 q^{-51} +17 q^{-53} -5 q^{-55} -20 q^{-57} -17 q^{-59} -3 q^{-61} +13 q^{-63} +17 q^{-65} +5 q^{-67} -6 q^{-69} -10 q^{-71} -6 q^{-73} + q^{-75} +6 q^{-77} +6 q^{-79} -3 q^{-83} -3 q^{-85} - q^{-87} +2 q^{-91} + q^{-93} - q^{-95} }[/math]
6 [math]\displaystyle{ q^{132}-q^{130}-2 q^{128}+q^{124}+3 q^{122}+q^{120}+3 q^{118}-2 q^{116}-8 q^{114}-5 q^{112}-q^{110}+6 q^{108}+8 q^{106}+14 q^{104}+q^{102}-14 q^{100}-19 q^{98}-15 q^{96}+13 q^{92}+38 q^{90}+25 q^{88}-4 q^{86}-29 q^{84}-41 q^{82}-28 q^{80}-q^{78}+49 q^{76}+53 q^{74}+26 q^{72}-18 q^{70}-53 q^{68}-58 q^{66}-30 q^{64}+39 q^{62}+64 q^{60}+53 q^{58}+7 q^{56}-43 q^{54}-66 q^{52}-49 q^{50}+18 q^{48}+53 q^{46}+57 q^{44}+22 q^{42}-24 q^{40}-53 q^{38}-47 q^{36}+4 q^{34}+33 q^{32}+43 q^{30}+21 q^{28}-9 q^{26}-32 q^{24}-33 q^{22}-3 q^{20}+14 q^{18}+23 q^{16}+15 q^{14}+q^{12}-13 q^{10}-17 q^8-7 q^6+2 q^4+11 q^2+13+11 q^{-2} +2 q^{-4} -7 q^{-6} -17 q^{-8} -13 q^{-10} + q^{-12} +15 q^{-14} +23 q^{-16} +14 q^{-18} -3 q^{-20} -33 q^{-22} -32 q^{-24} -9 q^{-26} +21 q^{-28} +43 q^{-30} +33 q^{-32} +4 q^{-34} -47 q^{-36} -53 q^{-38} -24 q^{-40} +22 q^{-42} +57 q^{-44} +53 q^{-46} +18 q^{-48} -49 q^{-50} -66 q^{-52} -43 q^{-54} +7 q^{-56} +53 q^{-58} +64 q^{-60} +39 q^{-62} -30 q^{-64} -58 q^{-66} -53 q^{-68} -18 q^{-70} +26 q^{-72} +53 q^{-74} +49 q^{-76} - q^{-78} -28 q^{-80} -41 q^{-82} -29 q^{-84} -4 q^{-86} +25 q^{-88} +38 q^{-90} +13 q^{-92} -15 q^{-96} -19 q^{-98} -14 q^{-100} + q^{-102} +14 q^{-104} +8 q^{-106} +6 q^{-108} - q^{-110} -5 q^{-112} -8 q^{-114} -2 q^{-116} +3 q^{-118} + q^{-120} +3 q^{-122} + q^{-124} -2 q^{-128} - q^{-130} + q^{-132} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{10}+2 q^2+1+2 q^{-2} - q^{-10} }[/math]
1,1 [math]\displaystyle{ q^{28}-2 q^{26}+4 q^{24}-6 q^{22}+7 q^{20}-10 q^{18}+8 q^{16}-8 q^{14}+4 q^{12}-4 q^8+10 q^6-11 q^4+18 q^2-14+18 q^{-2} -11 q^{-4} +10 q^{-6} -4 q^{-8} +4 q^{-12} -8 q^{-14} +8 q^{-16} -10 q^{-18} +7 q^{-20} -6 q^{-22} +4 q^{-24} -2 q^{-26} + q^{-28} }[/math]
2,0 [math]\displaystyle{ q^{26}-q^{22}-q^{20}-2 q^{14}+q^{10}+2 q^4+2 q^2+2+2 q^{-2} +2 q^{-4} + q^{-10} -2 q^{-14} - q^{-20} - q^{-22} + q^{-26} }[/math]
3,0 [math]\displaystyle{ -q^{48}+q^{44}+2 q^{42}+q^{40}-2 q^{38}-q^{36}+3 q^{32}-4 q^{28}-4 q^{26}-q^{24}+2 q^{22}-q^{20}-3 q^{18}+4 q^{14}+5 q^{12}+2 q^{10}+2 q^6+q^4-2+ q^{-4} +2 q^{-6} +2 q^{-10} +5 q^{-12} +4 q^{-14} -3 q^{-18} - q^{-20} +2 q^{-22} - q^{-24} -4 q^{-26} -4 q^{-28} +3 q^{-32} - q^{-36} -2 q^{-38} + q^{-40} +2 q^{-42} + q^{-44} - q^{-48} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{22}-q^{20}+q^{16}-3 q^{14}-q^{12}-2 q^8+3 q^4+3 q^2+4+3 q^{-2} +3 q^{-4} -2 q^{-8} - q^{-12} -3 q^{-14} + q^{-16} - q^{-20} + q^{-22} }[/math]
