10 123

10_122

10_124

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10_123 can be depicted with five-fold rotational symmetry (like 5 1).

Quasi-floral decorative knot.

Decorative pentagonal representation.

Symmetrical "flower".

Cylindrical depiction

Knot presentations

Planar diagram presentation	X₈₂₉₁ X_10,3,11,4 X_12,6,13,5 X_4,18,5,17 X_18,11,19,12 X_2,15,3,16 X_16,10,17,9 X_20,14,1,13 X_14,7,15,8 X_6,19,7,20
Gauss code	1, -6, 2, -4, 3, -10, 9, -1, 7, -2, 5, -3, 8, -9, 6, -7, 4, -5, 10, -8
Dowker-Thistlethwaite code	8 10 12 14 16 18 20 2 4 6
Conway Notation	[10*]

Three dimensional invariants

Symmetry type	Fully amphicheiral
Unknotting number	2
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	2
Maximal Thurston-Bennequin number	[-6][-6]
Hyperbolic Volume	17.0857
A-Polynomial	See Data:10 123/A-polynomial

[edit Notes for 10 123's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	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 0}
Topological 4 genus	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 0}
Concordance genus	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 0}
Rasmussen s-Invariant	0

[edit Notes for 10 123's four dimensional invariants]

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 t^4-6 t^3+15 t^2-24 t+29-24 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }
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 z^8+2 z^6-z^4-2 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 \left\{t^4-3 t^3+3 t^2-3 t+1\right\}}
Determinant and Signature	{ 121, 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^5+5 q^4-10 q^3+15 q^2-19 q+21-19 q^{-1} +15 q^{-2} -10 q^{-3} +5 q^{-4} - q^{-5} }
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-2z^{4}a^{-2}+3z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3$
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 4 a z^9+4 z^9 a^{-1} +10 a^2 z^8+10 z^8 a^{-2} +20 z^8+10 a^3 z^7+14 a z^7+14 z^7 a^{-1} +10 z^7 a^{-3} +5 a^4 z^6-11 a^2 z^6-11 z^6 a^{-2} +5 z^6 a^{-4} -32 z^6+a^5 z^5-15 a^3 z^5-38 a z^5-38 z^5 a^{-1} -15 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4-3 a^2 z^4-3 z^4 a^{-2} -5 z^4 a^{-4} +4 z^4+5 a^3 z^3+21 a z^3+21 z^3 a^{-1} +5 z^3 a^{-3} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-2 a z-2 z a^{-1} -2 a^2-2 a^{-2} -3}
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^{14}+3 q^{12}-2 q^{10}+3 q^8-3 q^4+4 q^2-5+4 q^{-2} -3 q^{-4} +3 q^{-8} -2 q^{-10} +3 q^{-12} - q^{-14} }
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}-4 q^{78}+10 q^{76}-20 q^{74}+26 q^{72}-25 q^{70}+10 q^{68}+30 q^{66}-80 q^{64}+140 q^{62}-180 q^{60}+158 q^{58}-71 q^{56}-100 q^{54}+308 q^{52}-473 q^{50}+528 q^{48}-391 q^{46}+69 q^{44}+343 q^{42}-693 q^{40}+822 q^{38}-656 q^{36}+239 q^{34}+267 q^{32}-647 q^{30}+750 q^{28}-495 q^{26}+29 q^{24}+435 q^{22}-675 q^{20}+551 q^{18}-133 q^{16}-414 q^{14}+836 q^{12}-944 q^{10}+710 q^8-173 q^6-472 q^4+970 q^2-1165+970 q^{-2} -472 q^{-4} -173 q^{-6} +710 q^{-8} -944 q^{-10} +836 q^{-12} -414 q^{-14} -133 q^{-16} +551 q^{-18} -675 q^{-20} +435 q^{-22} +29 q^{-24} -495 q^{-26} +750 q^{-28} -647 q^{-30} +267 q^{-32} +239 q^{-34} -656 q^{-36} +822 q^{-38} -693 q^{-40} +343 q^{-42} +69 q^{-44} -391 q^{-46} +528 q^{-48} -473 q^{-50} +308 q^{-52} -100 q^{-54} -71 q^{-56} +158 q^{-58} -180 q^{-60} +140 q^{-62} -80 q^{-64} +30 q^{-66} +10 q^{-68} -25 q^{-70} +26 q^{-72} -20 q^{-74} +10 q^{-76} -4 q^{-78} + q^{-80} }

Further Quantum Invariants

Further quantum knot invariants for 10_123.

