10 1

Polynomial invariants

Alexander polynomial	$-4t+9-4t^{-1}$
Conway polynomial	$1-4z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, 0 }
Jones polynomial	$q^{2}-q+2-2q^{-1}+2q^{-2}-2q^{-3}+2q^{-4}-2q^{-5}+q^{-6}-q^{-7}+q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$a^{8}-z^{2}a^{6}-a^{6}-z^{2}a^{4}-z^{2}a^{2}-z^{2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{7}z^{9}+a^{5}z^{9}+a^{8}z^{8}+2a^{6}z^{8}+a^{4}z^{8}-7a^{7}z^{7}-6a^{5}z^{7}+a^{3}z^{7}-7a^{8}z^{6}-12a^{6}z^{6}-4a^{4}z^{6}+a^{2}z^{6}+16a^{7}z^{5}+12a^{5}z^{5}-3a^{3}z^{5}+az^{5}+15a^{8}z^{4}+21a^{6}z^{4}+3a^{4}z^{4}-2a^{2}z^{4}+z^{4}-14a^{7}z^{3}-11a^{5}z^{3}+a^{3}z^{3}-az^{3}+z^{3}a^{-1}-10a^{8}z^{2}-11a^{6}z^{2}+z^{2}a^{-2}+4a^{7}z+4a^{5}z+a^{8}+a^{6}-a^{-2}$
The A2 invariant	$q^{26}+q^{24}-q^{18}-q^{16}+q^{-2}+q^{-6}+q^{-8}$
The G2 invariant	Data:10 1/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_1.

A1 Invariants.

Weight	Invariant
1	$q^{17}-q^{11}+q^{-1}+q^{-5}$
2	$q^{50}-q^{46}-q^{40}+q^{36}+q^{16}+q^{14}-q^{4}-q^{2}+q^{-2}+q^{-8}+q^{-14}$
3	$q^{99}-q^{95}-q^{93}+q^{89}-q^{85}+q^{81}+q^{79}-q^{75}+q^{49}+q^{47}-q^{43}-q^{37}-q^{35}-q^{29}+q^{25}+q^{23}+q^{15}+q^{13}+q^{11}-q^{9}+q^{5}+q^{3}-q-q^{-1}+q^{-3}+q^{-5}-q^{-7}-q^{-9}+q^{-19}+q^{-27}$

A2 Invariants.

Weight	Invariant
1,0	$q^{26}+q^{24}-q^{18}-q^{16}+q^{-2}+q^{-6}+q^{-8}$
2,0	$q^{68}+q^{66}+q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}-q^{52}+q^{50}+q^{48}+q^{46}+q^{24}+2q^{22}+q^{20}+q^{18}+q^{16}-q^{10}-2q^{8}-2q^{6}-q^{4}+q^{-4}+q^{-10}+q^{-12}+q^{-16}+q^{-18}+q^{-20}$

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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["10 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-4t+9-4t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$1-4z^{2}$

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]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{2}-q+2-2q^{-1}+2q^{-2}-2q^{-3}+2q^{-4}-2q^{-5}+q^{-6}-q^{-7}+q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{8}-z^{2}a^{6}-a^{6}-z^{2}a^{4}-z^{2}a^{2}-z^{2}+a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{7}z^{9}+a^{5}z^{9}+a^{8}z^{8}+2a^{6}z^{8}+a^{4}z^{8}-7a^{7}z^{7}-6a^{5}z^{7}+a^{3}z^{7}-7a^{8}z^{6}-12a^{6}z^{6}-4a^{4}z^{6}+a^{2}z^{6}+16a^{7}z^{5}+12a^{5}z^{5}-3a^{3}z^{5}+az^{5}+15a^{8}z^{4}+21a^{6}z^{4}+3a^{4}z^{4}-2a^{2}z^{4}+z^{4}-14a^{7}z^{3}-11a^{5}z^{3}+a^{3}z^{3}-az^{3}+z^{3}a^{-1}-10a^{8}z^{2}-11a^{6}z^{2}+z^{2}a^{-2}+4a^{7}z+4a^{5}z+a^{8}+a^{6}-a^{-2}$