3 1 Quantum Invariants: Difference between revisions

Latest revision as of 11:41, 28 June 2006

Further quantum knot invariants for 3_1.

The braid index of 3_1 is only 2, so it's easy to calculate lots of quantum invariants. A1 Invariants.

Weight	Invariant
1	$-q^{9}+q^{5}+q^{3}+q$
2	$q^{24}-q^{20}-q^{18}-q^{16}+q^{10}+q^{8}+q^{6}+q^{4}+q^{2}$
3	$-q^{45}+q^{41}+q^{39}+q^{37}-q^{31}-q^{29}-q^{27}-q^{25}-q^{23}+q^{15}+q^{13}+q^{11}+q^{9}+q^{7}+q^{5}+q^{3}$
4	$q^{72}-q^{68}-q^{66}-q^{64}+q^{58}+q^{56}+q^{54}+q^{52}+q^{50}-q^{42}-q^{40}-q^{38}-q^{36}-q^{34}-q^{32}-q^{30}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^{8}+q^{6}+q^{4}$
5	$-q^{105}+q^{101}+q^{99}+q^{97}-q^{91}-q^{89}-q^{87}-q^{85}-q^{83}+q^{75}+q^{73}+q^{71}+q^{69}+q^{67}+q^{65}+q^{63}-q^{53}-q^{51}-q^{49}-q^{47}-q^{45}-q^{43}-q^{41}-q^{39}-q^{37}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}+q^{7}+q^{5}$
6	$q^{144}-q^{140}-q^{138}-q^{136}+q^{130}+q^{128}+q^{126}+q^{124}+q^{122}-q^{114}-q^{112}-q^{110}-q^{108}-q^{106}-q^{104}-q^{102}+q^{92}+q^{90}+q^{88}+q^{86}+q^{84}+q^{82}+q^{80}+q^{78}+q^{76}-q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}-q^{46}-q^{44}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^{8}+q^{6}$
8	$q^{240}-q^{236}-q^{234}-q^{232}+q^{226}+q^{224}+q^{222}+q^{220}+q^{218}-q^{210}-q^{208}-q^{206}-q^{204}-q^{202}-q^{200}-q^{198}+q^{188}+q^{186}+q^{184}+q^{182}+q^{180}+q^{178}+q^{176}+q^{174}+q^{172}-q^{160}-q^{158}-q^{156}-q^{154}-q^{152}-q^{150}-q^{148}-q^{146}-q^{144}-q^{142}-q^{140}+q^{126}+q^{124}+q^{122}+q^{120}+q^{118}+q^{116}+q^{114}+q^{112}+q^{110}+q^{108}+q^{106}+q^{104}+q^{102}-q^{86}-q^{84}-q^{82}-q^{80}-q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^{8}$

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,0,1	$-q^{19}-q^{17}-q^{15}+q^{11}+2q^{9}+2q^{7}+q^{5}+q^{3}$
0,1,0	$q^{30}-q^{24}-2q^{22}-2q^{20}-2q^{18}+q^{14}+3q^{12}+3q^{10}+3q^{8}+q^{6}+q^{4}$
1,0,0	$-q^{19}-q^{17}-q^{15}+q^{11}+2q^{9}+2q^{7}+q^{5}+q^{3}$
1,0,1	$q^{48}+q^{38}+q^{36}+q^{34}-q^{32}-3q^{30}-5q^{28}-6q^{26}-6q^{24}-3q^{22}+q^{20}+4q^{18}+7q^{16}+8q^{14}+7q^{12}+5q^{10}+2q^{8}+q^{6}$

A4 Invariants.

Weight	Invariant
0,0,0,1	$-q^{24}-q^{22}-q^{20}-q^{18}+q^{14}+2q^{12}+2q^{10}+2q^{8}+q^{6}+q^{4}$
0,1,0,0	$q^{40}+q^{38}+q^{36}-q^{32}-3q^{30}-4q^{28}-4q^{26}-3q^{24}-q^{22}+q^{20}+4q^{18}+4q^{16}+5q^{14}+4q^{12}+3q^{10}+q^{8}+q^{6}$
1,0,0,0	$-q^{24}-q^{22}-q^{20}-q^{18}+q^{14}+2q^{12}+2q^{10}+2q^{8}+q^{6}+q^{4}$

A5 Invariants.

Weight	Invariant
0,0,0,0,1	$-q^{29}-q^{27}-q^{25}-q^{23}-q^{21}+q^{17}+2q^{15}+2q^{13}+2q^{11}+2q^{9}+q^{7}+q^{5}$
1,0,0,0,0	$-q^{29}-q^{27}-q^{25}-q^{23}-q^{21}+q^{17}+2q^{15}+2q^{13}+2q^{11}+2q^{9}+q^{7}+q^{5}$

A6 Invariants.