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}+4 q^9-5 q^7+5 q^5-4 q^3+2 q+2 q^{-1} -4 q^{-3} +5 q^{-5} -5 q^{-7} +4 q^{-9} - q^{-11} }
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}-4 q^{30}+q^{28}+15 q^{26}-21 q^{24}-12 q^{22}+53 q^{20}-27 q^{18}-51 q^{16}+75 q^{14}-q^{12}-76 q^{10}+49 q^8+31 q^6-53 q^4-q^2+45- q^{-2} -53 q^{-4} +31 q^{-6} +49 q^{-8} -76 q^{-10} - q^{-12} +75 q^{-14} -51 q^{-16} -27 q^{-18} +53 q^{-20} -12 q^{-22} -21 q^{-24} +15 q^{-26} + q^{-28} -4 q^{-30} + q^{-32} }
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}+4 q^{61}-q^{59}-11 q^{57}+q^{55}+27 q^{53}+12 q^{51}-73 q^{49}-38 q^{47}+131 q^{45}+126 q^{43}-184 q^{41}-289 q^{39}+185 q^{37}+493 q^{35}-82 q^{33}-682 q^{31}-127 q^{29}+783 q^{27}+387 q^{25}-754 q^{23}-619 q^{21}+601 q^{19}+772 q^{17}-376 q^{15}-810 q^{13}+130 q^{11}+751 q^9+102 q^7-636 q^5-298 q^3+478 q+478 q^{-1} -298 q^{-3} -636 q^{-5} +102 q^{-7} +751 q^{-9} +130 q^{-11} -810 q^{-13} -376 q^{-15} +772 q^{-17} +601 q^{-19} -619 q^{-21} -754 q^{-23} +387 q^{-25} +783 q^{-27} -127 q^{-29} -682 q^{-31} -82 q^{-33} +493 q^{-35} +185 q^{-37} -289 q^{-39} -184 q^{-41} +126 q^{-43} +131 q^{-45} -38 q^{-47} -73 q^{-49} +12 q^{-51} +27 q^{-53} + q^{-55} -11 q^{-57} - q^{-59} +4 q^{-61} - q^{-63} }
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 -q^{155}+4 q^{153}-q^{151}-11 q^{149}+5 q^{147}+11 q^{145}+7 q^{143}-3 q^{141}-28 q^{139}-43 q^{137}+23 q^{135}+137 q^{133}+116 q^{131}-95 q^{129}-381 q^{127}-416 q^{125}+53 q^{123}+958 q^{121}+1448 q^{119}+427 q^{117}-1828 q^{115}-3593 q^{113}-2582 q^{111}+1979 q^{109}+7323 q^{107}+7942 q^{105}+557 q^{103}-11096 q^{101}-17370 q^{99}-9355 q^{97}+11333 q^{95}+29603 q^{93}+26624 q^{91}-2590 q^{89}-39209 q^{87}-51313 q^{85}-19986 q^{83}+38304 q^{81}+77229 q^{79}+56261 q^{77}-19581 q^{75}-93770 q^{73}-99666 q^{71}-19217 q^{69}+90984 q^{67}+138225 q^{65}+72047 q^{63}-64007 q^{61}-159134 q^{59}-126635 q^{57}+16445 q^{55}+154719 q^{53}+168882 q^{51}+40575 q^{49}-125523 q^{47}-188457 q^{45}-93249 q^{43}+79592 q^{41}+182862 q^{39}+130390 q^{37}-29132 q^{35}-156911 q^{33}-146829 q^{31}-15052 q^{29}+119894 q^{27}+144524 q^{25}+46592 q^{23}-81740 q^{21}-129908 q^{19}-64506 q^{17}+49163 q^{15}+110867 q^{13}+72745 q^{11}-24867 q^9-94167 q^7-76937 q^5+7459 q^3+82883 q+82883 q^{-1} +7459 q^{-3} -76937 q^{-5} -94167 q^{-7} -24867 q^{-9} +72745 q^{-11} +110867 q^{-13} +49163 q^{-15} -64506 q^{-17} -129908 q^{-19} -81740 q^{-21} +46592 q^{-23} +144524 q^{-25} +119894 q^{-27} -15052 q^{-29} -146829 q^{-31} -156911 q^{-33} -29132 q^{-35} +130390 q^{-37} +182862 q^{-39} +79592 q^{-41} -93249 q^{-43} -188457 q^{-45} -125523 q^{-47} +40575 q^{-49} +168882 q^{-51} +154719 q^{-53} +16445 q^{-55} -126635 q^{-57} -159134 q^{-59} -64007 q^{-61} +72047 q^{-63} +138225 q^{-65} +90984 q^{-67} -19217 q^{-69} -99666 q^{-71} -93770 q^{-73} -19581 q^{-75} +56261 q^{-77} +77229 q^{-79} +38304 q^{-81} -19986 q^{-83} -51313 q^{-85} -39209 q^{-87} -2590 q^{-89} +26624 q^{-91} +29603 q^{-93} +11333 q^{-95} -9355 q^{-97} -17370 q^{-99} -11096 q^{-101} +557 q^{-103} +7942 q^{-105} +7323 q^{-107} +1979 q^{-109} -2582 q^{-111} -3593 q^{-113} -1828 q^{-115} +427 q^{-117} +1448 q^{-119} +958 q^{-121} +53 q^{-123} -416 q^{-125} -381 q^{-127} -95 q^{-129} +116 q^{-131} +137 q^{-133} +23 q^{-135} -43 q^{-137} -28 q^{-139} -3 q^{-141} +7 q^{-143} +11 q^{-145} +5 q^{-147} -11 q^{-149} - q^{-151} +4 q^{-153} - q^{-155} }
