T(27,2)

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Knot presentations

Planar diagram presentation	X_21,49,22,48 X_49,23,50,22 X_23,51,24,50 X_51,25,52,24 X_25,53,26,52 X_53,27,54,26 X_27,1,28,54 X_1,29,2,28 X_29,3,30,2 X_3,31,4,30 X_31,5,32,4 X_5,33,6,32 X_33,7,34,6 X_7,35,8,34 X_35,9,36,8 X_9,37,10,36 X_37,11,38,10 X_11,39,12,38 X_39,13,40,12 X_13,41,14,40 X_41,15,42,14 X_15,43,16,42 X_43,17,44,16 X_17,45,18,44 X_45,19,46,18 X_19,47,20,46 X_47,21,48,20
Gauss code	-8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 1, -2, 3, -4, 5, -6, 7
Dowker-Thistlethwaite code	28 30 32 34 36 38 40 42 44 46 48 50 52 54 2 4 6 8 10 12 14 16 18 20 22 24 26

Braid presentation

Polynomial invariants

Alexander polynomial	$t^{13}-t^{12}+t^{11}-t^{10}+t^{9}-t^{8}+t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}-t^{-8}+t^{-9}-t^{-10}+t^{-11}-t^{-12}+t^{-13}$
Conway polynomial	$z^{26}+25z^{24}+276z^{22}+1771z^{20}+7315z^{18}+20349z^{16}+38760z^{14}+50388z^{12}+43758z^{10}+24310z^{8}+8008z^{6}+1365z^{4}+91z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 27, 26 }
Jones polynomial	$-q^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}+q^{13}$
HOMFLY-PT polynomial (db, data sources)	$z^{26}a^{-26}-26z^{24}a^{-26}-z^{24}a^{-28}+300z^{22}a^{-26}+24z^{22}a^{-28}-2024z^{20}a^{-26}-253z^{20}a^{-28}+8855z^{18}a^{-26}+1540z^{18}a^{-28}-26334z^{16}a^{-26}-5985z^{16}a^{-28}+54264z^{14}a^{-26}+15504z^{14}a^{-28}-77520z^{12}a^{-26}-27132z^{12}a^{-28}+75582z^{10}a^{-26}+31824z^{10}a^{-28}-48620z^{8}a^{-26}-24310z^{8}a^{-28}+19448z^{6}a^{-26}+11440z^{6}a^{-28}-4368z^{4}a^{-26}-3003z^{4}a^{-28}+455z^{2}a^{-26}+364z^{2}a^{-28}-14a^{-26}-13a^{-28}$
Kauffman polynomial (db, data sources)	$z^{26}a^{-26}+z^{26}a^{-28}+z^{25}a^{-27}+z^{25}a^{-29}-26z^{24}a^{-26}-25z^{24}a^{-28}+z^{24}a^{-30}-24z^{23}a^{-27}-23z^{23}a^{-29}+z^{23}a^{-31}+300z^{22}a^{-26}+277z^{22}a^{-28}-22z^{22}a^{-30}+z^{22}a^{-32}+253z^{21}a^{-27}+231z^{21}a^{-29}-21z^{21}a^{-31}+z^{21}a^{-33}-2024z^{20}a^{-26}-1793z^{20}a^{-28}+210z^{20}a^{-30}-20z^{20}a^{-32}+z^{20}a^{-34}-1540z^{19}a^{-27}-1330z^{19}a^{-29}+190z^{19}a^{-31}-19z^{19}a^{-33}+z^{19}a^{-35}+8855z^{18}a^{-26}+7525z^{18}a^{-28}-1140z^{18}a^{-30}+171z^{18}a^{-32}-18z^{18}a^{-34}+z^{18}a^{-36}+5985z^{17}a^{-27}+4845z^{17}a^{-29}-969z^{17}a^{-31}+153z^{17}a^{-33}-17z^{17}a^{-35}+z^{17}a^{-37}-26334z^{16}a^{-26}-21489z^{16}a^{-28}+3876z^{16}a^{-30}-816z^{16}a^{-32}+136z^{16}a^{-34}-16z^{16}a^{-36}+z^{16}a^{-38}-15504z^{15}a^{-27}-11628z^{15}a^{-29}+3060z^{15}a^{-31}-680z^{15}a^{-33}+120z^{15}a^{-35}-15z^{15}