10 77

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^3-7 t^2+14 t-17+14 t^{-1} -7 t^{-2} +2 t^{-3} }
Conway polynomial	$2z^{6}+5z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 63, 2 }
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+8q^{5}-10q^{4}+11q^{3}-9q^{2}+8q-4+2q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+4z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+7z^{2}a^{-2}+2z^{2}a^{-4}-2z^{2}a^{-6}-3z^{2}+5a^{-2}-a^{-4}-a^{-6}-2$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+2z^{7}a^{-1}+2z^{7}a^{-3}+4z^{7}a^{-5}+4z^{7}a^{-7}-z^{6}a^{-2}-9z^{6}a^{-4}-3z^{6}a^{-6}+3z^{6}a^{-8}+2z^{6}+az^{5}-z^{5}a^{-1}-3z^{5}a^{-3}-9z^{5}a^{-5}-7z^{5}a^{-7}+z^{5}a^{-9}-3z^{4}a^{-2}+8z^{4}a^{-4}-6z^{4}a^{-8}-5z^{4}-3az^{3}-5z^{3}a^{-1}+6z^{3}a^{-5}+2z^{3}a^{-7}-2z^{3}a^{-9}+7z^{2}a^{-2}-z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}a^{-8}+4z^{2}+2az+4za^{-1}+3za^{-3}-za^{-5}-za^{-7}+za^{-9}-5a^{-2}-a^{-4}+a^{-6}-2$
The A2 invariant	$-q^{6}-q^{2}-1+3q^{-2}+4q^{-6}+2q^{-8}+q^{-12}-3q^{-14}+q^{-16}-q^{-18}-q^{-20}+q^{-22}-q^{-24}$
The G2 invariant	$q^{32}-q^{30}+3q^{28}-4q^{26}+3q^{24}-3q^{22}-3q^{20}+8q^{18}-14q^{16}+17q^{14}-19q^{12}+12q^{10}-q^{8}-16q^{6}+34q^{4}-51q^{2}+53-43q^{-2}+11q^{-4}+26q^{-6}-65q^{-8}+95q^{-10}-88q^{-12}+60q^{-14}-5q^{-16}-49q^{-18}+88q^{-20}-85q^{-22}+56q^{-24}-42q^{-28}+65q^{-30}-46q^{-32}+2q^{-34}+59q^{-36}-95q^{-38}+96q^{-40}-54q^{-42}-20q^{-44}+95q^{-46}-142q^{-48}+146q^{-50}-104q^{-52}+28q^{-54}+54q^{-56}-116q^{-58}+135q^{-60}-109q^{-62}+47q^{-64}+19q^{-66}-69q^{-68}+78q^{-70}-50q^{-72}-q^{-74}+52q^{-76}-77q^{-78}+62q^{-80}-17q^{-82}-47q^{-84}+94q^{-86}-108q^{-88}+84q^{-90}-33q^{-92}-27q^{-94}+73q^{-96}-90q^{-98}+81q^{-100}-48q^{-102}+9q^{-104}+20q^{-106}-40q^{-108}+39q^{-110}-29q^{-112}+17q^{-114}-3q^{-116}-5q^{-118}+8q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 10_77.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+q^{3}-2q+4q^{-1}-q^{-3}+2q^{-5}+q^{-7}-2q^{-9}+2q^{-11}-3q^{-13}+2q^{-15}-q^{-17}$
2	$q^{16}-q^{14}-q^{12}+3q^{10}-4q^{8}-3q^{6}+10q^{4}-7q^{2}-9+20q^{-2}-4q^{-4}-16q^{-6}+19q^{-8}+4q^{-10}-14q^{-12}+6q^{-14}+8q^{-16}-4q^{-18}-11q^{-20}+9q^{-22}+8q^{-24}-20q^{-26}+5q^{-28}+16q^{-30}-18q^{-32}-2q^{-34}+16q^{-36}-8q^{-38}-5q^{-40}+7q^{-42}-q^{-44}-2q^{-46}+q^{-48}$
