10 119

Polynomial invariants

Alexander polynomial	$-2t^{3}+10t^{2}-23t+31-23t^{-1}+10t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 101, 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 q^6-4 q^5+8 q^4-12 q^3+16 q^2-17 q+16-13 q^{-1} +9 q^{-2} -4 q^{-3} + q^{-4} }
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^6 a^{-2} -z^6+a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -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 3 z^9 a^{-1} +3 z^9 a^{-3} +15 z^8 a^{-2} +6 z^8 a^{-4} +9 z^8+12 a z^7+13 z^7 a^{-1} +5 z^7 a^{-3} +4 z^7 a^{-5} +9 a^2 z^6-31 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -7 z^6+4 a^3 z^5-17 a z^5-37 z^5 a^{-1} -26 z^5 a^{-3} -10 z^5 a^{-5} +a^4 z^4-9 a^2 z^4+13 z^4 a^{-2} +8 z^4 a^{-4} -2 z^4 a^{-6} -7 z^4-a^3 z^3+9 a z^3+22 z^3 a^{-1} +19 z^3 a^{-3} +7 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	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^{12}-2 q^{10}+2 q^8+2 q^6-3 q^4+3 q^2-3+ q^{-2} + q^{-4} - q^{-6} +4 q^{-8} -3 q^{-10} + q^{-12} + q^{-14} -2 q^{-16} + q^{-18} }
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^{66}-3 q^{64}+6 q^{62}-10 q^{60}+11 q^{58}-10 q^{56}+4 q^{54}+14 q^{52}-35 q^{50}+60 q^{48}-78 q^{46}+71 q^{44}-44 q^{42}-17 q^{40}+105 q^{38}-184 q^{36}+238 q^{34}-223 q^{32}+127 q^{30}+33 q^{28}-222 q^{26}+371 q^{24}-410 q^{22}+305 q^{20}-82 q^{18}-176 q^{16}+370 q^{14}-400 q^{12}+264 q^{10}-15 q^8-242 q^6+363 q^4-299 q^2+59+250 q^{-2} -475 q^{-4} +516 q^{-6} -332 q^{-8} - q^{-10} +350 q^{-12} -598 q^{-14} +642 q^{-16} -473 q^{-18} +155 q^{-20} +204 q^{-22} -467 q^{-24} +561 q^{-26} -441 q^{-28} +178 q^{-30} +115 q^{-32} -337 q^{-34} +382 q^{-36} -246 q^{-38} -7 q^{-40} +274 q^{-42} -413 q^{-44} +359 q^{-46} -130 q^{-48} -174 q^{-50} +415 q^{-52} -496 q^{-54} +387 q^{-56} -150 q^{-58} -116 q^{-60} +311 q^{-62} -370 q^{-64} +302 q^{-66} -149 q^{-68} -7 q^{-70} +108 q^{-72} -148 q^{-74} +125 q^{-76} -73 q^{-78} +27 q^{-80} +8 q^{-82} -22 q^{-84} +21 q^{-86} -16 q^{-88} +8 q^{-90} -3 q^{-92} + q^{-94} }

Further Quantum Invariants

Further quantum knot invariants for 10_119.

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^9-3 q^7+5 q^5-4 q^3+3 q- q^{-1} - q^{-3} +4 q^{-5} -4 q^{-7} +4 q^{-9} -3 q^{-11} + q^{-13} }
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^{26}-3 q^{24}+2 q^{22}+8 q^{20}-18 q^{18}+4 q^{16}+31 q^{14}-38 q^{12}-9 q^{10}+54 q^8-32 q^6-29 q^4+47 q^2-33 q^{-2} +10 q^{-4} +28 q^{-6} -19 q^{-8} -29 q^{-10} +41 q^{-12} +8 q^{-14} -52 q^{-16} +31 q^{-18} +31 q^{-20} -48 q^{-22} +4 q^{-24} +32 q^{-26} -20 q^{-28} -9 q^{-30} +13 q^{-32} - q^{-34} -3 q^{-36} + q^{-38} }
3	$q^{51}-3q^{49}+2q^{47}+5q^{45}-6q^{43}-11q^{41}+12q^{39}+32q^{37}-32q^{35}-68q^{33}+55q^{31}+132q^{29}-55q^{27}-237q^{25}+26q^{23}+343q^{21}+57q^{19}-417q^{17}-186q^{15}+425q^{13}+319q^{11}-350q^{9}-411q^{7}+215q^{5}+441q^{3}-53q-408q^{-1}-100q^{-3}+332q^{-5}+225q^{-7}-236q^{-9}-317q^{-11}+136q^{-13}+386q^{-15}-30q^{-17}-435q^{-19}-77q^{-21}+448q^{-23}+195q^{-25}-420q^{-27}-316q^{-29}+346q^{-31}+404q^{-33}-214q^{-35}-444q^{-37}+63q^{-39}+411q^{-41}+73q^{-43}-313q^{-45}-158q^{-47}+185q^{-49}+175q^{-51}-74q^{-53}-135q^{-55}+2q^{-57}+78q^{-59}+24q^{-61}-34q^{-63}-17q^{-65}+8q^{-67}+8q^{-69}-q^{-71}-3q^{-73}+q^{-75}$

A2 Invariants.