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 q^{216}-4 q^{214}+q^{212}+11 q^{210}-5 q^{208}-11 q^{206}-11 q^{204}+23 q^{202}+13 q^{200}-12 q^{198}+32 q^{196}-63 q^{194}-102 q^{192}-35 q^{190}+216 q^{188}+331 q^{186}+151 q^{184}-63 q^{182}-828 q^{180}-1302 q^{178}-822 q^{176}+1158 q^{174}+3193 q^{172}+3762 q^{170}+2167 q^{168}-3467 q^{166}-9707 q^{164}-11888 q^{162}-4605 q^{160}+9930 q^{158}+24650 q^{156}+29895 q^{154}+12734 q^{152}-22818 q^{150}-59258 q^{148}-66630 q^{146}-29644 q^{144}+45161 q^{142}+122100 q^{140}+139844 q^{138}+65184 q^{136}-85492 q^{134}-227185 q^{132}-263331 q^{130}-129523 q^{128}+140894 q^{126}+394984 q^{124}+456871 q^{122}+224564 q^{120}-215686 q^{118}-631328 q^{116}-729568 q^{114}-362254 q^{112}+323275 q^{110}+944757 q^{108}+1069701 q^{106}+528876 q^{104}-463516 q^{102}-1326402 q^{100}-1467670 q^{98}-686860 q^{96}+650850 q^{94}+1746741 q^{92}+1872976 q^{90}+805959 q^{88}-889284 q^{86}-2184458 q^{84}-2211519 q^{82}-835611 q^{80}+1167684 q^{78}+2577899 q^{76}+2429580 q^{74}+743587 q^{72}-1482269 q^{70}-2853591 q^{68}-2467337 q^{66}-527285 q^{64}+1780716 q^{62}+2971580 q^{60}+2302095 q^{58}+195777 q^{56}-1998465 q^{54}-2893466 q^{52}-1955828 q^{50}+181758 q^{48}+2103658 q^{46}+2622082 q^{44}+1470680 q^{42}-527658 q^{40}-2065674 q^{38}-2201421 q^{36}-938315 q^{34}+801680 q^{32}+1892462 q^{30}+1687640 q^{28}+437651 q^{26}-973947 q^{24}-1624896 q^{22}-1162065 q^{20}+1047 q^{18}+1054281 q^{16}+1308526 q^{14}+672400 q^{12}-368280 q^{10}-1077408 q^8-993482 q^6-218554 q^4+688126 q^2+1077985+688126 q^{-2} -218554 q^{-4} -993482 q^{-6} -1077408 q^{-8} -368280 q^{-10} +672400 q^{-12} +1308526 q^{-14} +1054281 q^{-16} +1047 q^{-18} -1162065 q^{-20} -1624896 q^{-22} -973947 q^{-24} +437651 q^{-26} +1687640 q^{-28} +1892462 q^{-30} +801680 q^{-32} -938315 q^{-34} -2201421 q^{-36} -2065674 q^{-38} -527658 q^{-40} +1470680 q^{-42} +2622082 q^{-44} +2103658 q^{-46} +181758 q^{-48} -1955828 q^{-50} -2893466 q^{-52} -1998465 q^{-54} +195777 q^{-56} +2302095 q^{-58} +2971580 q^{-60} +1780716 q^{-62} -527285 q^{-64} -2467337 q^{-66} -2853591 q^{-68} -1482269 q^{-70} +743587 q^{-72} +2429580 q^{-74} +2577899 q^{-76} +1167684 q^{-78} -835611 q^{-80} -2211519 q^{-82} -2184458 q^{-84} -889284 q^{-86} +805959 q^{-88} +1872976 q^{-90} +1746741 q^{-92} +650850 q^{-94} -686860 q^{-96} -1467670 q^{-98} -1326402 q^{-100} -463516 q^{-102} +528876 q^{-104} +1069701 q^{-106} +944757 q^{-108} +323275 q^{-110} -362254 q^{-112} -729568 q^{-114} -631328 q^{-116} -215686 q^{-118} +224564 q^{-120} +456871 q^{-122} +394984 q^{-124} +140894 q^{-126} -129523 q^{-128} -263331 q^{-130} -227185 q^{-132} -85492 q^{-134} +65184 q^{-136} +139844 q^{-138} +122100 q^{-140} +45161 q^{-142} -29644 q^{-144} -66630 q^{-146} -59258 q^{-148} -22818 q^{-150} +12734 q^{-152} +29895 q^{-154} +24650 q^{-156} +9930 q^{-158} -4605 q^{-160} -11888 q^{-162} -9707 q^{-164} -3467 q^{-166} +2167 q^{-168} +3762 q^{-170} +3193 q^{-172} +1158 q^{-174} -822 q^{-176} -1302 q^{-178} -828 q^{-180} -63 q^{-182} +151 q^{-184} +331 q^{-186} +216 q^{-188} -35 q^{-190} -102 q^{-192} -63 q^{-194} +32 q^{-196} -12 q^{-198} +13 q^{-200} +23 q^{-202} -11 q^{-204} -11 q^{-206} -5 q^{-208} +11 q^{-210} + q^{-212} -4 q^{-214} + q^{-216} }

