10 151

Polynomial invariants

Alexander polynomial	$t^{3}-4t^{2}+10t-13+10t^{-1}-4t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+2z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, 2 }
Jones polynomial	$-2q^{6}+4q^{5}-6q^{4}+8q^{3}-7q^{2}+7q-5+3q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+6z^{2}a^{-2}-z^{2}a^{-4}-2z^{2}+3a^{-2}-a^{-6}-1$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+5z^{7}a^{-3}+2z^{7}a^{-5}+5z^{6}a^{-2}+3z^{6}a^{-4}+z^{6}a^{-6}+3z^{6}+az^{5}-4z^{5}a^{-1}-7z^{5}a^{-3}-2z^{5}a^{-5}-15z^{4}a^{-2}-6z^{4}a^{-4}+2z^{4}a^{-6}-7z^{4}-2az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}+5z^{3}a^{-5}+3z^{3}a^{-7}+10z^{2}a^{-2}+4z^{2}a^{-4}-2z^{2}a^{-6}+4z^{2}+az+2za^{-1}+za^{-3}-3za^{-5}-3za^{-7}-3a^{-2}+a^{-6}-1$
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+q^4-q^2+2 q^{-2} - q^{-4} +3 q^{-6} +2 q^{-10} + q^{-12} - q^{-14} + q^{-16} -2 q^{-18} - q^{-20} }
The G2 invariant	$q^{32}-2q^{30}+5q^{28}-8q^{26}+7q^{24}-4q^{22}-6q^{20}+19q^{18}-30q^{16}+34q^{14}-27q^{12}+q^{10}+27q^{8}-52q^{6}+62q^{4}-46q^{2}+15+23q^{-2}-51q^{-4}+55q^{-6}-34q^{-8}+33q^{-12}-46q^{-14}+35q^{-16}-35q^{-20}+62q^{-22}-62q^{-24}+40q^{-26}-q^{-28}-43q^{-30}+75q^{-32}-80q^{-34}+64q^{-36}-24q^{-38}-18q^{-40}+56q^{-42}-68q^{-44}+57q^{-46}-25q^{-48}-11q^{-50}+42q^{-52}-45q^{-54}+24q^{-56}+13q^{-58}-42q^{-60}+55q^{-62}-42q^{-64}+5q^{-66}+29q^{-68}-56q^{-70}+60q^{-72}-45q^{-74}+15q^{-76}+13q^{-78}-34q^{-80}+34q^{-82}-28q^{-84}+15q^{-86}-2q^{-88}-7q^{-90}+7q^{-92}-8q^{-94}+6q^{-96}-2q^{-98}+q^{-100}+q^{-102}$

