9 5

Polynomial invariants

Alexander polynomial	$6t-11+6t^{-1}$
Conway polynomial	$6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 23, 2 }
Jones polynomial	$-q^{10}+q^{9}-2q^{8}+3q^{7}-3q^{6}+4q^{5}-3q^{4}+3q^{3}-2q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{-2}+2z^{2}a^{-4}+2z^{2}a^{-6}+z^{2}a^{-8}+a^{-4}+a^{-6}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+2z^{7}a^{-7}+3z^{7}a^{-9}+z^{7}a^{-11}+3z^{6}a^{-6}-2z^{6}a^{-8}-5z^{6}a^{-10}+3z^{5}a^{-5}-5z^{5}a^{-7}-14z^{5}a^{-9}-6z^{5}a^{-11}+3z^{4}a^{-4}-7z^{4}a^{-6}-3z^{4}a^{-8}+7z^{4}a^{-10}+2z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+18z^{3}a^{-9}+11z^{3}a^{-11}+z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+4z^{2}a^{-8}-3z^{2}a^{-10}-6za^{-9}-6za^{-11}+a^{-4}-a^{-6}+a^{-10}$
The A2 invariant	$q^{-2}-q^{-4}+q^{-8}+q^{-12}+q^{-14}+q^{-16}+q^{-18}+q^{-22}-q^{-26}-q^{-30}-q^{-32}$
The G2 invariant	$q^{-10}-q^{-12}+q^{-14}-q^{-16}-q^{-22}+3q^{-24}-2q^{-26}+2q^{-28}-q^{-30}+q^{-34}-q^{-36}+3q^{-38}-2q^{-40}+q^{-42}+2q^{-48}-q^{-50}+q^{-52}-q^{-54}+q^{-56}-q^{-60}+q^{-62}+q^{-66}+q^{-72}+q^{-74}+2q^{-78}-2q^{-80}+5q^{-82}-q^{-84}-q^{-86}+5q^{-88}-4q^{-90}+6q^{-92}-2q^{-94}-q^{-96}+3q^{-98}-2q^{-100}+4q^{-102}-3q^{-104}-q^{-110}+q^{-112}-2q^{-114}-q^{-116}+q^{-118}-3q^{-120}-2q^{-124}-2q^{-126}+3q^{-128}-6q^{-130}+3q^{-132}-2q^{-134}-2q^{-136}+4q^{-138}-5q^{-140}+3q^{-142}-q^{-144}+q^{-148}-2q^{-150}+2q^{-152}+q^{-156}$

Further Quantum Invariants

Further quantum knot invariants for 9_5.

A1 Invariants.

