4 1

Polynomial invariants

Alexander polynomial	$-t-t^{-1}+3$
Conway polynomial	$1-z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, 0 }
Jones polynomial	$q^{2}+q^{-2}-q-q^{-1}+1$
HOMFLY-PT polynomial (db, data sources)	$a^{2}+a^{-2}-z^{2}-1$
Kauffman polynomial (db, data sources)	$a^{2}z^{2}+z^{2}a^{-2}-a^{2}-a^{-2}+az^{3}+z^{3}a^{-1}-az-za^{-1}+2z^{2}-1$
The A2 invariant	$q^{8}+q^{6}-1+q^{-6}+q^{-8}$
The G2 invariant	$q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^{4}-1-q^{-4}-q^{-10}+q^{-16}+q^{-18}+q^{-24}+q^{-26}+q^{-28}-q^{-30}+q^{-34}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 4_1.

A1 Invariants.

Weight	Invariant
1	$q^{5}+q^{-5}$
2	$q^{14}-q^{10}+q^{2}+1+q^{-2}-q^{-10}+q^{-14}$
3	$q^{27}-q^{23}-q^{21}+q^{17}+q^{11}+q^{9}+q^{-9}+q^{-11}+q^{-17}-q^{-21}-q^{-23}+q^{-27}$
4	$q^{44}-q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}-q^{26}+q^{24}+q^{22}-q^{18}-q^{16}+q^{4}+q^{2}+1+q^{-2}+q^{-4}-q^{-16}-q^{-18}+q^{-22}+q^{-24}-q^{-26}+q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}-q^{-40}+q^{-44}$
5	$q^{65}-q^{61}-q^{59}-q^{57}+q^{53}+2q^{51}+q^{49}-q^{45}-q^{43}+q^{39}+q^{37}-q^{35}-2q^{33}-q^{31}+q^{27}+q^{25}+q^{17}+q^{15}+q^{13}+q^{-13}+q^{-15}+q^{-17}+q^{-25}+q^{-27}-q^{-31}-2q^{-33}-q^{-35}+q^{-37}+q^{-39}-q^{-43}-q^{-45}+q^{-49}+2q^{-51}+q^{-53}-q^{-57}-q^{-59}-q^{-61}+q^{-65}$
6	$q^{90}-q^{86}-q^{84}-q^{82}+2q^{76}+2q^{74}+q^{72}-q^{68}-2q^{66}-2q^{64}+q^{62}+q^{60}+q^{58}-q^{54}-2q^{52}-2q^{50}+q^{48}+2q^{46}+2q^{44}+q^{42}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}-q^{26}-q^{24}-q^{22}+q^{6}+q^{4}+q^{2}+1+q^{-2}+q^{-4}+q^{-6}-q^{-22}-q^{-24}-q^{-26}+q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-42}+2q^{-44}+2q^{-46}+q^{-48}-2q^{-50}-2q^{-52}-q^{-54}+q^{-58}+q^{-60}+q^{-62}-2q^{-64}-2q^{-66}-q^{-68}+q^{-72}+2q^{-74}+2q^{-76}-q^{-82}-q^{-84}-q^{-86}+q^{-90}$

A2 Invariants.