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}+3q^{12}-2q^{10}+3q^{8}-3q^{4}+4q^{2}-5+4q^{-2}-3q^{-4}+3q^{-8}-2q^{-10}+3q^{-12}-q^{-14}$
1,1	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^{44}-8 q^{42}+32 q^{40}-88 q^{38}+196 q^{36}-392 q^{34}+716 q^{32}-1194 q^{30}+1831 q^{28}-2600 q^{26}+3434 q^{24}-4172 q^{22}+4631 q^{20}-4638 q^{18}+4072 q^{16}-2860 q^{14}+1036 q^{12}+1192 q^{10}-3588 q^8+5840 q^6-7694 q^4+8922 q^2-9330+8922 q^{-2} -7694 q^{-4} +5840 q^{-6} -3588 q^{-8} +1192 q^{-10} +1036 q^{-12} -2860 q^{-14} +4072 q^{-16} -4638 q^{-18} +4631 q^{-20} -4172 q^{-22} +3434 q^{-24} -2600 q^{-26} +1831 q^{-28} -1194 q^{-30} +716 q^{-32} -392 q^{-34} +196 q^{-36} -88 q^{-38} +32 q^{-40} -8 q^{-42} + q^{-44} }
2,0	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^{38}-3 q^{36}-q^{34}+9 q^{32}-6 q^{30}-9 q^{28}+14 q^{26}+9 q^{24}-13 q^{22}-10 q^{20}+20 q^{18}+10 q^{16}-35 q^{14}+2 q^{12}+24 q^{10}-20 q^8-14 q^6+21 q^4+11 q^2-14+11 q^{-2} +21 q^{-4} -14 q^{-6} -20 q^{-8} +24 q^{-10} +2 q^{-12} -35 q^{-14} +10 q^{-16} +20 q^{-18} -10 q^{-20} -13 q^{-22} +9 q^{-24} +14 q^{-26} -9 q^{-28} -6 q^{-30} +9 q^{-32} - q^{-34} -3 q^{-36} + q^{-38} }

