3 1

The trefoil knot has only three crossings!

Polynomial invariants

Alexander polynomial	$t+t^{-1}-1$
Conway polynomial	$z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 3, -2 }
Jones polynomial	$-q^{-4}+q^{-3}+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-a^{4}+a^{2}z^{2}+2a^{2}$
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^5 z+a^4 z^2-a^4+a^3 z+a^2 z^2-2 a^2}
The A2 invariant	$-q^{14}-q^{12}+q^{8}+2q^{6}+q^{4}+q^{2}$
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{72}-q^{64}-q^{62}-q^{56}-2 q^{54}-q^{52}+q^{50}-q^{46}-2 q^{44}+2 q^{40}+q^{38}-q^{36}+2 q^{32}+2 q^{30}+q^{28}+2 q^{22}+2 q^{20}+q^{14}+q^{12}+q^{10}}

Further Quantum Invariants

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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}-2 q^{24}-2 q^{22}-3 q^{20}-2 q^{18}+2 q^{14}+3 q^{12}+4 q^{10}+4 q^8+2 q^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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{19}-q^{17}-q^{15}+q^{11}+2 q^9+2 q^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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{19}-q^{17}-q^{15}+q^{11}+2 q^9+2 q^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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{24}-q^{22}-q^{20}-q^{18}+q^{14}+2 q^{12}+2 q^{10}+2 q^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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{29}-q^{27}-q^{25}-q^{23}-q^{21}+q^{17}+2 q^{15}+2 q^{13}+2 q^{11}+2 q^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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{30}-q^{24}+q^{14}+q^{12}+q^{10}+q^8+q^6+q^4}
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-q^{38}-q^{36}-q^{34}-q^{32}-q^{30}-q^{28}+q^{22}+q^{20}+2 q^{18}+q^{16}+2 q^{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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{96}-q^{78}-q^{74}-q^{70}-q^{68}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}-q^{54}+q^{46}+2 q^{42}+2 q^{38}+q^{36}+2 q^{34}+q^{32}+2 q^{30}+q^{28}+2 q^{26}+2 q^{22}+q^{18}+q^{14}}

B5 Invariants.

Weight

Invariant

1,0,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}+2 q^{54}+2 q^{50}+2 q^{46}+q^{44}+2 q^{42}+q^{40}+2 q^{38}+q^{36}+2 q^{34}+2 q^{30}+2 q^{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	$q^{72}-q^{64}-q^{62}-q^{56}-2q^{54}-q^{52}+q^{50}-q^{46}-2q^{44}+2q^{40}+q^{38}-q^{36}+2q^{32}+2q^{30}+q^{28}+2q^{22}+2q^{20}+q^{14}+q^{12}+q^{10}$

.

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["3 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t+t^{-1}-1$

In[5]:=

Conway[K][z]

Out[5]=

$z^{2}+1$

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]=

{ 3, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{-4}+q^{-3}+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{4}+a^{2}z^{2}+2a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^5 z+a^4 z^2-a^4+a^3 z+a^2 z^2-2 a^2}