Further Quantum Invariants

Further quantum knot invariants for 10_151.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+2q^{3}-2q+2q^{-1}+q^{-5}+2q^{-7}-2q^{-9}+2q^{-11}-2q^{-13}$
2	$q^{16}-2q^{14}-2q^{12}+7q^{10}-2q^{8}-10q^{6}+10q^{4}+5q^{2}-14+5q^{-2}+10q^{-4}-9q^{-6}-q^{-8}+9q^{-10}-6q^{-14}+2q^{-16}+10q^{-18}-10q^{-20}-6q^{-22}+15q^{-24}-6q^{-26}-10q^{-28}+9q^{-30}-5q^{-34}+2q^{-36}+q^{-38}$
3	$-q^{33}+2q^{31}+2q^{29}-3q^{27}-7q^{25}+2q^{23}+17q^{21}+2q^{19}-25q^{17}-17q^{15}+28q^{13}+38q^{11}-22q^{9}-55q^{7}+5q^{5}+64q^{3}+19q-67q^{-1}-37q^{-3}+56q^{-5}+52q^{-7}-40q^{-9}-55q^{-11}+26q^{-13}+57q^{-15}-10q^{-17}-47q^{-19}-5q^{-21}+40q^{-23}+21q^{-25}-31q^{-27}-39q^{-29}+17q^{-31}+56q^{-33}-3q^{-35}-65q^{-37}-20q^{-39}+71q^{-41}+35q^{-43}-61q^{-45}-50q^{-47}+41q^{-49}+52q^{-51}-20q^{-53}-45q^{-55}+5q^{-57}+28q^{-59}+6q^{-61}-14q^{-63}-6q^{-65}+7q^{-67}+2q^{-69}-2q^{-73}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+q^{4}-q^{2}+2q^{-2}-q^{-4}+3q^{-6}+2q^{-10}+q^{-12}-q^{-14}+q^{-16}-2q^{-18}-q^{-20}$
1,1	$q^{20}-4q^{18}+12q^{16}-28q^{14}+52q^{12}-86q^{10}+128q^{8}-170q^{6}+196q^{4}-206q^{2}+188-134q^{-2}+52q^{-4}+48q^{-6}-152q^{-8}+254q^{-10}-324q^{-12}+378q^{-14}-380q^{-16}+360q^{-18}-295q^{-20}+208q^{-22}-108q^{-24}+92q^{-28}-166q^{-30}+200q^{-32}-208q^{-34}+188q^{-36}-154q^{-38}+108q^{-40}-70q^{-42}+40q^{-44}-18q^{-46}+6q^{-48}-2q^{-50}+2q^{-54}$
2,0	$q^{18}-q^{16}-2q^{14}+2q^{12}+2q^{10}-2q^{8}-4q^{6}+2q^{4}+5q^{2}-5-3q^{-2}+6q^{-4}+q^{-6}-3q^{-8}+5q^{-12}+q^{-16}+6q^{-18}+3q^{-20}-2q^{-22}+4q^{-24}+4q^{-26}-7q^{-28}-3q^{-30}+3q^{-32}-q^{-34}-6q^{-36}-4q^{-38}+3q^{-40}-q^{-42}-3q^{-44}+q^{-46}+2q^{-48}+2q^{-50}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-2q^{12}+q^{10}+3q^{8}-7q^{6}+3q^{4}+3q^{2}-11+6q^{-2}+5q^{-4}-8q^{-6}+6q^{-8}+7q^{-10}-q^{-12}+q^{-14}+4q^{-16}+4q^{-18}-q^{-20}-3q^{-22}+9q^{-24}-6q^{-26}-9q^{-28}+8q^{-30}-7q^{-32}-7q^{-34}+7q^{-36}-q^{-38}-2q^{-40}+3q^{-42}$
1,0,0	$-q^{7}+q^{5}-2q^{3}+q-q^{-1}+2q^{-3}+2q^{-7}+2q^{-9}+q^{-11}+2q^{-13}+2q^{-17}-2q^{-19}+q^{-21}-2q^{-23}-q^{-25}-q^{-27}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{16}-q^{14}+3q^{10}-2q^{8}-4q^{6}+4q^{4}-q^{2}-9-q^{-2}+6q^{-4}-3q^{-6}-7q^{-8}+9q^{-10}+10q^{-12}-4q^{-14}+2q^{-16}+16q^{-18}-q^{-20}-2q^{-22}+12q^{-24}+5q^{-26}-6q^{-28}+3q^{-30}+6q^{-32}-8q^{-34}-11q^{-36}-2q^{-40}-12q^{-42}-3q^{-44}+5q^{-46}-q^{-50}+3q^{-52}+2q^{-54}+q^{-56}$
1,0,0,0	$-q^{8}+q^{6}-2q^{4}-q^{-2}+2q^{-4}+3q^{-8}+q^{-10}+3q^{-12}+q^{-14}+2q^{-16}+q^{-20}+q^{-22}-2q^{-24}+q^{-26}-2q^{-28}-q^{-30}-q^{-32}-q^{-34}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+2q^{12}-5q^{10}+7q^{8}-9q^{6}+11q^{4}-11q^{2}+9-6q^{-2}+3q^{-4}+4q^{-6}-8q^{-8}+15q^{-10}-17q^{-12}+21q^{-14}-20q^{-16}+18q^{-18}-13q^{-20}+9q^{-22}-3q^{-24}-2q^{-26}+7q^{-28}-10q^{-30}+11q^{-32}-11q^{-34}+9q^{-36}-7q^{-38}+4q^{-40}-3q^{-42}$
1,0	$q^{24}-2q^{20}-2q^{18}+3q^{16}+5q^{14}-2q^{12}-8q^{10}-3q^{8}+9q^{6}+7q^{4}-8q^{2}-11+2q^{-2}+12q^{-4}+4q^{-6}-10q^{-8}-5q^{-10}+7q^{-12}+8q^{-14}-3q^{-16}-5q^{-18}+3q^{-20}+9q^{-22}+q^{-24}-7q^{-26}-q^{-28}+9q^{-30}+5q^{-32}-6q^{-34}-7q^{-36}+6q^{-38}+9q^{-40}-4q^{-42}-13q^{-44}-2q^{-46}+10q^{-48}+4q^{-50}-9q^{-52}-10q^{-54}+2q^{-56}+8q^{-58}+2q^{-60}-4q^{-62}-3q^{-64}+q^{-66}+3q^{-68}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{18}-2q^{16}+3q^{14}-4q^{12}+6q^{10}-8q^{8}+7q^{6}-10q^{4}+8q^{2}-9+4q^{-2}-4q^{-4}+3q^{-6}+3q^{-8}-4q^{-10}+12q^{-12}-6q^{-14}+16q^{-16}-13q^{-18}+17q^{-20}-13q^{-22}+17q^{-24}-12q^{-26}+10q^{-28}-7q^{-30}+7q^{-32}-q^{-34}-4q^{-36}+q^{-38}-9q^{-40}+7q^{-42}-11q^{-44}+5q^{-46}-10q^{-48}+8q^{-50}-4q^{-52}+4q^{-54}-3q^{-56}+3q^{-58}$

