10 135

Polynomial invariants

Alexander polynomial	$3t^{2}-9t+13-9t^{-1}+3t^{-2}$
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 3 z^4+3 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	{ 37, 0 }
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 -2 q^3+4 q^2-5 q+7-6 q^{-1} +6 q^{-2} -4 q^{-3} +2 q^{-4} - q^{-5} }
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^2 a^4-a^4+z^4 a^2+z^2 a^2+2 z^4+5 z^2+4-2 z^2 a^{-2} -2 a^{-2} }
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 a^2 z^8+z^8+2 a^3 z^7+4 a z^7+2 z^7 a^{-1} +2 a^4 z^6+a^2 z^6+z^6 a^{-2} +a^5 z^5-3 a^3 z^5-8 a z^5-4 z^5 a^{-1} -5 a^4 z^4-4 a^2 z^4+2 z^4 a^{-2} +3 z^4-3 a^5 z^3-a^3 z^3+8 a z^3+9 z^3 a^{-1} +3 z^3 a^{-3} +3 a^4 z^2+a^2 z^2-4 z^2 a^{-2} -6 z^2+2 a^5 z+a^3 z-4 a z-6 z a^{-1} -3 z a^{-3} -a^4+2 a^{-2} +4}
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^{16}-2 q^{10}+q^8+q^4+3 q^2+1+3 q^{-2} - q^{-4} -2 q^{-10} }
The G2 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^{80}-q^{78}+3 q^{76}-4 q^{74}+3 q^{72}-2 q^{70}-3 q^{68}+9 q^{66}-15 q^{64}+17 q^{62}-15 q^{60}+3 q^{58}+9 q^{56}-25 q^{54}+35 q^{52}-32 q^{50}+17 q^{48}+3 q^{46}-25 q^{44}+35 q^{42}-31 q^{40}+14 q^{38}+7 q^{36}-24 q^{34}+26 q^{32}-14 q^{30}-9 q^{28}+30 q^{26}-37 q^{24}+31 q^{22}-10 q^{20}-18 q^{18}+42 q^{16}-50 q^{14}+46 q^{12}-24 q^{10}-q^8+30 q^6-41 q^4+45 q^2-27+8 q^{-2} +18 q^{-4} -27 q^{-6} +25 q^{-8} -6 q^{-10} -12 q^{-12} +30 q^{-14} -29 q^{-16} +14 q^{-18} +7 q^{-20} -29 q^{-22} +39 q^{-24} -37 q^{-26} +19 q^{-28} -20 q^{-32} +26 q^{-34} -26 q^{-36} +16 q^{-38} -5 q^{-40} -4 q^{-42} +5 q^{-44} -9 q^{-46} +6 q^{-48} -2 q^{-50} + q^{-52} + q^{-54} }

Further Quantum Invariants

Further quantum knot invariants for 10_135.

A1 Invariants.

Weight	Invariant
1	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^{11}+q^9-2 q^7+2 q^5+q+2 q^{-1} - q^{-3} +2 q^{-5} -2 q^{-7} }
2	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^{32}-q^{30}-q^{28}+4 q^{26}-2 q^{24}-6 q^{22}+7 q^{20}+q^{18}-11 q^{16}+5 q^{14}+6 q^{12}-8 q^{10}+q^8+7 q^6-q^4-3 q^2+3+7 q^{-2} -7 q^{-4} -2 q^{-6} +11 q^{-8} -6 q^{-10} -6 q^{-12} +8 q^{-14} -2 q^{-16} -5 q^{-18} +2 q^{-20} + q^{-22} }
3	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^{63}+q^{61}+q^{59}-q^{57}-3 q^{55}+2 q^{53}+7 q^{51}-q^{49}-12 q^{47}-3 q^{45}+17 q^{43}+13 q^{41}-19 q^{39}-24 q^{37}+15 q^{35}+34 q^{33}-5 q^{31}-45 q^{29}-7 q^{27}+41 q^{25}+19 q^{23}-38 q^{21}-26 q^{19}+29 q^{17}+30 q^{15}-19 q^{13}-26 q^{11}+8 q^9+27 q^7+4 q^5-21 q^3-14 q+18 q^{-1} +28 q^{-3} -12 q^{-5} -36 q^{-7} +4 q^{-9} +46 q^{-11} +4 q^{-13} -44 q^{-15} -17 q^{-17} +38 q^{-19} +22 q^{-21} -27 q^{-23} -26 q^{-25} +15 q^{-27} +19 q^{-29} -3 q^{-31} -14 q^{-33} -2 q^{-35} +8 q^{-37} +2 q^{-39} -2 q^{-43} }

