10 147

Polynomial invariants

Alexander polynomial	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 -2 t^2+7 t-9+7 t^{-1} -2 t^{-2} }
Conway polynomial	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 -2 z^4-z^2+1}
2nd Alexander ideal (db, data sources)	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 \{1\}}
Determinant and Signature	{ 27, 2 }
Jones polynomial	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^5-3 q^4+4 q^3-4 q^2+5 q-4+3 q^{-1} -2 q^{-2} + q^{-3} }
HOMFLY-PT polynomial (db, data sources)	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 -z^4 a^{-2} -z^4+a^2 z^2-z^2 a^{-2} +z^2 a^{-4} -2 z^2+a^2+ a^{-2} -1}
Kauffman polynomial (db, data sources)	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 z^8 a^{-2} +z^8+2 a z^7+4 z^7 a^{-1} +2 z^7 a^{-3} +a^2 z^6-z^6 a^{-2} +z^6 a^{-4} -z^6-8 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} -4 a^2 z^4-2 z^4 a^{-2} -6 z^4+8 a z^3+13 z^3 a^{-1} +8 z^3 a^{-3} +3 z^3 a^{-5} +4 a^2 z^2+z^2 a^{-2} +z^2 a^{-6} +6 z^2-2 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} -a^2- a^{-2} -1}
The A2 invariant	$q^{10}+q^{4}-q^{2}+2q^{-6}+q^{-10}-q^{-12}-q^{-14}+q^{-16}$
The G2 invariant	$q^{46}-q^{44}+3q^{42}-4q^{40}+3q^{38}-q^{36}-3q^{34}+10q^{32}-11q^{30}+12q^{28}-6q^{26}-5q^{24}+12q^{22}-14q^{20}+11q^{18}-5q^{16}-4q^{14}+12q^{12}-9q^{10}+3q^{8}+5q^{6}-13q^{4}+14q^{2}-7-5q^{-2}+10q^{-4}-14q^{-6}+18q^{-8}-11q^{-10}+4q^{-12}+4q^{-14}-13q^{-16}+15q^{-18}-13q^{-20}+7q^{-22}-7q^{-26}+11q^{-28}-6q^{-30}+3q^{-32}+6q^{-34}-14q^{-36}+11q^{-38}-q^{-40}-7q^{-42}+13q^{-44}-15q^{-46}+11q^{-48}+q^{-50}-6q^{-52}+7q^{-54}-11q^{-56}+7q^{-58}-3q^{-62}+2q^{-64}-2q^{-66}+q^{-68}+q^{-70}+2q^{-72}-q^{-74}-q^{-78}-q^{-84}+q^{-86}$

Further Quantum Invariants

Further quantum knot invariants for 10_147.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+q^{3}-q+q^{-1}+q^{-3}+q^{-7}-2q^{-9}+q^{-11}$
2	$q^{22}-q^{20}-2q^{18}+3q^{16}+q^{14}-4q^{12}+2q^{10}+4q^{8}-3q^{6}-2q^{4}+3q^{2}-1-3q^{-2}+3q^{-4}+3q^{-6}-2q^{-8}+q^{-10}+4q^{-12}-q^{-14}-5q^{-16}+2q^{-18}+q^{-20}-4q^{-22}+2q^{-24}+2q^{-26}-q^{-28}$
3	$q^{45}-q^{43}-2q^{41}+4q^{37}+3q^{35}-5q^{33}-6q^{31}+3q^{29}+9q^{27}+2q^{25}-10q^{23}-8q^{21}+6q^{19}+12q^{17}+q^{15}-13q^{13}-7q^{11}+11q^{9}+13q^{7}-6q^{5}-15q^{3}+3q+14q^{-1}+q^{-3}-15q^{-5}-q^{-7}+14q^{-9}+3q^{-11}-11q^{-13}-3q^{-15}+10q^{-17}+7q^{-19}-5q^{-21}-11q^{-23}-q^{-25}+10q^{-27}+8q^{-29}-12q^{-31}-14q^{-33}+7q^{-35}+19q^{-37}-q^{-39}-15q^{-41}-3q^{-43}+9q^{-45}+6q^{-47}-6q^{-49}-4q^{-51}+q^{-53}+2q^{-55}+q^{-57}-q^{-59}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}+q^{4}-q^{2}+2q^{-6}+q^{-10}-q^{-12}-q^{-14}+q^{-16}$
1,1	$q^{28}-2q^{26}+6q^{24}-12q^{22}+19q^{20}-28q^{18}+36q^{16}-38q^{14}+33q^{12}-24q^{10}+12q^{8}+14q^{6}-31q^{4}+48q^{2}-60+62q^{-2}-69q^{-4}+56q^{-6}-42q^{-8}+32q^{-10}-3q^{-12}-6q^{-14}+28q^{-16}-38q^{-18}+38q^{-20}-40q^{-22}+24q^{-24}-18q^{-26}+11q^{-28}-2q^{-30}-2q^{-32}+2q^{-34}+2q^{-36}-2q^{-42}+q^{-44}$
2,0	$q^{28}-q^{24}-q^{22}+q^{20}+2q^{18}-q^{16}-q^{14}+q^{12}+2q^{10}+q^{8}-2q^{6}+q^{2}-1-2q^{-2}+q^{-6}+q^{-8}+2q^{-10}+2q^{-12}+2q^{-14}+2q^{-16}+2q^{-18}-q^{-20}-5q^{-22}-q^{-24}-2q^{-28}-q^{-30}+2q^{-32}+3q^{-34}-q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-q^{18}+q^{16}+q^{14}-2q^{12}+2q^{10}-q^{6}+3q^{4}-1+2q^{-2}-2q^{-6}+q^{-8}+q^{-12}-q^{-14}+2q^{-18}-2q^{-20}+q^{-22}+3q^{-24}-2q^{-26}+2q^{-30}-2q^{-32}-q^{-34}+q^{-36}$
1,0,0	$q^{13}+q^{9}+q^{5}-q^{3}-q^{-1}+q^{-7}+2q^{-9}+q^{-13}-q^{-15}-q^{-19}+q^{-21}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{26}+2q^{20}+q^{18}-2q^{16}+q^{12}-2q^{10}-2q^{8}+3q^{6}+4q^{4}-q^{2}+2+6q^{-2}-4q^{-6}-q^{-10}-5q^{-12}-q^{-14}+2q^{-16}+q^{-18}-q^{-20}+4q^{-22}+3q^{-24}-2q^{-26}+3q^{-30}-2q^{-34}+q^{-36}+q^{-38}-q^{-40}-2q^{-42}+q^{-46}$
1,0,0,0	$q^{16}+q^{12}+q^{10}+q^{6}-q^{4}-1-q^{-2}+q^{-8}+q^{-10}+2q^{-12}+q^{-16}-q^{-18}-q^{-24}+q^{-26}$