1,0,0 [math]\displaystyle{ -q^{13}-q^9+2 q^3+2 q+2 q^{-1} +2 q^{-3} - q^{-9} - q^{-13} }[/math]
1,0,1 [math]\displaystyle{ q^{36}-2 q^{34}+3 q^{32}-q^{30}-3 q^{28}+7 q^{26}-9 q^{24}+6 q^{22}-q^{20}-9 q^{18}+10 q^{16}-15 q^{14}+5 q^{12}-q^{10}-7 q^8+11 q^6+8 q^2+9+8 q^{-2} +11 q^{-6} -7 q^{-8} - q^{-10} +5 q^{-12} -15 q^{-14} +10 q^{-16} -9 q^{-18} - q^{-20} +6 q^{-22} -9 q^{-24} +7 q^{-26} -3 q^{-28} - q^{-30} +3 q^{-32} -2 q^{-34} + q^{-36} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{28}+q^{22}-3 q^{18}-3 q^{16}-2 q^{14}-3 q^{12}-3 q^{10}+4 q^6+3 q^4+6 q^2+8+6 q^{-2} +3 q^{-4} +4 q^{-6} -3 q^{-10} -3 q^{-12} -2 q^{-14} -3 q^{-16} -3 q^{-18} + q^{-22} + q^{-28} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{16}-q^{12}-q^{10}+2 q^4+2 q^2+3+2 q^{-2} +2 q^{-4} - q^{-10} - q^{-12} - q^{-16} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{22}+q^{20}-2 q^{18}+q^{16}-q^{14}+q^{12}+2 q^6-q^4+3 q^2-2+3 q^{-2} - q^{-4} +2 q^{-6} + q^{-12} - q^{-14} + q^{-16} -2 q^{-18} + q^{-20} - q^{-22} }[/math]
1,0 [math]\displaystyle{ q^{36}-q^{32}-q^{30}+q^{28}+q^{26}-q^{24}-2 q^{22}-q^{20}+q^{18}-q^{14}-q^{12}+q^{10}+q^8+q^6+2 q^2+3+2 q^{-2} + q^{-6} + q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-18} - q^{-20} -2 q^{-22} - q^{-24} + q^{-26} + q^{-28} - q^{-30} - q^{-32} + q^{-36} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{30}-q^{28}+q^{26}-q^{24}+q^{22}-2 q^{20}-q^{18}-2 q^{16}-q^{14}-q^{12}-2 q^{10}+q^8+q^6+5 q^4+2 q^2+6+2 q^{-2} +5 q^{-4} + q^{-6} + q^{-8} -2 q^{-10} - q^{-12} - q^{-14} -2 q^{-16} - q^{-18} -2 q^{-20} + q^{-22} - q^{-24} + q^{-26} - q^{-28} + q^{-30} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-q^{44}+q^{42}-3 q^{40}+4 q^{38}-4 q^{36}+q^{34}-3 q^{30}+3 q^{28}-3 q^{26}+q^{24}+q^{22}-2 q^{20}+q^{18}+q^{16}-q^{14}+4 q^{12}-3 q^{10}+3 q^8+q^6-q^4+6 q^2-5+6 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +4 q^{-12} - q^{-14} + q^{-16} + q^{-18} -2 q^{-20} + q^{-22} + q^{-24} -3 q^{-26} +3 q^{-28} -3 q^{-30} + q^{-34} -4 q^{-36} +4 q^{-38} -3 q^{-40} + q^{-42} - q^{-44} -2 q^{-46} +2 q^{-48} - q^{-50} + q^{-52} }[/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["6 3"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^2-3 t+5-3 t^{-1} + t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^4+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]= { 13, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^3+2 q^2-2 q+3-2 q^{-1} +2 q^{-2} - q^{-3} }[/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 z^2-z^2 a^{-2} +3 z^2-a^2- a^{-2} +3 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^5+z^5 a^{-1} +2 a^2 z^4+2 z^4 a^{-2} +4 z^4+a^3 z^3+a z^3+z^3 a^{-1} +z^3 a^{-3} -3 a^2 z^2-3 z^2 a^{-2} -6 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3 }[/math]