3	$-q^{33}+q^{31}+q^{29}-2q^{25}+2q^{23}+2q^{21}-4q^{19}-4q^{17}+8q^{15}+8q^{13}-16q^{11}-18q^{9}+21q^{7}+34q^{5}-26q^{3}-53q+17q^{-1}+83q^{-3}-6q^{-5}-93q^{-7}-18q^{-9}+103q^{-11}+43q^{-13}-93q^{-15}-60q^{-17}+75q^{-19}+70q^{-21}-46q^{-23}-69q^{-25}+16q^{-27}+67q^{-29}+11q^{-31}-54q^{-33}-47q^{-35}+43q^{-37}+66q^{-39}-27q^{-41}-92q^{-43}+11q^{-45}+102q^{-47}+13q^{-49}-104q^{-51}-36q^{-53}+95q^{-55}+54q^{-57}-73q^{-59}-64q^{-61}+46q^{-63}+65q^{-65}-20q^{-67}-55q^{-69}+3q^{-71}+36q^{-73}+8q^{-75}-21q^{-77}-9q^{-79}+11q^{-81}+5q^{-83}-4q^{-85}-3q^{-87}+q^{-89}+2q^{-91}-q^{-93}$
4	$q^{56}-q^{54}-q^{52}-q^{48}+4q^{46}-2q^{44}+3q^{40}-6q^{38}+4q^{36}-7q^{34}+4q^{32}+19q^{30}-7q^{28}-5q^{26}-37q^{24}-2q^{22}+61q^{20}+30q^{18}-2q^{16}-118q^{14}-73q^{12}+101q^{10}+147q^{8}+98q^{6}-205q^{4}-265q^{2}+19+282q^{-2}+365q^{-4}-143q^{-6}-484q^{-8}-253q^{-10}+248q^{-12}+638q^{-14}+125q^{-16}-495q^{-18}-533q^{-20}+q^{-22}+669q^{-24}+391q^{-26}-266q^{-28}-575q^{-30}-247q^{-32}+436q^{-34}+454q^{-36}+15q^{-38}-400q^{-40}-353q^{-42}+127q^{-44}+365q^{-46}+226q^{-48}-169q^{-50}-366q^{-52}-164q^{-54}+248q^{-56}+401q^{-58}+54q^{-60}-358q^{-62}-433q^{-64}+108q^{-66}+535q^{-68}+300q^{-70}-271q^{-72}-646q^{-74}-116q^{-76}+519q^{-78}+515q^{-80}-36q^{-82}-661q^{-84}-356q^{-86}+274q^{-88}+545q^{-90}+251q^{-92}-419q^{-94}-421q^{-96}-42q^{-98}+329q^{-100}+355q^{-102}-101q^{-104}-257q^{-106}-184q^{-108}+65q^{-110}+233q^{-112}+49q^{-114}-56q^{-116}-119q^{-118}-42q^{-120}+77q^{-122}+36q^{-124}+16q^{-126}-35q^{-128}-28q^{-130}+17q^{-132}+5q^{-134}+10q^{-136}-6q^{-138}-8q^{-140}+4q^{-142}+3q^{-146}-q^{-148}-2q^{-150}+q^{-152}$
5	$-q^{85}+q^{83}+q^{81}+q^{77}-q^{75}-4q^{73}+2q^{69}-q^{67}+5q^{65}+5q^{63}-5q^{61}-7q^{59}-6q^{57}-7q^{55}+10q^{53}+27q^{51}+16q^{49}-13q^{47}-40q^{45}-48q^{43}-2q^{41}+71q^{39}+106q^{37}+40q^{35}-95q^{33}-192q^{31}-137q^{29}+83q^{27}+312q^{25}+319q^{23}+q^{21}-428q^{19}-597q^{17}-241q^{15}+475q^{13}+948q^{11}+668q^{9}-328q^{7}-1310q^{5}-1302q^{3}-63q+1503q^{-1}+2026q^{-3}+824q^{-5}-1426q^{-7}-2748q^{-9}-1748q^{-11}+953q^{-13}+3162q^{-15}+2820q^{-17}-116q^{-19}-3236q^{-21}-3677q^{-23}-899q^{-25}+2809q^{-27}+4208q^{-29}+1925q^{-31}-2061q^{-33}-4257q^{-35}-2715q^{-37}+1139q^{-39}+3861q^{-41}+3146q^{-43}-227q^{-45}-3169q^{-47}-3204q^{-49}-520q^{-51}+2345q^{-53}+2963q^{-55}+1055q^{-57}-1535q^{-59}-2576q^{-61}-1397q^{-63}+803q^{-65}+2174q^{-67}+1648q^{-69}-213q^{-71}-1814q^{-73}-1871q^{-75}-366q^{-77}+1546q^{-79}+2186q^{-81}+883q^{-83}-1300q^{-85}-2505q^{-87}-1536q^{-89}+1021q^{-91}+2888q^{-93}+2206q^{-95}-606q^{-97}-3121q^{-99}-2953q^{-101}-7q^{-103}+3176q^{-105}+3615q^{-107}+782q^{-109}-2875q^{-111}-4075q^{-113}-1659q^{-115}+2233q^{-117}+4178q^{-119}+2483q^{-121}-1309q^{-123}-3848q^{-125}-3048q^{-127}