10 142

Polynomial invariants

Alexander polynomial	$2t^{3}-3t^{2}+2t-1+2t^{-1}-3t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+9z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 15, 6 }
Jones polynomial	$-2q^{10}+2q^{9}-2q^{8}+3q^{7}-2q^{6}+2q^{5}-q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+z^{6}a^{-8}+5z^{4}a^{-6}+5z^{4}a^{-8}-z^{4}a^{-10}+6z^{2}a^{-6}+7z^{2}a^{-8}-5z^{2}a^{-10}+a^{-6}+4a^{-8}-5a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-7}+2z^{7}a^{-9}+z^{7}a^{-11}+z^{6}a^{-6}-5z^{6}a^{-8}-6z^{6}a^{-10}-4z^{5}a^{-7}-9z^{5}a^{-9}-5z^{5}a^{-11}-5z^{4}a^{-6}+9z^{4}a^{-8}+15z^{4}a^{-10}+z^{4}a^{-12}+3z^{3}a^{-7}+12z^{3}a^{-9}+9z^{3}a^{-11}+6z^{2}a^{-6}-10z^{2}a^{-8}-17z^{2}a^{-10}-z^{2}a^{-12}-6za^{-9}-4za^{-11}+2za^{-13}-a^{-6}+4a^{-8}+5a^{-10}+a^{-12}$
The A2 invariant	$q^{-10}+q^{-14}+q^{-18}+q^{-20}+2q^{-22}+3q^{-24}-q^{-28}-3q^{-30}-2q^{-32}-q^{-34}+q^{-38}$
The G2 invariant	$q^{-50}+q^{-54}-q^{-56}+q^{-58}+3q^{-64}-2q^{-66}+4q^{-68}-q^{-70}-q^{-72}+3q^{-74}-2q^{-76}+2q^{-78}-q^{-82}+3q^{-84}+3q^{-90}-4q^{-92}+4q^{-94}-2q^{-98}+5q^{-100}-2q^{-102}+4q^{-104}+q^{-106}+3q^{-108}+2q^{-110}-2q^{-112}+q^{-114}+3q^{-118}-q^{-120}-3q^{-122}-q^{-124}+q^{-126}-q^{-130}-9q^{-132}+q^{-136}-4q^{-138}-7q^{-142}+q^{-144}+3q^{-146}-3q^{-148}-2q^{-150}-2q^{-152}+q^{-154}+3q^{-156}-q^{-158}-q^{-160}+2q^{-162}+2q^{-166}+2q^{-168}-q^{-170}+q^{-172}$

Further Quantum Invariants

Further quantum knot invariants for 10_142.

A1 Invariants.