Weight	Invariant
1	$q^{-1}-q^{-3}+q^{-5}+q^{-9}+q^{-11}+q^{-15}-q^{-17}-q^{-21}$
2	$q^{-2}-q^{-4}+2q^{-8}-q^{-10}+q^{-12}+q^{-14}-q^{-16}+q^{-18}+q^{-20}+2q^{-26}-q^{-30}+q^{-34}-q^{-36}-q^{-38}+q^{-40}-q^{-42}-2q^{-44}+q^{-46}-2q^{-50}+q^{-52}+q^{-54}-q^{-56}+q^{-60}$
3	$q^{-3}-q^{-5}+q^{-9}+q^{-11}-q^{-15}+q^{-17}+q^{-19}+q^{-21}-q^{-25}+q^{-27}+2q^{-29}+q^{-31}-3q^{-33}+2q^{-37}+2q^{-39}-q^{-43}-q^{-45}+q^{-47}+3q^{-49}+q^{-51}-3q^{-53}-2q^{-55}+2q^{-57}-q^{-59}-3q^{-61}-2q^{-63}+q^{-65}-q^{-71}+q^{-73}+q^{-75}-2q^{-79}-q^{-81}+q^{-83}+2q^{-85}-q^{-87}-2q^{-89}+3q^{-93}+2q^{-95}-2q^{-97}-2q^{-99}+q^{-101}+3q^{-103}-2q^{-107}-q^{-109}+q^{-111}+q^{-113}-q^{-117}$
4	$q^{-4}-q^{-6}+q^{-10}+2q^{-14}-2q^{-16}+q^{-20}+q^{-22}+4q^{-24}-4q^{-26}-2q^{-28}+q^{-30}+4q^{-32}+5q^{-34}-5q^{-36}-5q^{-38}+7q^{-42}+8q^{-44}-4q^{-46}-8q^{-48}-3q^{-50}+6q^{-52}+10q^{-54}-7q^{-58}-8q^{-60}-q^{-62}+7q^{-64}+4q^{-66}+2q^{-68}-3q^{-70}-6q^{-72}-q^{-74}+4q^{-76}+6q^{-78}+2q^{-80}-6q^{-82}-6q^{-84}-q^{-86}+4q^{-88}+3q^{-90}-4q^{-92}-6q^{-94}+3q^{-98}+q^{-100}-4q^{-102}-5q^{-104}+2q^{-106}+5q^{-108}+3q^{-110}-3q^{-112}-5q^{-114}+2q^{-116}+5q^{-118}+5q^{-120}+q^{-122}-5q^{-124}-2q^{-126}-q^{-128}+3q^{-130}+4q^{-132}-2q^{-134}-2q^{-136}-4q^{-138}-q^{-140}+5q^{-142}+3q^{-144}+3q^{-146}-3q^{-148}-5q^{-150}-q^{-152}+q^{-154}+6q^{-156}+2q^{-158}-3q^{-160}-4q^{-162}-4q^{-164}+3q^{-166}+4q^{-168}+2q^{-170}-q^{-172}-5q^{-174}-q^{-176}+q^{-178}+2q^{-180}+2q^{-182}-q^{-184}-q^{-186}-q^{-188}+q^{-192}$
5	$q^{-5}-q^{-7}+q^{-11}+q^{-15}-q^{-19}+2q^{-23}+2q^{-25}-2q^{-29}-3q^{-31}+q^{-33}+5q^{-35}+6q^{-37}-2q^{-39}-7q^{-41}-5q^{-43}+3q^{-45}+11q^{-47}+8q^{-49}-4q^{-51}-13q^{-53}-7q^{-55}+7q^{-57}+14q^{-59}+8q^{-61}-7q^{-63}-17q^{-65}-11q^{-67}+8q^{-69}+21q^{-71}+13q^{-73}-4q^{-75}-17q^{-77}-18q^{-79}-q^{-81}+17q^{-83}+17q^{-85}+5q^{-87}-8q^{-89}-17q^{-91}-12q^{-93}-q^{-95}+10q^{-97}+14q^{-99}+8q^{-101}-2q^{-103}-13q^{-105}-15q^{-107}-6q^{-109}+8q^{-111}+15q^{-113}+10q^{-115}-3q^{-117}-13q^{-119}-11q^{-121}+q^{-123}+8q^{-125}+7q^{-127}-2q^{-129}-8q^{-131}-5q^{-133}+2q^{-135}+5q^{-137}+4q^{-139}-6q^{-141}-10q^{-143}-3q^{-145}+7q^{-147}+12q^{-149}+8q^{-151}-6q^{-153}-14q^{-155}-8q^{-157}+5q^{-159}+15q^{-161}+14q^{-163}+q^{-165}-13q^{-167}-14q^{-169}-3q^{-171}+10q^{-173}+17q^{-175}+8q^{-177}-6q^{-179}-14q^{-181}-12q^{-183}+q^{-185}+10q^{-187}+11q^{-189}+4q^{-191}-5q^{-193}-11q^{-195}-10q^{-197}-q^{-199}+6q^{-201}+8q^{-203}+7q^{-205}+q^{-207}-6q^{-209}-8q^{-211}-4q^{-213}+6q^{-217}+8q^{-219}+5q^{-221}-2q^{-223}-7q^{-225}-8q^{-227}-5q^{-229}+2q^{-231}+8q^{-233}+8q^{-235}+3q^{-237}-4q^{-239}-8q^{-241}-7q^{-243}-q^{-245}+5q^{-247}+8q^{-249}+5q^{-251}-q^{-253}-5q^{-255}-6q^{-257}-3q^{-259}+2q^{-261}+5q^{-263}+3q^{-265}+q^{-267}-q^{-269}-3q^{-271}-2q^{-273}+q^{-277}+q^{-279}+q^{-281}-q^{-285}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-2}-q^{-4}+q^{-8}+q^{-12}+q^{-14}+q^{-16}+q^{-18}+q^{-22}-q^{-26}-q^{-30}-q^{-32}$