Vassiliev invariants

V2 and V3: (1, 0)
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{ 4 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{14}{3} }[/math] [math]\displaystyle{ -\frac{14}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{56}{3} }[/math] [math]\displaystyle{ -\frac{56}{3} }[/math] [math]\displaystyle{ \frac{511}{30} }[/math] [math]\displaystyle{ \frac{418}{15} }[/math] [math]\displaystyle{ -\frac{1858}{45} }[/math] [math]\displaystyle{ \frac{65}{18} }[/math] [math]\displaystyle{ -\frac{449}{30} }[/math]

V2,1 through V6,9 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]0 is the signature of 6 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -3-2-10123χ
7      1-1
5     1 1
3    11 0
1   21  1
-1  12   1
-3 11    0
-5 1     1
-71      -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[6, 3]]
Out[2]=  
6
In[3]:=
PD[Knot[6, 3]]
Out[3]=  
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 9, 1, 10], X[10, 5, 11, 6], 
  X[6, 11, 7, 12], X[2, 8, 3, 7]]
In[4]:=
GaussCode[Knot[6, 3]]
Out[4]=  
GaussCode[1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3]
In[5]:=
BR[Knot[6, 3]]
Out[5]=  
BR[3, {-1, -1, 2, -1, 2, 2}]
In[6]:=
alex = Alexander[Knot[6, 3]][t]
Out[6]=  
     -2   3          2

5 + t   - - - 3 t + t


          t
In[7]:=
Conway[Knot[6, 3]][z]
Out[7]=  
     2    4
1 + z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[6, 3], Knot[11, NonAlternating, 12]}
In[9]:=
{KnotDet[Knot[6, 3]], KnotSignature[Knot[6, 3]]}
Out[9]=  
{13, 0}
In[10]:=
J=Jones[Knot[6, 3]][q]
Out[10]=  
     -3   2    2            2    3

3 - q   + -- - - - 2 q + 2 q  - q



          2   q


          q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[6, 3]}
In[12]:=
A2Invariant[Knot[6, 3]][q]
Out[12]=  
     -10   2       2    10

1 - q    + -- + 2 q  - q



           2


           q
In[13]:=
Kauffman[Knot[6, 3]][a, z]
Out[13]=  
                                                   2              3

    -2    2   z    2 z            3        2   3 z       2  2   z



3 + a   + a  - -- - --- - 2 a z - a  z - 6 z  - ---- - 3 a  z  + -- + 



               3    a                            2               3
              a                                 a               a

  3                            4              5
 z       3    3  3      4   2 z       2  4   z       5
 -- + a z  + a  z  + 4 z  + ---- + 2 a  z  + -- + a z
 a                            2              a


                              a
In[14]:=
{Vassiliev[2][Knot[6, 3]], Vassiliev[3][Knot[6, 3]]}
Out[14]=  
{0, 0}
In[15]:=
Kh[Knot[6, 3]][q, t]
Out[15]=  
2           1       1       1      1      1           3      3  2

- + 2 q + ----- + ----- + ----- + ---- + --- + q t + q  t + q  t  + 
q          7  3    5  2    3  2    3     q t



         q  t    q  t    q  t    q  t

  5  2    7  3


  q  t  + q  t
Retrieved from "https://katlas.org/index.php?title=6_3&oldid=26022"