a^{-37}+z^{15}a^{-39}+54264z^{14}a^{-26}+42636z^{14}a^{-28}-8568z^{14}a^{-30}+2380z^{14}a^{-32}-560z^{14}a^{-34}+105z^{14}a^{-36}-14z^{14}a^{-38}+z^{14}a^{-40}+27132z^{13}a^{-27}+18564z^{13}a^{-29}-6188z^{13}a^{-31}+1820z^{13}a^{-33}-455z^{13}a^{-35}+91z^{13}a^{-37}-13z^{13}a^{-39}+z^{13}a^{-41}-77520z^{12}a^{-26}-58956z^{12}a^{-28}+12376z^{12}a^{-30}-4368z^{12}a^{-32}+1365z^{12}a^{-34}-364z^{12}a^{-36}+78z^{12}a^{-38}-12z^{12}a^{-40}+z^{12}a^{-42}-31824z^{11}a^{-27}-19448z^{11}a^{-29}+8008z^{11}a^{-31}-3003z^{11}a^{-33}+1001z^{11}a^{-35}-286z^{11}a^{-37}+66z^{11}a^{-39}-11z^{11}a^{-41}+z^{11}a^{-43}+75582z^{10}a^{-26}+56134z^{10}a^{-28}-11440z^{10}a^{-30}+5005z^{10}a^{-32}-2002z^{10}a^{-34}+715z^{10}a^{-36}-220z^{10}a^{-38}+55z^{10}a^{-40}-10z^{10}a^{-42}+z^{10}a^{-44}+24310z^{9}a^{-27}+12870z^{9}a^{-29}-6435z^{9}a^{-31}+3003z^{9}a^{-33}-1287z^{9}a^{-35}+495z^{9}a^{-37}-165z^{9}a^{-39}+45z^{9}a^{-41}-9z^{9}a^{-43}+z^{9}a^{-45}-48620z^{8}a^{-26}-35750z^{8}a^{-28}+6435z^{8}a^{-30}-3432z^{8}a^{-32}+1716z^{8}a^{-34}-792z^{8}a^{-36}+330z^{8}a^{-38}-120z^{8}a^{-40}+36z^{8}a^{-42}-8z^{8}a^{-44}+z^{8}a^{-46}-11440z^{7}a^{-27}-5005z^{7}a^{-29}+3003z^{7}a^{-31}-1716z^{7}a^{-33}+924z^{7}a^{-35}-462z^{7}a^{-37}+210z^{7}a^{-39}-84z^{7}a^{-41}+28z^{7}a^{-43}-7z^{7}a^{-45}+z^{7}a^{-47}+19448z^{6}a^{-26}+14443z^{6}a^{-28}-2002z^{6}a^{-30}+1287z^{6}a^{-32}-792z^{6}a^{-34}+462z^{6}a^{-36}-252z^{6}a^{-38}+126z^{6}a^{-40}-56z^{6}a^{-42}+21z^{6}a^{-44}-6z^{6}a^{-46}+z^{6}a^{-48}+3003z^{5}a^{-27}+1001z^{5}a^{-29}-715z^{5}a^{-31}+495z^{5}a^{-33}-330z^{5}a^{-35}+210z^{5}a^{-37}-126z^{5}a^{-39}+70z^{5}a^{-41}-35z^{5}a^{-43}+15z^{5}a^{-45}-5z^{5}a^{-47}+z^{5}a^{-49}-4368z^{4}a^{-26}-3367z^{4}a^{-28}+286z^{4}a^{-30}-220z^{4}a^{-32}+165z^{4}a^{-34}-120z^{4}a^{-36}+84z^{4}a^{-38}-56z^{4}a^{-40}+35z^{4}a^{-42}-20z^{4}a^{-44}+10z^{4}a^{-46}-4z^{4}a^{-48}+z^{4}a^{-50}-364z^{3}a^{-27}-78z^{3}a^{-29}+66z^{3}a^{-31}-55z^{3}a^{-33}+45z^{3}a^{-35}-36z^{3}a^{-37}+28z^{3}a^{-39}-21z^{3}a^{-41}+15z^{3}a^{-43}-10z^{3}a^{-45}+6z^{3}a^{-47}-3z^{3}a^{-49}+z^{3}a^{-51}+455z^{2}a^{-26}+377z^{2}a^{-28}-12z^{2}a^{-30}+11z^{2}a^{-32}-10z^{2}a^{-34}+9z^{2}a^{-36}-8z^{2}a^{-38}+7z^{2}a^{-40}-6z^{2}a^{-42}+5z^{2}a^{-44}-4z^{2}a^{-46}+3z^{2}a^{-48}-2z^{2}a^{-50}+z^{2}a^{-52}+13za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-14a^{-26}-13a^{-28}$
The A2 invariant	Data:T(27,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(27,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants[show]

Computer Talk[show]

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.