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-2q^{30}+5q^{28}-8q^{26}+7q^{24}-4q^{22}-6q^{20}+19q^{18}-30q^{16}+34q^{14}-27q^{12}+q^{10}+27q^{8}-52q^{6}+62q^{4}-46q^{2}+15+23q^{-2}-51q^{-4}+55q^{-6}-34q^{-8}+33q^{-12}-46q^{-14}+35q^{-16}-35q^{-20}+62q^{-22}-62q^{-24}+40q^{-26}-q^{-28}-43q^{-30}+75q^{-32}-80q^{-34}+64q^{-36}-24q^{-38}-18q^{-40}+56q^{-42}-68q^{-44}+57q^{-46}-25q^{-48}-11q^{-50}+42q^{-52}-45q^{-54}+24q^{-56}+13q^{-58}-42q^{-60}+55q^{-62}-42q^{-64}+5q^{-66}+29q^{-68}-56q^{-70}+60q^{-72}-45q^{-74}+15q^{-76}+13q^{-78}-34q^{-80}+34q^{-82}-28q^{-84}+15q^{-86}-2q^{-88}-7q^{-90}+7q^{-92}-8q^{-94}+6q^{-96}-2q^{-98}+q^{-100}+q^{-102}$

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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["10 151"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-4t^{2}+10t-13+10t^{-1}-4t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+2z^{4}+3z^{2}+1$

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]=

{ 43, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+6z^{2}a^{-2}-z^{2}a^{-4}-2z^{2}+3a^{-2}-a^{-6}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+5z^{7}a^{-3}+2z^{7}a^{-5}+5z^{6}a^{-2}+3z^{6}a^{-4}+z^{6}a^{-6}+3z^{6}+az^{5}-4z^{5}a^{-1}-7z^{5}a^{-3}-2z^{5}a^{-5}-15z^{4}a^{-2}-6z^{4}a^{-4}+2z^{4}a^{-6}-7z^{4}-2az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}+5z^{3}a^{-5}+3z^{3}a^{-7}+10z^{2}a^{-2}+4z^{2}a^{-4}-2z^{2}a^{-6}+4z^{2}+az+2za^{-1}+za^{-3}-3za^{-5}-3za^{-7}-3a^{-2}+a^{-6}-1$