A3 Invariants.

Weight	Invariant
0,1,0	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^{34}-4 q^{32}+2 q^{30}+11 q^{28}-19 q^{26}+4 q^{24}+29 q^{22}-43 q^{20}+6 q^{18}+49 q^{16}-54 q^{14}+5 q^{12}+50 q^{10}-40 q^8-8 q^6+28 q^4-6 q^2-16-6 q^{-2} +28 q^{-4} -8 q^{-6} -40 q^{-8} +50 q^{-10} +5 q^{-12} -54 q^{-14} +49 q^{-16} +6 q^{-18} -43 q^{-20} +29 q^{-22} +4 q^{-24} -19 q^{-26} +11 q^{-28} +2 q^{-30} -4 q^{-32} + q^{-34} }
1,0,0	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^{17}+3 q^{15}-3 q^{13}+6 q^{11}-3 q^9+4 q^7-3 q^5+q^3-2 q-2 q^{-1} + q^{-3} -3 q^{-5} +4 q^{-7} -3 q^{-9} +6 q^{-11} -3 q^{-13} +3 q^{-15} - q^{-17} }
1,0,1	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^{56}-8 q^{54}+28 q^{52}-52 q^{50}+38 q^{48}+65 q^{46}-253 q^{44}+409 q^{42}-326 q^{40}-151 q^{38}+912 q^{36}-1524 q^{34}+1436 q^{32}-334 q^{30}-1493 q^{28}+3199 q^{26}-3741 q^{24}+2539 q^{22}+148 q^{20}-3147 q^{18}+5022 q^{16}-4816 q^{14}+2595 q^{12}+368 q^{10}-2626 q^8+3081 q^6-1935 q^4+375 q^2+395+375 q^{-2} -1935 q^{-4} +3081 q^{-6} -2626 q^{-8} +368 q^{-10} +2595 q^{-12} -4816 q^{-14} +5022 q^{-16} -3147 q^{-18} +148 q^{-20} +2539 q^{-22} -3741 q^{-24} +3199 q^{-26} -1493 q^{-28} -334 q^{-30} +1436 q^{-32} -1524 q^{-34} +912 q^{-36} -151 q^{-38} -326 q^{-40} +409 q^{-42} -253 q^{-44} +65 q^{-46} +38 q^{-48} -52 q^{-50} +28 q^{-52} -8 q^{-54} + q^{-56} }

A4 Invariants.