Weight	Invariant
0,0,0,0,0,1	$-q^{34}-q^{32}-q^{30}-q^{28}-q^{26}-q^{24}+q^{20}+2q^{18}+2q^{16}+2q^{14}+2q^{12}+2q^{10}+q^{8}+q^{6}$
1,0,0,0,0,0	$-q^{34}-q^{32}-q^{30}-q^{28}-q^{26}-q^{24}+q^{20}+2q^{18}+2q^{16}+2q^{14}+2q^{12}+2q^{10}+q^{8}+q^{6}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{30}-q^{24}+q^{14}+q^{12}+q^{10}+q^{8}+q^{6}+q^{4}$
1,0	$q^{48}-q^{38}-q^{36}-q^{34}-q^{32}-q^{30}-q^{28}+q^{22}+q^{20}+2q^{18}+q^{16}+2q^{14}+q^{12}+q^{10}+q^{6}$

B3 Invariants.

Weight	Invariant
1,0,0	$q^{72}-q^{58}-q^{54}-q^{52}-q^{50}-q^{48}-q^{46}-q^{44}-q^{42}+q^{34}+2q^{30}+q^{28}+2q^{26}+q^{24}+2q^{22}+q^{20}+2q^{18}+q^{14}+q^{10}$

B4 Invariants.

Weight	Invariant
1,0,0,0	$q^{96}-q^{78}-q^{74}-q^{70}-q^{68}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}-q^{54}+q^{46}+2q^{42}+2q^{38}+q^{36}+2q^{34}+q^{32}+2q^{30}+q^{28}+2q^{26}+2q^{22}+q^{18}+q^{14}$

B5 Invariants.

Weight	Invariant
1,0,0,0,0	$q^{120}-q^{98}-q^{94}-q^{90}-q^{86}-q^{84}-q^{82}-q^{80}-q^{78}-q^{76}-q^{74}-q^{70}-q^{66}+q^{58}+2q^{54}+2q^{50}+2q^{46}+q^{44}+2q^{42}+q^{40}+2q^{38}+q^{36}+2q^{34}+2q^{30}+2q^{26}+q^{22}+q^{18}$

C3 Invariants.

Weight	Invariant
1,0,0	$-q^{42}-q^{34}-q^{32}-q^{24}+q^{20}+2q^{18}+q^{16}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}$

C4 Invariants.

Weight	Invariant
1,0,0,0	$-q^{54}-q^{44}-q^{42}-q^{40}-q^{32}-q^{30}+q^{26}+2q^{24}+2q^{22}+q^{20}+q^{18}+q^{16}+2q^{14}+2q^{12}+q^{10}+q^{8}$

D4 Invariants.

Weight	Invariant
0,1,0,0	$q^{72}-q^{64}-q^{62}+2q^{56}+4q^{54}+5q^{52}+4q^{50}+3q^{48}-q^{46}-5q^{44}-9q^{42}-13q^{40}-14q^{38}-13q^{36}-9q^{34}-4q^{32}+2q^{30}+7q^{28}+12q^{26}+12q^{24}+14q^{22}+11q^{20}+9q^{18}+6q^{16}+4q^{14}+q^{12}+q^{10}$
1,0,0,0	$q^{42}-q^{34}-q^{32}-2q^{30}-2q^{28}-2q^{26}-q^{24}+q^{20}+2q^{18}+3q^{16}+3q^{14}+3q^{12}+2q^{10}+q^{8}+q^{6}$

G2 Invariants.

Weight	Invariant
0,1	$q^{144}-q^{126}+q^{122}-q^{116}+2q^{112}+q^{110}-q^{108}+2q^{104}+q^{102}-q^{98}-q^{96}+q^{94}-2q^{90}-2q^{88}-q^{86}-q^{84}-2q^{82}-3q^{80}-2q^{78}-2q^{76}-2q^{74}-2q^{72}-2q^{70}-q^{68}-q^{64}-q^{62}+q^{60}+q^{58}+q^{56}+2q^{54}+q^{52}+2q^{50}+3q^{48}+2q^{46}+2q^{44}+3q^{42}+2q^{40}+2q^{38}+3q^{36}+2q^{34}+q^{32}+2q^{30}+q^{28}+q^{26}+q^{24}+q^{18}$
1,0	$q^{72}-q^{64}-q^{62}-q^{56}-2q^{54}-q^{52}+q^{50}-q^{46}-2q^{44}+2q^{40}+q^{38}-q^{36}+2q^{32}+2q^{30}+q^{28}+2q^{22}+2q^{20}+q^{14}+q^{12}+q^{10}$