In[3]:= K = Knot["T(27,2)"];

In[4]:= Alexander[K][t]

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

Out[4]= $t^{13}-t^{12}+t^{11}-t^{10}+t^{9}-t^{8}+t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}-t^{-8}+t^{-9}-t^{-10}+t^{-11}-t^{-12}+t^{-13}$

In[5]:= Conway[K][z]

Out[5]= $z^{26}+25z^{24}+276z^{22}+1771z^{20}+7315z^{18}+20349z^{16}+38760z^{14}+50388z^{12}+43758z^{10}+24310z^{8}+8008z^{6}+1365z^{4}+91z^{2}+1$

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]= { 27, 26 }

In[8]:= Jones[K][q]

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

Out[8]= $-q^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}+q^{13}$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $z^{26}a^{-26}-26z^{24}a^{-26}-z^{24}a^{-28}+300z^{22}a^{-26}+24z^{22}a^{-28}-2024z^{20}a^{-26}-253z^{20}a^{-28}+8855z^{18}a^{-26}+1540z^{18}a^{-28}-26334z^{16}a^{-26}-5985z^{16}a^{-28}+54264z^{14}a^{-26}+15504z^{14}a^{-28}-77520z^{12}a^{-26}-27132z^{12}a^{-28}+75582z^{10}a^{-26}+31824z^{10}a^{-28}-48620z^{8}a^{-26}-24310z^{8}a^{-28}+19448z^{6}a^{-26}+11440z^{6}a^{-28}-4368z^{4}a^{-26}-3003z^{4}a^{-28}+455z^{2}a^{-26}+364z^{2}a^{-28}-14a^{-26}-13a^{-28}$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $z^{26}a^{-26}+z^{26}a^{-28}+z^{25}a^{-27}+z^{25}a^{-29}-26z^{24}a^{-26}-25z^{24}a^{-28}+z^{24}a^{-30}-24z^{23}a^{-27}-23z^{23}a^{-29}+z^{23}a^{-31}+300z^{22}a^{-26}+277z^{22}a^{-28}-22z^{22}a^{-30}+z^{22}a^{-32}+253z^{21}a^{-27}+231z^{21}a^{-29}-21z^{21}a^{-31}+z^{21}a^{-33}-2024z^{20}a^{-26}-1793z^{20}a^{-28}+210z^{20}a^{-30}-20z^{20}a^{-32}+z^{20}a^{-34}-1540z^{19}a^{-27}-1330z^{19}a^{-29}+190z^{19}a^{-31}-19z^{19}a^{-33}+z^{19}a^{-35}+8855z^{18}a^{-26}+7525z^{18}a^{-28}-1140z^{18}a^{-30}+171z^{18}a^{-32}-18z^{18}a^{-34}+z^{18}a^{-36}+5985z^{17}a^{-27}+4845z^{17}a^{-29}-969z^{17}a^{-31}+153z^{17}a^{-33}-17z^{17}a^{-35}+z^{17}a^{-37}-26334z^{16}a^{-26}-21489z^{16}a^{-28}+3876z^{16}a^{-30}-816z^{16}a^{-32}+136z^{16}a^{-34}-16z^{16}a^{-36}+z^{16}a^{-38}-15504z^{15}a^{-27}-11628z^{15}a^{-29}+3060z^{15}a^{-31}-680z^{15}a^{-33}+120z^{15}a^{-35}-15z^{15}a^{-37}+z^{15}a^{-39}+54264z^{14}a^{-26}+42636z^{14}a^{-28}-8568z^{14}a^{-30}+2380z^{14}a^{-32}-560z^{14}a^{-34}+105z^{14}a^{-36}-14z^{14}a^{-38}+z^{14}a^{-40}+27132z^{13}a^{-27}+18564z^{13}a^{-29}-6188z^{13}a^{-31}+1820z^{13}a^{-33}-455z^{13}a^{-35}+91z^{13}a^{-37}-13z^{13}a^{-39}+z^{13}a^{-41}-77520z^{12}a^{-26}-58956z^{12}a^{-28}+12376z^{12}a^{-30}-4368z^{12}a^{-32}+1365z^{12}a^{-34}-364z^{12}a^{-36}+78z^{12}a^{-38}-12z^{12}a^{-40}+z^{12}a^{-42}-31824z^{11}a^{-27}-19448z^{11}a^{-29}+8008z^{11}a^{-31}-3003z^{11}a^{-33}+1001z^{11}a^{-35}-286z^{11}a^{-37}+66z^{11}a^{-39}-11z^{11}a^{-41}+z^{11}a^{-43}+75582z^{10}a^{-26}+56134z^{10}a^{-28}-11440z^{10}a^{-30}+5005z^{10}a^{-32}-2002z^{10}a^{-34}+715z^{10}a^{-36}-220z^{10}a^{-38}+55z^{10}a^{-40}-10z^{10}a^{-42}+z^{10}a^{-44}+24310z^{9}a^{-27}+12870z^{9}a^{-29}-6435z^{9}a^{-31}+3003z^{9}a^{-33}-1287z^{9}a^{-35}+495z^{9}a^{-37}-165z^{9}a^{-39}+45z^{9}a^{-41}-9z^{9}a^{-43}+z^{9}a^{-45}-48620z^{8}a^{-26}-35750z^{8}a^{-28}+6435z^{8}a^{-30}-3432z^{8}a^{-32}+1716z^{8}a^{-34}-792z^{8}a^{-36}+330z^{8}a^{-38}-120z^{8}a^{-40}+36z^{8}a^{-42}-8z^{8}a^{-44}+z^{8}a^{-46}-11440z^{7}a^{-27}-5005z^{7}a^{-29}+3003z^{7}a^{-31}-1716z^{7}a^{-33}+924z^{7}a^{-35}-462z^{7}a^{-37}+210z^{7}a^{-39}-84z^{7}a^{-41}+28z^{7}a^{-43}-7z^{7}a^{-45}+z^{7}a^{-47}+19448z^{6}a^{-26}+14443z^{6}a^{-28}-2002z^{6}a^{-30}+1287z^{6}a^{-32}-792z^{6}a^{-34}+462z^{6}a^{-36}-252z^{6}a^{-38}+126z^{6}a^{-40}-56z^{6}a^{-42}+21z^{6}a^{-44}-6z^{6}a^{-46}+z^{6}a^{-48}+3003z^{5}a^{-27}+1001z^{5}a^{-29}-715z^{5}a^{-31}+495z^{5}a^{-33}-330z^{5}a^{-35}+210z^{5}a^{-37}-126z^{5}a^{-39}+70z^{5}a^{-41}-35z^{5}a^{-43}+15z^{5}a^{-45}-5z^{5}a^{-47}+z^{5}a^{-49}-4368z^{4}a^{-26}-3367z^{4}a^{-28}+286z^{4}a^{-30}-220z^{4}a^{-32}+165z^{4}a^{-34}-120z^{4}a^{-36}+84z^{4}a^{-38}-56z^{4}a^{-40}+35z^{4}a^{-42}-20z^{4}a^{-44}+10z^{4}a^{-46}-4z^{4}a^{-48}+z^{4}a^{-50}-364z^{3}a^{-27}-78z^{3}a^{-29}+66z^{3}a^{-31}-55z^{3}a^{-33}+45z^{3}a^{-35}-36z^{3}a^{-37}+28z^{3}a^{-39}-21z^{3}a^{-41}+15z^{3}a^{-43}-10z^{3}a^{-45}+6z^{3}a^{-47}-3z^{3}a^{-49}+z^{3}a^{-51}+455z^{2}a^{-26}+377z^{2}a^{-28}-12z^{2}a^{-30}+11z^{2}a^{-32}-10z^{2}a^{-34}+9z^{2}a^{-36}-8z^{2}a^{-38}+7z^{2}a^{-40}-6z^{2}a^{-42}+5z^{2}a^{-44}-4z^{2}a^{-46}+3z^{2}a^{-48}-2z^{2}a^{-50}+z^{2}a^{-52}+13za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-14a^{-26}-13a^{-28}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {}

