10 136

Polynomial invariants

Alexander polynomial	$-t^{2}+4t-5+4t^{-1}-t^{-2}$
Conway polynomial	$1-z^{4}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 15, 2 }
Jones polynomial	$-q^{4}+2q^{3}-2q^{2}+3q-2+2q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}+a^{2}z^{2}+2z^{2}a^{-2}-3z^{2}+a^{2}+3a^{-2}-a^{-4}-2$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}+2az^{7}+3z^{7}a^{-1}+z^{7}a^{-3}+a^{2}z^{6}-4z^{6}a^{-2}-3z^{6}-9az^{5}-14z^{5}a^{-1}-5z^{5}a^{-3}-4a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}+9az^{3}+16z^{3}a^{-1}+7z^{3}a^{-3}+3a^{2}z^{2}+4z^{2}a^{-2}+z^{2}a^{-4}+6z^{2}-2az-4za^{-1}-2za^{-3}-a^{2}-3a^{-2}-a^{-4}-2$
The A2 invariant	$q^{10}-q^{2}+q^{-4}+2q^{-6}+q^{-8}+q^{-10}-q^{-12}-q^{-14}$
The G2 invariant	$q^{46}-q^{44}+2q^{42}-2q^{40}+q^{38}-2q^{34}+6q^{32}-4q^{30}+3q^{28}-2q^{24}+3q^{22}-2q^{20}-q^{18}+3q^{16}-3q^{14}+3q^{10}-6q^{8}+6q^{6}-7q^{4}+3-5q^{-2}+4q^{-4}-3q^{-6}+3q^{-8}+q^{-12}-q^{-14}+2q^{-18}+q^{-20}+q^{-24}+3q^{-26}+4q^{-30}-6q^{-32}+6q^{-34}-2q^{-36}+4q^{-40}-7q^{-42}+6q^{-44}-2q^{-48}+q^{-50}-2q^{-52}-2q^{-54}+3q^{-56}-3q^{-58}+q^{-60}+q^{-62}-3q^{-64}+3q^{-66}-3q^{-68}+q^{-70}+q^{-72}-2q^{-74}+q^{-76}$

Further Quantum Invariants

Further quantum knot invariants for 10_136.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+q^{-1}+q^{-3}+q^{-7}-q^{-9}$
2	$q^{22}-q^{20}-2q^{18}+2q^{16}+q^{14}-q^{12}+q^{8}+q^{6}-2q^{4}+2-q^{-2}+2q^{-6}-q^{-8}+2q^{-12}+q^{-14}-q^{-16}-q^{-18}+q^{-20}-2q^{-24}+q^{-28}$
3	$q^{45}-q^{43}-2q^{41}+3q^{37}+3q^{35}-3q^{33}-3q^{31}+2q^{27}+2q^{25}-2q^{21}-4q^{19}+q^{17}+6q^{15}+2q^{13}-6q^{11}-4q^{9}+5q^{7}+6q^{5}-3q^{3}-5q+2q^{-1}+6q^{-3}-q^{-5}-4q^{-7}+q^{-9}+3q^{-11}-q^{-13}-2q^{-15}+q^{-17}+q^{-19}+2q^{-21}-4q^{-25}-q^{-27}+7q^{-29}+5q^{-31}-7q^{-33}-10q^{-35}+5q^{-37}+8q^{-39}-2q^{-41}-7q^{-43}-q^{-45}+5q^{-47}+2q^{-49}-q^{-51}-q^{-53}$

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-q^{18}+q^{14}+q^{10}+q^{6}-q^{4}-q^{2}-q^{-2}+q^{-6}+3q^{-8}+q^{-10}+3q^{-12}+q^{-14}-q^{-18}-q^{-20}-q^{-24}$
1,0,0	$q^{13}+q^{9}-q^{3}-q-q^{-1}+q^{-5}+2q^{-7}+3q^{-9}+q^{-11}+q^{-13}-q^{-15}-q^{-17}-q^{-19}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-q^{18}+2q^{16}-q^{14}+2q^{12}-q^{10}-q^{6}-q^{4}+q^{2}-4+3q^{-2}-2q^{-4}+3q^{-6}-q^{-8}+3q^{-10}+q^{-12}+q^{-14}+2q^{-16}-q^{-18}+q^{-20}-2q^{-22}+q^{-24}-2q^{-26}$
1,0	$q^{34}-q^{30}-q^{28}+q^{26}+q^{24}-q^{22}-q^{20}+2q^{18}+2q^{16}-q^{14}-q^{12}+2q^{8}-2q^{4}-q^{2}+1-q^{-4}+2q^{-10}+q^{-12}+3q^{-18}+2q^{-20}-2q^{-24}+q^{-26}+q^{-28}-q^{-30}-2q^{-32}+q^{-36}-q^{-40}-q^{-42}+q^{-44}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-q^{44}+2q^{42}-2q^{40}+q^{38}-2q^{34}+6q^{32}-4q^{30}+3q^{28}-2q^{24}+3q^{22}-2q^{20}-q^{18}+3q^{16}-3q^{14}+3q^{10}-6q^{8}+6q^{6}-7q^{4}+3-5q^{-2}+4q^{-4}-3q^{-6}+3q^{-8}+q^{-12}-q^{-14}+2q^{-18}+q^{-20}+q^{-24}+3q^{-26}+4q^{-30}-6q^{-32}+6q^{-34}-2q^{-36}+4q^{-40}-7q^{-42}+6q^{-44}-2q^{-48}+q^{-50}-2q^{-52}-2q^{-54}+3q^{-56}-3q^{-58}+q^{-60}+q^{-62}-3q^{-64}+3q^{-66}-3q^{-68}+q^{-70}+q^{-72}-2q^{-74}+q^{-76}$

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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["10 136"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{2}+4t-5+4t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-z^{4}$

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]=

{ 15, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+2q^{3}-2q^{2}+3q-2+2q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}+a^{2}z^{2}+2z^{2}a^{-2}-3z^{2}+a^{2}+3a^{-2}-a^{-4}-2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-2}+z^{8}+2az^{7}+3z^{7}a^{-1}+z^{7}a^{-3}+a^{2}z^{6}-4z^{6}a^{-2}-3z^{6}-9az^{5}-14z^{5}a^{-1}-5z^{5}a^{-3}-4a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}+9az^{3}+16z^{3}a^{-1}+7z^{3}a^{-3}+3a^{2}z^{2}+4z^{2}a^{-2}+z^{2}a^{-4}+6z^{2}-2az-4za^{-1}-2za^{-3}-a^{2}-3a^{-2}-a^{-4}-2$