@@ Line 8: / Line 8: @@
 {{Quantum invariant table entry|knot=3_1|algebra=A1|weight=4}}
 {{Quantum invariant table entry|knot=3_1|algebra=A1|weight=5}}
+{{Quantum invariant table entry|knot=3_1|algebra=A1|weight=6}}
+{{Quantum invariant table entry|knot=3_1|algebra=A1|weight=8}}
 {{Quantum invariant table end|knot=3_1|algebra=A1}}
 {{Quantum invariant table start|knot=3_1|algebra=A2}}
+{{Quantum invariant table entry|knot=3_1|algebra=A2|weight=0,1}}
+{{Quantum invariant table entry|knot=3_1|algebra=A2|weight=0,2}}
 {{Quantum invariant table entry|knot=3_1|algebra=A2|weight=1,0}}
 {{Quantum invariant table entry|knot=3_1|algebra=A2|weight=1,1}}
@@ Line 16: / Line 20: @@
 {{Quantum invariant table end|knot=3_1|algebra=A2}}
 {{Quantum invariant table start|knot=3_1|algebra=A3}}
+{{Quantum invariant table entry|knot=3_1|algebra=A3|weight=0,0,1}}
 {{Quantum invariant table entry|knot=3_1|algebra=A3|weight=0,1,0}}
 {{Quantum invariant table entry|knot=3_1|algebra=A3|weight=1,0,0}}
@@ Line 21: / Line 26: @@
 {{Quantum invariant table end|knot=3_1|algebra=A3}}
 {{Quantum invariant table start|knot=3_1|algebra=A4}}
+{{Quantum invariant table entry|knot=3_1|algebra=A4|weight=0,0,0,1}}
 {{Quantum invariant table entry|knot=3_1|algebra=A4|weight=0,1,0,0}}
 {{Quantum invariant table entry|knot=3_1|algebra=A4|weight=1,0,0,0}}
 {{Quantum invariant table end|knot=3_1|algebra=A4}}
+{{Quantum invariant table start|knot=3_1|algebra=A5}}
+{{Quantum invariant table entry|knot=3_1|algebra=A5|weight=0,0,0,0,1}}
+{{Quantum invariant table entry|knot=3_1|algebra=A5|weight=1,0,0,0,0}}
+{{Quantum invariant table end|knot=3_1|algebra=A5}}
+{{Quantum invariant table start|knot=3_1|algebra=A6}}
+{{Quantum invariant table entry|knot=3_1|algebra=A6|weight=0,0,0,0,0,1}}
+{{Quantum invariant table entry|knot=3_1|algebra=A6|weight=1,0,0,0,0,0}}
+{{Quantum invariant table end|knot=3_1|algebra=A6}}
 {{Quantum invariant table start|knot=3_1|algebra=B2}}
 {{Quantum invariant table entry|knot=3_1|algebra=B2|weight=0,1}}
 {{Quantum invariant table entry|knot=3_1|algebra=B2|weight=1,0}}
 {{Quantum invariant table end|knot=3_1|algebra=B2}}
+{{Quantum invariant table start|knot=3_1|algebra=B3}}
+{{Quantum invariant table entry|knot=3_1|algebra=B3|weight=1,0,0}}
+{{Quantum invariant table end|knot=3_1|algebra=B3}}
+{{Quantum invariant table start|knot=3_1|algebra=B4}}
+{{Quantum invariant table entry|knot=3_1|algebra=B4|weight=1,0,0,0}}
+{{Quantum invariant table end|knot=3_1|algebra=B4}}
+{{Quantum invariant table start|knot=3_1|algebra=B5}}
+{{Quantum invariant table entry|knot=3_1|algebra=B5|weight=1,0,0,0,0}}
+{{Quantum invariant table end|knot=3_1|algebra=B5}}
+{{Quantum invariant table start|knot=3_1|algebra=C3}}
+{{Quantum invariant table entry|knot=3_1|algebra=C3|weight=1,0,0}}
+{{Quantum invariant table end|knot=3_1|algebra=C3}}
+{{Quantum invariant table start|knot=3_1|algebra=C4}}
+{{Quantum invariant table entry|knot=3_1|algebra=C4|weight=1,0,0,0}}
+{{Quantum invariant table end|knot=3_1|algebra=C4}}
 {{Quantum invariant table start|knot=3_1|algebra=D4}}
 {{Quantum invariant table entry|knot=3_1|algebra=D4|weight=0,1,0,0}}

3 1 Quantum Invariants: Difference between revisions

Latest revision as of 11:41, 28 June 2006

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