B2 Invariants.

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

D4 Invariants.

Weight

Invariant

1,0,0,0

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^{26}-q^{24}+2 q^{22}-2 q^{20}+4 q^{18}-3 q^{16}+3 q^{14}-3 q^{12}+4 q^{10}-2 q^8+q^6+3-3 q^{-2} +4 q^{-4} -5 q^{-6} +5 q^{-8} -6 q^{-10} +6 q^{-12} -6 q^{-14} +5 q^{-16} -4 q^{-18} +3 q^{-20} - q^{-22} + q^{-24} + q^{-26} - q^{-28} +3 q^{-30} -2 q^{-32} +4 q^{-34} -3 q^{-36} +2 q^{-38} -3 q^{-40} +3 q^{-42} -2 q^{-44} - q^{-48} + q^{-50} }

G2 Invariants.

Weight

Invariant

1,0

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^{46}-q^{44}+3 q^{42}-4 q^{40}+3 q^{38}-q^{36}-3 q^{34}+10 q^{32}-11 q^{30}+12 q^{28}-6 q^{26}-5 q^{24}+12 q^{22}-14 q^{20}+11 q^{18}-5 q^{16}-4 q^{14}+12 q^{12}-9 q^{10}+3 q^8+5 q^6-13 q^4+14 q^2-7-5 q^{-2} +10 q^{-4} -14 q^{-6} +18 q^{-8} -11 q^{-10} +4 q^{-12} +4 q^{-14} -13 q^{-16} +15 q^{-18} -13 q^{-20} +7 q^{-22} -7 q^{-26} +11 q^{-28} -6 q^{-30} +3 q^{-32} +6 q^{-34} -14 q^{-36} +11 q^{-38} - q^{-40} -7 q^{-42} +13 q^{-44} -15 q^{-46} +11 q^{-48} + q^{-50} -6 q^{-52} +7 q^{-54} -11 q^{-56} +7 q^{-58} -3 q^{-62} +2 q^{-64} -2 q^{-66} + q^{-68} + q^{-70} +2 q^{-72} - q^{-74} - q^{-78} - q^{-84} + q^{-86} }

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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["10 147"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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 -2 t^2+7 t-9+7 t^{-1} -2 t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

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 -2 z^4-z^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]=

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 \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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^5-3 q^4+4 q^3-4 q^2+5 q-4+3 q^{-1} -2 q^{-2} + q^{-3} }

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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 -z^4 a^{-2} -z^4+a^2 z^2-z^2 a^{-2} +z^2 a^{-4} -2 z^2+a^2+ a^{-2} -1}

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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 z^8 a^{-2} +z^8+2 a z^7+4 z^7 a^{-1} +2 z^7 a^{-3} +a^2 z^6-z^6 a^{-2} +z^6 a^{-4} -z^6-8 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} -4 a^2 z^4-2 z^4 a^{-2} -6 z^4+8 a z^3+13 z^3 a^{-1} +8 z^3 a^{-3} +3 z^3 a^{-5} +4 a^2 z^2+z^2 a^{-2} +z^2 a^{-6} +6 z^2-2 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} -a^2- a^{-2} -1}