Weight	Invariant
1,0	$q^{12}-2q^{10}+2q^{8}+2q^{6}-3q^{4}+3q^{2}-3+q^{-2}+q^{-4}-q^{-6}+4q^{-8}-3q^{-10}+q^{-12}+q^{-14}-2q^{-16}+q^{-18}$
1,1	$q^{36}-6q^{34}+18q^{32}-38q^{30}+77q^{28}-150q^{26}+258q^{24}-404q^{22}+617q^{20}-892q^{18}+1198q^{16}-1522q^{14}+1824q^{12}-2026q^{10}+2036q^{8}-1802q^{6}+1301q^{4}-514q^{2}-480+1584q^{-2}-2643q^{-4}+3530q^{-6}-4132q^{-8}+4358q^{-10}-4205q^{-12}+3680q^{-14}-2842q^{-16}+1796q^{-18}-674q^{-20}-386q^{-22}+1298q^{-24}-1946q^{-26}+2274q^{-28}-2298q^{-30}+2086q^{-32}-1722q^{-34}+1284q^{-36}-874q^{-38}+540q^{-40}-296q^{-42}+145q^{-44}-62q^{-46}+22q^{-48}-6q^{-50}+q^{-52}$
2,0	$q^{32}-2q^{30}+6q^{26}-3q^{24}-10q^{22}+5q^{20}+17q^{18}-6q^{16}-23q^{14}+8q^{12}+24q^{10}-15q^{8}-19q^{6}+19q^{4}+13q^{2}-10-6q^{-2}+13q^{-4}-4q^{-6}-10q^{-8}+11q^{-10}-2q^{-12}-16q^{-14}+9q^{-16}+20q^{-18}-14q^{-20}-14q^{-22}+16q^{-24}+15q^{-26}-15q^{-28}-16q^{-30}+15q^{-32}+9q^{-34}-10q^{-36}-7q^{-38}+4q^{-40}+6q^{-42}-2q^{-44}-2q^{-46}+q^{-48}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{28}-3q^{26}+10q^{22}-11q^{20}-7q^{18}+27q^{16}-18q^{14}-17q^{12}+41q^{10}-19q^{8}-23q^{6}+38q^{4}-11q^{2}-18+16q^{-2}+5q^{-4}-6q^{-6}-13q^{-8}+16q^{-10}+12q^{-12}-33q^{-14}+15q^{-16}+25q^{-18}-38q^{-20}+12q^{-22}+23q^{-24}-30q^{-26}+10q^{-28}+11q^{-30}-14q^{-32}+5q^{-34}+2q^{-36}-3q^{-38}+q^{-40}$
1,0,0	$q^{15}-2q^{13}+3q^{11}-q^{9}+3q^{7}-3q^{5}+3q^{3}-3q+2q^{-7}-q^{-9}+4q^{-11}-3q^{-13}+2q^{-15}-2q^{-17}+2q^{-19}-2q^{-21}+q^{-23}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{34}-2q^{32}-2q^{30}+7q^{28}+q^{26}-11q^{24}+q^{22}+17q^{20}-2q^{18}-22q^{16}+10q^{14}+26q^{12}-20q^{10}-25q^{8}+30q^{6}+13q^{4}-31q^{2}+4+31q^{-2}-12q^{-4}-23q^{-6}+24q^{-8}+9q^{-10}-36q^{-12}+6q^{-14}+35q^{-16}-23q^{-18}-24q^{-20}+34q^{-22}+13q^{-24}-30q^{-26}-3q^{-28}+26q^{-30}-3q^{-32}-20q^{-34}+8q^{-36}+12q^{-38}-9q^{-40}-4q^{-42}+6q^{-44}-q^{-46}-2q^{-48}+q^{-50}$
1,0,0,0	$q^{18}-2q^{16}+3q^{14}+3q^{8}-3q^{6}+3q^{4}-3q^{2}-q^{-2}+q^{-8}+2q^{-10}-q^{-12}+4q^{-14}-3q^{-16}+2q^{-18}-q^{-20}-q^{-22}+2q^{-24}-2q^{-26}+q^{-28}$

B2 Invariants.