+245q^{-129}+3108q^{-131}+3230q^{-133}+729q^{-135}-2110q^{-137}-2971q^{-139}-1404q^{-141}+1053q^{-143}+2346q^{-145}+1681q^{-147}-151q^{-149}-1574q^{-151}-1564q^{-153}-411q^{-155}+811q^{-157}+1185q^{-159}+648q^{-161}-239q^{-163}-753q^{-165}-594q^{-167}-70q^{-169}+357q^{-171}+417q^{-173}+184q^{-175}-115q^{-177}-241q^{-179}-153q^{-181}+5q^{-183}+102q^{-185}+92q^{-187}+31q^{-189}-35q^{-191}-50q^{-193}-16q^{-195}+10q^{-197}+15q^{-199}+7q^{-201}+2q^{-203}-8q^{-205}-6q^{-207}+5q^{-209}+3q^{-211}-q^{-213}-3q^{-219}+q^{-221}+2q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}-q^{2}-1+3q^{-2}+4q^{-6}+2q^{-8}+q^{-12}-3q^{-14}+q^{-16}-q^{-18}-q^{-20}+q^{-22}-q^{-24}$
1,1	$q^{20}-2q^{18}+6q^{16}-12q^{14}+23q^{12}-38q^{10}+58q^{8}-92q^{6}+128q^{4}-182q^{2}+234-300q^{-2}+352q^{-4}-382q^{-6}+392q^{-8}-338q^{-10}+257q^{-12}-96q^{-14}-70q^{-16}+282q^{-18}-482q^{-20}+652q^{-22}-784q^{-24}+832q^{-26}-831q^{-28}+740q^{-30}-598q^{-32}+410q^{-34}-198q^{-36}-8q^{-38}+188q^{-40}-324q^{-42}+406q^{-44}-436q^{-46}+416q^{-48}-360q^{-50}+291q^{-52}-218q^{-54}+148q^{-56}-92q^{-58}+54q^{-60}-28q^{-62}+12q^{-64}-4q^{-66}+q^{-68}$
2,0	$q^{18}-q^{14}+q^{12}+2q^{10}-3q^{8}-4q^{6}+2q^{4}+2q^{2}-8-5q^{-2}+9q^{-4}+q^{-6}-7q^{-8}+4q^{-10}+12q^{-12}+3q^{-14}+q^{-16}+11q^{-18}+7q^{-20}-7q^{-22}+q^{-26}-10q^{-28}-7q^{-30}+5q^{-32}-q^{-34}-9q^{-36}+7q^{-40}-q^{-42}-7q^{-44}+4q^{-46}+5q^{-48}-2q^{-50}-2q^{-52}+q^{-54}+2q^{-56}-q^{-58}-q^{-60}+q^{-62}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-q^{12}+q^{10}+2q^{8}-5q^{6}-q^{4}+3q^{2}-13-2q^{-2}+14q^{-4}-11q^{-6}+5q^{-8}+23q^{-10}-4q^{-12}+12q^{-16}-4q^{-18}-7q^{-20}-2q^{-22}+5q^{-24}-4q^{-26}-11q^{-28}+11q^{-30}+q^{-32}-17q^{-34}+11q^{-36}+5q^{-38}-14q^{-40}+8q^{-42}+3q^{-44}-7q^{-46}+4q^{-48}+q^{-50}-2q^{-52}+q^{-54}$
1,0,0	$-q^{7}-2q^{3}-2q^{-1}+3q^{-3}+5q^{-7}+3q^{-9}+3q^{-11}+q^{-13}-q^{-15}-3q^{-19}+q^{-21}-2q^{-23}+q^{-25}-2q^{-27}+q^{-29}-q^{-31}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{16}+q^{12}+2q^{10}+q^{8}-2q^{6}-2q^{4}-3q^{2}-9-12q^{-2}-4q^{-4}+3q^{-6}-7q^{-8}+4q^{-10}+23q^{-12}+17q^{-14}+3q^{-16}+15q^{-18}+17q^{-20}-4q^{-22}-10q^{-24}+4q^{-26}-q^{-28}-17q^{-30}+8q^{-34}-12q^{-36}-8q^{-38}+10q^{-40}-3q^{-42}-14q^{-44}+2q^{-46}+9q^{-48}-5q^{-50}-7q^{-52}+7q^{-54}+5q^{-56}-5q^{-58}+4q^{-62}-q^{-64}-q^{-66}+q^{-68}$
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^8-2 q^4-q^2-1-2 q^{-2} +3 q^{-4} +5 q^{-8} +4 q^{-10} +4 q^{-12} +3 q^{-14} + q^{-16} -2 q^{-20} -3 q^{-24} + q^{-26} -2 q^{-28} -2 q^{-34} + q^{-36} - q^{-38} }