Weight

Invariant

0,1,0,0

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^{40}-3 q^{38}-q^{36}+10 q^{34}-8 q^{32}-9 q^{30}+25 q^{28}-5 q^{26}-30 q^{24}+23 q^{22}+26 q^{20}-36 q^{18}-11 q^{16}+43 q^{14}+q^{12}-48 q^{10}+18 q^8+46 q^6-50 q^4-18 q^2+62-18 q^{-2} -50 q^{-4} +46 q^{-6} +18 q^{-8} -48 q^{-10} + q^{-12} +43 q^{-14} -11 q^{-16} -36 q^{-18} +26 q^{-20} +23 q^{-22} -30 q^{-24} -5 q^{-26} +25 q^{-28} -9 q^{-30} -8 q^{-32} +10 q^{-34} - q^{-36} -3 q^{-38} + q^{-40} }

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^{20}+3 q^{18}-3 q^{16}+5 q^{14}+q^{10}+3 q^8-3 q^6+2 q^4-5 q^2+1-5 q^{-2} +2 q^{-4} -3 q^{-6} +3 q^{-8} + q^{-10} +5 q^{-14} -3 q^{-16} +3 q^{-18} - q^{-20} }

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^{34}+4 q^{32}-10 q^{30}+19 q^{28}-31 q^{26}+46 q^{24}-59 q^{22}+69 q^{20}-70 q^{18}+65 q^{16}-48 q^{14}+23 q^{12}+10 q^{10}-46 q^8+80 q^6-110 q^4+130 q^2-138+130 q^{-2} -110 q^{-4} +80 q^{-6} -46 q^{-8} +10 q^{-10} +23 q^{-12} -48 q^{-14} +65 q^{-16} -70 q^{-18} +69 q^{-20} -59 q^{-22} +46 q^{-24} -31 q^{-26} +19 q^{-28} -10 q^{-30} +4 q^{-32} - q^{-34} }

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^{56}-4 q^{52}-4 q^{50}+6 q^{48}+15 q^{46}+q^{44}-25 q^{42}-20 q^{40}+25 q^{38}+44 q^{36}-6 q^{34}-62 q^{32}-28 q^{30}+57 q^{28}+61 q^{26}-29 q^{24}-75 q^{22}-4 q^{20}+71 q^{18}+32 q^{16}-52 q^{14}-45 q^{12}+32 q^{10}+48 q^8-17 q^6-49 q^4+5 q^2+49+5 q^{-2} -49 q^{-4} -17 q^{-6} +48 q^{-8} +32 q^{-10} -45 q^{-12} -52 q^{-14} +32 q^{-16} +71 q^{-18} -4 q^{-20} -75 q^{-22} -29 q^{-24} +61 q^{-26} +57 q^{-28} -28 q^{-30} -62 q^{-32} -6 q^{-34} +44 q^{-36} +25 q^{-38} -20 q^{-40} -25 q^{-42} + q^{-44} +15 q^{-46} +6 q^{-48} -4 q^{-50} -4 q^{-52} + q^{-56} }

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^{46}-4 q^{44}+6 q^{42}-8 q^{40}+16 q^{38}-25 q^{36}+30 q^{34}-36 q^{32}+48 q^{30}-56 q^{28}+53 q^{26}-53 q^{24}+56 q^{22}-42 q^{20}+26 q^{18}-12 q^{16}+2 q^{14}+30 q^{12}-51 q^{10}+61 q^8-79 q^6+98 q^4-106 q^2+98-106 q^{-2} +98 q^{-4} -79 q^{-6} +61 q^{-8} -51 q^{-10} +30 q^{-12} +2 q^{-14} -12 q^{-16} +26 q^{-18} -42 q^{-20} +56 q^{-22} -53 q^{-24} +53 q^{-26} -56 q^{-28} +48 q^{-30} -36 q^{-32} +30 q^{-34} -25 q^{-36} +16 q^{-38} -8 q^{-40} +6 q^{-42} -4 q^{-44} + q^{-46} }