Computer Talk[show]

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["T(27,2)"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]= { $t^{13}-t^{12}+t^{11}-t^{10}+t^{9}-t^{8}+t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}-t^{-8}+t^{-9}-t^{-10}+t^{-11}-t^{-12}+t^{-13}$ , $-q^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}+q^{13}$ }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {}

Vassiliev invariants

V₂ and V₃:

(91, 819)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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=

26 is the signature of T(27,2). 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

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

χ

81

1

-1

79

0

77

1

0

75

0

73

1

0

71

0

69

1

0

67

0

65

1

0

63

0

61

1

0

59

0

57

1

0

55

0

53

1

0

51

0

49

1

0

47

0

45

1

0

43

0

41

1

0

39

0

37

1

0

35

0

33

1

0

31

0

29

1

27

1

25

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=25$	$i=27$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=1$
$r=2$	${\mathbb {Z} }$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=4$	${\mathbb {Z} }$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=6$	${\mathbb {Z} }$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=8$	${\mathbb {Z} }$
$r=9$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=10$	${\mathbb {Z} }$
$r=11$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=12$	${\mathbb {Z} }$
$r=13$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=14$	${\mathbb {Z} }$
$r=15$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=16$	${\mathbb {Z} }$
$r=17$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=18$	${\mathbb {Z} }$
$r=19$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=20$	${\mathbb {Z} }$
$r=21$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=22$	${\mathbb {Z} }$
$r=23$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=24$	${\mathbb {Z} }$
$r=25$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=26$	${\mathbb {Z} }$
$r=27$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$