Weight	Invariant
1	$q^{-5}+q^{-9}+q^{-13}+q^{-15}-2q^{-21}$
2	$q^{-10}+2q^{-16}+q^{-18}-q^{-20}+q^{-22}+2q^{-24}-q^{-28}-q^{-34}+q^{-36}+2q^{-38}-q^{-40}-q^{-46}-3q^{-48}-q^{-50}-q^{-54}+q^{-56}+q^{-58}+q^{-60}$
3	$q^{-15}+q^{-21}+2q^{-23}+q^{-25}-q^{-27}-q^{-29}+2q^{-31}+3q^{-33}+q^{-35}-2q^{-37}-2q^{-39}+q^{-41}+3q^{-43}+q^{-45}-2q^{-47}-3q^{-49}+q^{-51}+4q^{-53}+q^{-55}-4q^{-57}-3q^{-59}+2q^{-61}+2q^{-63}-3q^{-65}-2q^{-67}+3q^{-69}+q^{-71}-2q^{-73}-q^{-75}+q^{-77}-q^{-81}-3q^{-83}-2q^{-85}+3q^{-89}-2q^{-91}-4q^{-93}+7q^{-97}+4q^{-99}-2q^{-101}-2q^{-103}+q^{-105}+4q^{-107}-2q^{-111}-2q^{-113}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-10}+q^{-14}+q^{-18}+q^{-20}+2q^{-22}+3q^{-24}-q^{-28}-3q^{-30}-2q^{-32}-q^{-34}+q^{-38}$
1,1	$q^{-20}+2q^{-24}-2q^{-26}+6q^{-28}-2q^{-30}+12q^{-32}-2q^{-34}+7q^{-36}+6q^{-42}-10q^{-44}+10q^{-46}-10q^{-48}+8q^{-50}-11q^{-52}+2q^{-54}-8q^{-56}-4q^{-58}-3q^{-60}-6q^{-62}+6q^{-64}-4q^{-66}+10q^{-68}+2q^{-72}+2q^{-74}-2q^{-76}-2q^{-78}-4q^{-80}-2q^{-82}+4q^{-84}+2q^{-88}$
2,0	$q^{-20}+q^{-26}+2q^{-28}+q^{-30}+q^{-32}+2q^{-34}+3q^{-36}+2q^{-38}+q^{-40}+q^{-42}-q^{-46}+q^{-52}+2q^{-54}+3q^{-56}+2q^{-58}+q^{-60}-4q^{-62}-6q^{-64}-10q^{-66}-9q^{-68}-5q^{-70}-q^{-72}+3q^{-74}+4q^{-76}+6q^{-78}+5q^{-80}+4q^{-82}-2q^{-88}-2q^{-90}-q^{-92}+q^{-96}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-20}+q^{-24}+q^{-26}+q^{-30}+2q^{-32}+3q^{-34}+5q^{-36}+4q^{-38}+4q^{-40}+2q^{-42}-3q^{-46}-3q^{-48}-5q^{-50}-4q^{-52}-4q^{-54}-3q^{-56}-q^{-58}-q^{-60}+q^{-62}+q^{-64}+q^{-66}+q^{-68}+2q^{-70}$
1,0,0	$q^{-15}+q^{-19}+q^{-23}+2q^{-27}+3q^{-29}+3q^{-31}+3q^{-33}-q^{-37}-4q^{-39}-3q^{-41}-3q^{-43}-q^{-45}+q^{-49}+q^{-51}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-30}+q^{-34}+q^{-36}+q^{-38}+2q^{-42}+2q^{-44}+4q^{-46}+5q^{-48}+5q^{-50}+7q^{-52}+7q^{-54}+5q^{-56}+4q^{-58}+4q^{-60}+2q^{-62}-2q^{-64}-4q^{-66}-6q^{-68}-10q^{-70}-14q^{-72}-12q^{-74}-12q^{-76}-10q^{-78}-2q^{-80}+3q^{-82}+6q^{-84}+8q^{-86}+10q^{-88}+6q^{-90}+2q^{-92}-2q^{-98}-2q^{-100}$
1,0,0,0	$q^{-20}+q^{-24}+q^{-28}+q^{-32}+2q^{-34}+3q^{-36}+4q^{-38}+3q^{-40}+3q^{-42}-q^{-46}-4q^{-48}-4q^{-50}-4q^{-52}-3q^{-54}-q^{-56}+q^{-60}+q^{-62}+q^{-64}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-20}+q^{-24}-q^{-26}+2q^{-28}-q^{-30}+2q^{-32}+q^{-34}+q^{-36}+2q^{-38}+2q^{-42}-2q^{-44}+3q^{-46}-3q^{-48}+q^{-50}-2q^{-52}-q^{-56}-q^{-58}+q^{-60}-q^{-62}+q^{-64}-q^{-66}+q^{-68}-2q^{-70}$
1,0	$q^{-30}+q^{-38}+q^{-40}-q^{-44}+q^{-46}+2q^{-48}+q^{-50}-q^{-52}+2q^{-54}+3q^{-56}+3q^{-58}+q^{-60}+q^{-62}+q^{-64}+2q^{-66}-q^{-70}-q^{-72}-q^{-76}-2q^{-78}-2q^{-80}-q^{-82}-2q^{-86}-3q^{-88}-2q^{-90}-q^{-94}-2q^{-96}-q^{-98}+q^{-100}+q^{-102}+q^{-108}+q^{-110}+2q^{-112}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-30}+q^{-34}+2q^{-38}-q^{-40}+2q^{-42}+3q^{-46}+3q^{-48}+3q^{-50}+5q^{-52}+5q^{-54}+6q^{-56}+3q^{-58}+4q^{-60}-2q^{-62}-6q^{-66}-4q^{-68}-8q^{-70}-5q^{-72}-6q^{-74}-3q^{-76}-2q^{-78}-q^{-80}+q^{-82}+2q^{-86}+2q^{-90}+q^{-94}+2q^{-98}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-50}+q^{-54}-q^{-56}+q^{-58}+3q^{-64}-2q^{-66}+4q^{-68}-q^{-70}-q^{-72}+3q^{-74}-2q^{-76}+2q^{-78}-q^{-82}+3q^{-84}+3q^{-90}-4q^{-92}+4q^{-94}-2q^{-98}+5q^{-100}-2q^{-102}+4q^{-104}+q^{-106}+3q^{-108}+2q^{-110}-2q^{-112}+q^{-114}+3q^{-118}-q^{-120}-3q^{-122}-q^{-124}+q^{-126}-q^{-130}-9q^{-132}+q^{-136}-4q^{-138}-7q^{-142}+q^{-144}+3q^{-146}-3q^{-148}-2q^{-150}-2q^{-152}+q^{-154}+3q^{-156}-q^{-158}-q^{-160}+2q^{-162}+2q^{-166}+2q^{-168}-q^{-170}+q^{-172}$

.

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["10 142"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-3t^{2}+2t-1+2t^{-1}-3t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+9z^{4}+8z^{2}+1$

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 15, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-2q^{10}+2q^{9}-2q^{8}+3q^{7}-2q^{6}+2q^{5}-q^{4}+q^{3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-6}+z^{6}a^{-8}+5z^{4}a^{-6}+5z^{4}a^{-8}-z^{4}a^{-10}+6z^{2}a^{-6}+7z^{2}a^{-8}-5z^{2}a^{-10}+a^{-6}+4a^{-8}-5a^{-10}+a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-7}+2z^{7}a^{-9}+z^{7}a^{-11}+z^{6}a^{-6}-5z^{6}a^{-8}-6z^{6}a^{-10}-4z^{5}a^{-7}-9z^{5}a^{-9}-5z^{5}a^{-11}-5z^{4}a^{-6}+9z^{4}a^{-8}+15z^{4}a^{-10}+z^{4}a^{-12}+3z^{3}a^{-7}+12z^{3}a^{-9}+9z^{3}a^{-11}+6z^{2}a^{-6}-10z^{2}a^{-8}-17z^{2}a^{-10}-z^{2}a^{-12}-6za^{-9}-4za^{-11}+2za^{-13}-a^{-6}+4a^{-8}+5a^{-10}+a^{-12}$