B2 Invariants.

Weight	Invariant
0,1	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^{14}+q^{12}-3 q^{10}+4 q^8-7 q^6+9 q^4-13 q^2+15-16 q^{-2} +18 q^{-4} -13 q^{-6} +11 q^{-8} - q^{-10} -4 q^{-12} +16 q^{-14} -22 q^{-16} +30 q^{-18} -33 q^{-20} +34 q^{-22} -33 q^{-24} +26 q^{-26} -19 q^{-28} +9 q^{-30} - q^{-32} -7 q^{-34} +13 q^{-36} -17 q^{-38} +18 q^{-40} -18 q^{-42} +15 q^{-44} -11 q^{-46} +8 q^{-48} -5 q^{-50} +2 q^{-52} - q^{-54} }
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^{24}-q^{20}-q^{18}+2 q^{16}+3 q^{14}-q^{12}-6 q^{10}-4 q^8+4 q^6+8 q^4-3 q^2-15-8 q^{-2} +11 q^{-4} +17 q^{-6} -4 q^{-8} -17 q^{-10} -3 q^{-12} +20 q^{-14} +15 q^{-16} -6 q^{-18} -12 q^{-20} +7 q^{-22} +15 q^{-24} + q^{-26} -13 q^{-28} -3 q^{-30} +10 q^{-32} +4 q^{-34} -11 q^{-36} -9 q^{-38} +8 q^{-40} +9 q^{-42} -8 q^{-44} -14 q^{-46} +4 q^{-48} +16 q^{-50} +2 q^{-52} -18 q^{-54} -12 q^{-56} +13 q^{-58} +18 q^{-60} -4 q^{-62} -18 q^{-64} -6 q^{-66} +13 q^{-68} +10 q^{-70} -5 q^{-72} -9 q^{-74} - q^{-76} +6 q^{-78} +3 q^{-80} -2 q^{-82} -2 q^{-84} + q^{-88} }

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^{18}-q^{16}+2 q^{14}-2 q^{12}+4 q^{10}-6 q^8+4 q^6-10 q^4+7 q^2-16+8 q^{-2} -14 q^{-4} +15 q^{-6} -10 q^{-8} +13 q^{-10} + q^{-12} +13 q^{-14} +10 q^{-16} -3 q^{-18} +14 q^{-20} -14 q^{-22} +22 q^{-24} -26 q^{-26} +21 q^{-28} -29 q^{-30} +26 q^{-32} -24 q^{-34} +19 q^{-36} -20 q^{-38} +11 q^{-40} -6 q^{-42} + q^{-44} - q^{-46} -8 q^{-48} +11 q^{-50} -12 q^{-52} +13 q^{-54} -15 q^{-56} +15 q^{-58} -12 q^{-60} +10 q^{-62} -9 q^{-64} +7 q^{-66} -4 q^{-68} +3 q^{-70} -2 q^{-72} + q^{-74} }

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^{32}-q^{30}+3 q^{28}-4 q^{26}+3 q^{24}-3 q^{22}-3 q^{20}+8 q^{18}-14 q^{16}+17 q^{14}-19 q^{12}+12 q^{10}-q^8-16 q^6+34 q^4-51 q^2+53-43 q^{-2} +11 q^{-4} +26 q^{-6} -65 q^{-8} +95 q^{-10} -88 q^{-12} +60 q^{-14} -5 q^{-16} -49 q^{-18} +88 q^{-20} -85 q^{-22} +56 q^{-24} -42 q^{-28} +65 q^{-30} -46 q^{-32} +2 q^{-34} +59 q^{-36} -95 q^{-38} +96 q^{-40} -54 q^{-42} -20 q^{-44} +95 q^{-46} -142 q^{-48} +146 q^{-50} -104 q^{-52} +28 q^{-54} +54 q^{-56} -116 q^{-58} +135 q^{-60} -109 q^{-62} +47 q^{-64} +19 q^{-66} -69 q^{-68} +78 q^{-70} -50 q^{-72} - q^{-74} +52 q^{-76} -77 q^{-78} +62 q^{-80} -17 q^{-82} -47 q^{-84} +94 q^{-86} -108 q^{-88} +84 q^{-90} -33 q^{-92} -27 q^{-94} +73 q^{-96} -90 q^{-98} +81 q^{-100} -48 q^{-102} +9 q^{-104} +20 q^{-106} -40 q^{-108} +39 q^{-110} -29 q^{-112} +17 q^{-114} -3 q^{-116} -5 q^{-118} +8 q^{-120} -8 q^{-122} +5 q^{-124} -2 q^{-126} + q^{-128} }

.

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 77"];

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^3-7 t^2+14 t-17+14 t^{-1} -7 t^{-2} +2 t^{-3} }

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+5z^{4}+4z^{2}+1$

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]=

{ 63, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+3q^{7}-6q^{6}+8q^{5}-10q^{4}+11q^{3}-9q^{2}+8q-4+2q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+z^{6}a^{-4}+4z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+7z^{2}a^{-2}+2z^{2}a^{-4}-2z^{2}a^{-6}-3z^{2}+5a^{-2}-a^{-4}-a^{-6}-2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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