Weight	Invariant
0,1	$q^{28}-3q^{26}+6q^{24}-12q^{22}+21q^{20}-29q^{18}+39q^{16}-46q^{14}+49q^{12}-45q^{10}+35q^{8}-19q^{6}-2q^{4}+27q^{2}-52+72q^{-2}-87q^{-4}+94q^{-6}-91q^{-8}+80q^{-10}-60q^{-12}+37q^{-14}-11q^{-16}-11q^{-18}+30q^{-20}-42q^{-22}+49q^{-24}-48q^{-26}+42q^{-28}-35q^{-30}+24q^{-32}-15q^{-34}+8q^{-36}-3q^{-38}+q^{-40}$
1,0	$q^{46}-3q^{42}-3q^{40}+3q^{38}+11q^{36}+4q^{34}-16q^{32}-18q^{30}+8q^{28}+33q^{26}+11q^{24}-35q^{22}-33q^{20}+21q^{18}+49q^{16}+3q^{14}-49q^{12}-24q^{10}+37q^{8}+37q^{6}-22q^{4}-40q^{2}+8+39q^{-2}+4q^{-4}-35q^{-6}-11q^{-8}+30q^{-10}+17q^{-12}-26q^{-14}-23q^{-16}+24q^{-18}+32q^{-20}-17q^{-22}-44q^{-24}+3q^{-26}+50q^{-28}+20q^{-30}-43q^{-32}-41q^{-34}+24q^{-36}+50q^{-38}-42q^{-42}-20q^{-44}+25q^{-46}+26q^{-48}-8q^{-50}-19q^{-52}-3q^{-54}+10q^{-56}+5q^{-58}-3q^{-60}-3q^{-62}+q^{-66}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{38}-3q^{36}+3q^{34}-4q^{32}+11q^{30}-16q^{28}+16q^{26}-21q^{24}+33q^{22}-35q^{20}+33q^{18}-37q^{16}+41q^{14}-30q^{12}+22q^{10}-18q^{8}+5q^{6}+15q^{4}-25q^{2}+35-52q^{-2}+64q^{-4}-66q^{-6}+70q^{-8}-75q^{-10}+69q^{-12}-58q^{-14}+52q^{-16}-42q^{-18}+25q^{-20}-8q^{-22}-q^{-24}+12q^{-26}-25q^{-28}+35q^{-30}-36q^{-32}+38q^{-34}-39q^{-36}+36q^{-38}-30q^{-40}+25q^{-42}-20q^{-44}+13q^{-46}-8q^{-48}+5q^{-50}-3q^{-52}+q^{-54}$

G2 Invariants.

Weight

Invariant

1,0

$q^{66}-3q^{64}+6q^{62}-10q^{60}+11q^{58}-10q^{56}+4q^{54}+14q^{52}-35q^{50}+60q^{48}-78q^{46}+71q^{44}-44q^{42}-17q^{40}+105q^{38}-184q^{36}+238q^{34}-223q^{32}+127q^{30}+33q^{28}-222q^{26}+371q^{24}-410q^{22}+305q^{20}-82q^{18}-176q^{16}+370q^{14}-400q^{12}+264q^{10}-15q^{8}-242q^{6}+363q^{4}-299q^{2}+59+250q^{-2}-475q^{-4}+516q^{-6}-332q^{-8}-q^{-10}+350q^{-12}-598q^{-14}+642q^{-16}-473q^{-18}+155q^{-20}+204q^{-22}-467q^{-24}+561q^{-26}-441q^{-28}+178q^{-30}+115q^{-32}-337q^{-34}+382q^{-36}-246q^{-38}-7q^{-40}+274q^{-42}-413q^{-44}+359q^{-46}-130q^{-48}-174q^{-50}+415q^{-52}-496q^{-54}+387q^{-56}-150q^{-58}-116q^{-60}+311q^{-62}-370q^{-64}+302q^{-66}-149q^{-68}-7q^{-70}+108q^{-72}-148q^{-74}+125q^{-76}-73q^{-78}+27q^{-80}+8q^{-82}-22q^{-84}+21q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}$

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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["10 119"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+10t^{2}-23t+31-23t^{-1}+10t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 101, 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 q^6-4 q^5+8 q^4-12 q^3+16 q^2-17 q+16-13 q^{-1} +9 q^{-2} -4 q^{-3} + q^{-4} }

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^6 a^{-2} -z^6+a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -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 3 z^9 a^{-1} +3 z^9 a^{-3} +15 z^8 a^{-2} +6 z^8 a^{-4} +9 z^8+12 a z^7+13 z^7 a^{-1} +5 z^7 a^{-3} +4 z^7 a^{-5} +9 a^2 z^6-31 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -7 z^6+4 a^3 z^5-17 a z^5-37 z^5 a^{-1} -26 z^5 a^{-3} -10 z^5 a^{-5} +a^4 z^4-9 a^2 z^4+13 z^4 a^{-2} +8 z^4 a^{-4} -2 z^4 a^{-6} -7 z^4-a^3 z^3+9 a z^3+22 z^3 a^{-1} +19 z^3 a^{-3} +7 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}