A2 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^{16}-2 q^{10}+q^8+q^4+3 q^2+1+3 q^{-2} - q^{-4} -2 q^{-10} }
1,1	$q^{44}-2q^{42}+6q^{40}-12q^{38}+23q^{36}-38q^{34}+58q^{32}-80q^{30}+100q^{28}-116q^{26}+114q^{24}-102q^{22}+64q^{20}-20q^{18}-44q^{16}+104q^{14}-158q^{12}+202q^{10}-220q^{8}+228q^{6}-199q^{4}+170q^{2}-108+54q^{-2}+12q^{-4}-64q^{-6}+102q^{-8}-124q^{-10}+123q^{-12}-112q^{-14}+86q^{-16}-64q^{-18}+36q^{-20}-20q^{-22}+8q^{-24}-2q^{-26}+2q^{-30}$
2,0	$q^{42}-q^{38}+3q^{34}+2q^{32}-3q^{30}-3q^{28}+2q^{26}-7q^{22}-5q^{20}+2q^{18}+2q^{16}-4q^{14}+5q^{10}+q^{8}+2q^{6}+6q^{4}+4q^{2}+3+5q^{-2}+4q^{-4}-5q^{-6}-3q^{-8}+3q^{-10}-8q^{-14}-2q^{-16}+3q^{-18}-q^{-20}-3q^{-22}-q^{-24}+3q^{-26}+q^{-28}$

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}+2q^{38}+2q^{36}-2q^{34}-2q^{32}+2q^{30}-q^{28}-8q^{26}-3q^{24}+4q^{22}-5q^{20}-10q^{18}+q^{14}-7q^{12}-2q^{10}+9q^{8}+8q^{6}+6q^{4}+18q^{2}+17+5q^{-2}+5q^{-4}+8q^{-6}-8q^{-8}-14q^{-10}-6q^{-12}-4q^{-14}-10q^{-16}-6q^{-18}+4q^{-20}+3q^{-22}-q^{-24}+q^{-26}+3q^{-28}$
1,0,0,0	$-q^{26}-q^{22}-q^{20}-2q^{16}+q^{14}-q^{12}+q^{8}+q^{6}+3q^{4}+3q^{2}+4+2q^{-2}+3q^{-4}-q^{-6}-2q^{-10}-2q^{-12}-2q^{-16}$

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-q^{44}+2q^{42}-2q^{40}+4q^{38}-5q^{36}+4q^{34}-7q^{32}+5q^{30}-8q^{28}+3q^{26}-5q^{24}+2q^{22}-q^{20}-4q^{18}+4q^{16}-6q^{14}+8q^{12}-11q^{10}+12q^{8}-7q^{6}+17q^{4}-4q^{2}+15-q^{-2}+11q^{-4}+q^{-6}-q^{-8}-2q^{-10}-7q^{-12}+2q^{-14}-10q^{-16}+2q^{-18}-9q^{-20}+6q^{-22}-4q^{-24}+4q^{-26}-3q^{-28}+3q^{-30}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-q^{78}+3q^{76}-4q^{74}+3q^{72}-2q^{70}-3q^{68}+9q^{66}-15q^{64}+17q^{62}-15q^{60}+3q^{58}+9q^{56}-25q^{54}+35q^{52}-32q^{50}+17q^{48}+3q^{46}-25q^{44}+35q^{42}-31q^{40}+14q^{38}+7q^{36}-24q^{34}+26q^{32}-14q^{30}-9q^{28}+30q^{26}-37q^{24}+31q^{22}-10q^{20}-18q^{18}+42q^{16}-50q^{14}+46q^{12}-24q^{10}-q^{8}+30q^{6}-41q^{4}+45q^{2}-27+8q^{-2}+18q^{-4}-27q^{-6}+25q^{-8}-6q^{-10}-12q^{-12}+30q^{-14}-29q^{-16}+14q^{-18}+7q^{-20}-29q^{-22}+39q^{-24}-37q^{-26}+19q^{-28}-20q^{-32}+26q^{-34}-26q^{-36}+16q^{-38}-5q^{-40}-4q^{-42}+5q^{-44}-9q^{-46}+6q^{-48}-2q^{-50}+q^{-52}+q^{-54}$

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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["10 135"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-9t+13-9t^{-1}+3t^{-2}$

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 3 z^4+3 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]=

{ 37, 0 }

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

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^2 a^4-a^4+z^4 a^2+z^2 a^2+2 z^4+5 z^2+4-2 z^2 a^{-2} -2 a^{-2} }

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 a^2 z^8+z^8+2 a^3 z^7+4 a z^7+2 z^7 a^{-1} +2 a^4 z^6+a^2 z^6+z^6 a^{-2} +a^5 z^5-3 a^3 z^5-8 a z^5-4 z^5 a^{-1} -5 a^4 z^4-4 a^2 z^4+2 z^4 a^{-2} +3 z^4-3 a^5 z^3-a^3 z^3+8 a z^3+9 z^3 a^{-1} +3 z^3 a^{-3} +3 a^4 z^2+a^2 z^2-4 z^2 a^{-2} -6 z^2+2 a^5 z+a^3 z-4 a z-6 z a^{-1} -3 z a^{-3} -a^4+2 a^{-2} +4}