1,1	$q^{-4}-2q^{-6}+2q^{-8}-2q^{-10}+5q^{-12}-4q^{-14}+4q^{-16}-2q^{-18}+5q^{-20}-4q^{-22}+4q^{-24}+6q^{-28}+4q^{-32}+2q^{-34}+q^{-36}+2q^{-38}-4q^{-40}+2q^{-42}-11q^{-44}+10q^{-46}-16q^{-48}+12q^{-50}-16q^{-52}+14q^{-54}-10q^{-56}+8q^{-58}-3q^{-60}+2q^{-62}+4q^{-64}-8q^{-66}+7q^{-68}-12q^{-70}+10q^{-72}-10q^{-74}+6q^{-76}-4q^{-78}+4q^{-80}+q^{-84}$
2,0	$q^{-4}-q^{-6}-q^{-8}+2q^{-10}+q^{-12}-q^{-14}+2q^{-18}+q^{-20}-2q^{-22}+3q^{-26}+2q^{-28}+q^{-30}+2q^{-32}+q^{-34}+2q^{-36}+q^{-38}-q^{-42}+q^{-46}-2q^{-50}-q^{-52}-q^{-56}-3q^{-58}-2q^{-60}-2q^{-66}-q^{-68}+q^{-70}+q^{-72}-q^{-76}+q^{-78}+q^{-80}+q^{-82}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-4}-q^{-6}-q^{-8}+2q^{-10}-q^{-14}+2q^{-16}+q^{-18}+2q^{-22}+2q^{-24}+q^{-28}+q^{-30}+q^{-32}+2q^{-36}+3q^{-38}+q^{-40}+q^{-44}-3q^{-46}-3q^{-48}-3q^{-50}-4q^{-52}-2q^{-54}+q^{-60}+2q^{-62}+q^{-66}$
1,0,0	$q^{-3}-q^{-5}+q^{-11}+q^{-15}+q^{-17}+q^{-19}+q^{-21}+q^{-23}+q^{-25}+q^{-29}-q^{-35}-q^{-39}-q^{-41}-q^{-43}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-4}-q^{-6}+q^{-8}-2q^{-10}+2q^{-12}-q^{-14}+2q^{-16}-q^{-18}+2q^{-20}+2q^{-26}-q^{-28}+3q^{-30}-3q^{-32}+4q^{-34}-4q^{-36}+3q^{-38}-3q^{-40}+2q^{-42}-q^{-44}+q^{-46}+q^{-48}-q^{-50}+2q^{-52}-2q^{-54}+2q^{-56}-2q^{-58}+q^{-60}-2q^{-62}-q^{-66}$
1,0	$q^{-6}-q^{-10}-q^{-12}+2q^{-16}+q^{-18}-q^{-20}-q^{-22}+2q^{-26}+q^{-28}-q^{-32}+q^{-34}+2q^{-36}+q^{-38}-q^{-40}+q^{-44}+2q^{-46}-q^{-50}+2q^{-54}+q^{-56}+q^{-60}+2q^{-62}+q^{-64}-q^{-66}-q^{-68}+q^{-70}+q^{-72}-q^{-74}-4q^{-76}-2q^{-78}-2q^{-84}-3q^{-86}-q^{-88}+q^{-90}+q^{-92}-q^{-94}-q^{-96}+q^{-98}+2q^{-100}+q^{-108}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-10}-q^{-12}+q^{-14}-q^{-16}-q^{-22}+3q^{-24}-2q^{-26}+2q^{-28}-q^{-30}+q^{-34}-q^{-36}+3q^{-38}-2q^{-40}+q^{-42}+2q^{-48}-q^{-50}+q^{-52}-q^{-54}+q^{-56}-q^{-60}+q^{-62}+q^{-66}+q^{-72}+q^{-74}+2q^{-78}-2q^{-80}+5q^{-82}-q^{-84}-q^{-86}+5q^{-88}-4q^{-90}+6q^{-92}-2q^{-94}-q^{-96}+3q^{-98}-2q^{-100}+4q^{-102}-3q^{-104}-q^{-110}+q^{-112}-2q^{-114}-q^{-116}+q^{-118}-3q^{-120}-2q^{-124}-2q^{-126}+3q^{-128}-6q^{-130}+3q^{-132}-2q^{-134}-2q^{-136}+4q^{-138}-5q^{-140}+3q^{-142}-q^{-144}+q^{-148}-2q^{-150}+2q^{-152}+q^{-156}$

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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["9 5"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$6t-11+6t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$6z^{2}+1$

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]=

{ 23, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-8}+z^{8}a^{-10}+2z^{7}a^{-7}+3z^{7}a^{-9}+z^{7}a^{-11}+3z^{6}a^{-6}-2z^{6}a^{-8}-5z^{6}a^{-10}+3z^{5}a^{-5}-5z^{5}a^{-7}-14z^{5}a^{-9}-6z^{5}a^{-11}+3z^{4}a^{-4}-7z^{4}a^{-6}-3z^{4}a^{-8}+7z^{4}a^{-10}+2z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+18z^{3}a^{-9}+11z^{3}a^{-11}+z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+4z^{2}a^{-8}-3z^{2}a^{-10}-6za^{-9}-6za^{-11}+a^{-4}-a^{-6}+a^{-10}$