Weight	Invariant
1,0	$q^{8}+q^{6}-1+q^{-6}+q^{-8}$
1,1	$q^{20}+2q^{16}-2q^{10}-2q^{8}+4q^{2}+2+4q^{-2}-2q^{-8}-2q^{-10}+2q^{-16}+q^{-20}$
2,0	$q^{20}+q^{18}+q^{16}-q^{14}-q^{12}-q^{10}-q^{8}+q^{4}+2q^{2}+2+2q^{-2}+q^{-4}-q^{-8}-q^{-10}-q^{-12}-q^{-14}+q^{-16}+q^{-18}+q^{-20}$
3,0	$q^{36}+q^{34}+q^{32}-2q^{28}-2q^{26}-2q^{24}+q^{18}+2q^{16}+3q^{14}+3q^{12}+2q^{10}+q^{8}-q^{4}-2q^{2}-2-2q^{-2}-q^{-4}+q^{-8}+2q^{-10}+3q^{-12}+3q^{-14}+2q^{-16}+q^{-18}-2q^{-24}-2q^{-26}-2q^{-28}+q^{-32}+q^{-34}+q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{16}+q^{12}+q^{10}+q^{-10}+q^{-12}+q^{-16}$
1,0,0	$q^{11}+q^{9}+q^{7}-q-q^{-1}+q^{-7}+q^{-9}+q^{-11}$
1,0,1	$q^{26}+2q^{22}+2q^{20}+q^{18}+2q^{16}-2q^{14}-2q^{12}-4q^{10}-4q^{8}+2q^{4}+6q^{2}+7+6q^{-2}+2q^{-4}-4q^{-8}-4q^{-10}-2q^{-12}-2q^{-14}+2q^{-16}+q^{-18}+2q^{-20}+2q^{-22}+q^{-26}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-q^{10}-q^{8}+q^{2}+2+q^{-2}-q^{-8}-q^{-10}+q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}$
1,0,0,0	$q^{14}+q^{12}+q^{10}+q^{8}-q^{2}-1-q^{-2}+q^{-8}+q^{-10}+q^{-12}+q^{-14}$

B2 Invariants.

Weight	Invariant
0,1	$q^{16}+q^{12}+q^{10}-2+q^{-10}+q^{-12}+q^{-16}$
1,0	$q^{26}+q^{18}+1+q^{-18}+q^{-26}$

B3 Invariants.

Weight	Invariant
1,0,0	$q^{38}+q^{30}+q^{26}+q^{22}-q^{2}+1-q^{-2}+q^{-22}+q^{-26}+q^{-30}+q^{-38}$

B4 Invariants.

Weight	Invariant
1,0,0,0	$q^{50}+q^{42}+q^{38}+q^{34}+q^{30}+q^{26}-q^{6}-q^{2}+1-q^{-2}-q^{-6}+q^{-26}+q^{-30}+q^{-34}+q^{-38}+q^{-42}+q^{-50}$

C3 Invariants.

Weight	Invariant
1,0,0	$q^{22}+q^{18}+q^{16}+q^{14}+q^{12}-q^{2}-2-q^{-2}+q^{-12}+q^{-14}+q^{-16}+q^{-18}+q^{-22}$

C4 Invariants.

Weight	Invariant
1,0,0,0	$q^{28}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-q^{4}-q^{2}-2-q^{-2}-q^{-4}+q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}+q^{-24}+q^{-28}$

D4 Invariants.

Weight	Invariant
0,1,0,0	$q^{38}+q^{34}+3q^{32}+2q^{30}+q^{28}+4q^{26}+q^{24}-2q^{18}-5q^{16}-3q^{14}-3q^{12}-4q^{10}+3q^{6}+5q^{4}+6q^{2}+8+6q^{-2}+5q^{-4}+3q^{-6}-4q^{-10}-3q^{-12}-3q^{-14}-5q^{-16}-2q^{-18}+q^{-24}+4q^{-26}+q^{-28}+2q^{-30}+3q^{-32}+q^{-34}+q^{-38}$
1,0,0,0	$q^{22}+q^{18}+q^{16}+q^{14}+q^{12}-q^{2}-q^{-2}+q^{-12}+q^{-14}+q^{-16}+q^{-18}+q^{-22}$

G2 Invariants.

Weight	Invariant
1,0	$q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^{4}-1-q^{-4}-q^{-10}+q^{-16}+q^{-18}+q^{-24}+q^{-26}+q^{-28}-q^{-30}+q^{-34}+q^{-38}$

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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["4 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t-t^{-1}+3$

In[5]:=

Conway[K][z]

Out[5]=

$1-z^{2}$

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]=

{ 5, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{2}+q^{-2}-q-q^{-1}+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{2}+a^{-2}-z^{2}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{2}+z^{2}a^{-2}-a^{2}-a^{-2}+az^{3}+z^{3}a^{-1}-az-za^{-1}+2z^{2}-1$