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^{80}-4 q^{78}+10 q^{76}-20 q^{74}+26 q^{72}-25 q^{70}+10 q^{68}+30 q^{66}-80 q^{64}+140 q^{62}-180 q^{60}+158 q^{58}-71 q^{56}-100 q^{54}+308 q^{52}-473 q^{50}+528 q^{48}-391 q^{46}+69 q^{44}+343 q^{42}-693 q^{40}+822 q^{38}-656 q^{36}+239 q^{34}+267 q^{32}-647 q^{30}+750 q^{28}-495 q^{26}+29 q^{24}+435 q^{22}-675 q^{20}+551 q^{18}-133 q^{16}-414 q^{14}+836 q^{12}-944 q^{10}+710 q^8-173 q^6-472 q^4+970 q^2-1165+970 q^{-2} -472 q^{-4} -173 q^{-6} +710 q^{-8} -944 q^{-10} +836 q^{-12} -414 q^{-14} -133 q^{-16} +551 q^{-18} -675 q^{-20} +435 q^{-22} +29 q^{-24} -495 q^{-26} +750 q^{-28} -647 q^{-30} +267 q^{-32} +239 q^{-34} -656 q^{-36} +822 q^{-38} -693 q^{-40} +343 q^{-42} +69 q^{-44} -391 q^{-46} +528 q^{-48} -473 q^{-50} +308 q^{-52} -100 q^{-54} -71 q^{-56} +158 q^{-58} -180 q^{-60} +140 q^{-62} -80 q^{-64} +30 q^{-66} +10 q^{-68} -25 q^{-70} +26 q^{-72} -20 q^{-74} +10 q^{-76} -4 q^{-78} + q^{-80} }

.

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.

In[3]:= K = Knot["10 123"];

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 t^4-6 t^3+15 t^2-24 t+29-24 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }

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 z^8+2 z^6-z^4-2 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 \left\{t^4-3 t^3+3 t^2-3 t+1\right\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 121, 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^5+5 q^4-10 q^3+15 q^2-19 q+21-19 q^{-1} +15 q^{-2} -10 q^{-3} +5 q^{-4} - q^{-5} }

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $z^{8}-a^{2}z^{6}-z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-2z^{4}a^{-2}+3z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3$

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 4 a z^9+4 z^9 a^{-1} +10 a^2 z^8+10 z^8 a^{-2} +20 z^8+10 a^3 z^7+14 a z^7+14 z^7 a^{-1} +10 z^7 a^{-3} +5 a^4 z^6-11 a^2 z^6-11 z^6 a^{-2} +5 z^6 a^{-4} -32 z^6+a^5 z^5-15 a^3 z^5-38 a z^5-38 z^5 a^{-1} -15 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4-3 a^2 z^4-3 z^4 a^{-2} -5 z^4 a^{-4} +4 z^4+5 a^3 z^3+21 a z^3+21 z^3 a^{-1} +5 z^3 a^{-3} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-2 a z-2 z a^{-1} -2 a^2-2 a^{-2} -3}

Vassiliev invariants

V₂ and V₃:

(-2, 0)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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 -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 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 32}

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 \frac{164}{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 \frac{76}{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 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 -\frac{256}{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 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 -\frac{1312}{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 -\frac{608}{3}}

-{\frac {6271}{15}}

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 \frac{1484}{15}}

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 -\frac{16684}{45}}

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 \frac{31}{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 -\frac{1471}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials 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 t^rq^j} are shown, along with their alternating sums 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 \chi} (fixed 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 j} , alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where 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 j-2r=s+1} or 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 j-2r=s-1} , where 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 s=} 0 is the signature of 10 123. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

4

7

6

1

-5

5

9

4

5

3

10

6

-4

1

11

9

2

-1

9

11

2

-3

6

10

-4

-5

4

9

5

-7

1

6

-5

-9

4

-11

1

-1

Integral Khovanov Homology

(db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	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 i=-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 i=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 r=-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 {\mathbb Z}}
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 r=-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 {\mathbb Z}^{4}\oplus{\mathbb Z}_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 {\mathbb Z}}
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 r=-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 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-2}	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{11}}
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 r=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 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}_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 {\mathbb Z}}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

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 \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 123]]
Out[2]=	10
In[3]:=	PD[Knot[10, 123]]
Out[3]=	PD[X[8, 2, 9, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[4, 18, 5, 17], X[18, 11, 19, 12], X[2, 15, 3, 16], X[16, 10, 17, 9], X[20, 14, 1, 13], X[14, 7, 15, 8], X[6, 19, 7, 20]]
In[4]:=	GaussCode[Knot[10, 123]]
Out[4]=	GaussCode[1, -6, 2, -4, 3, -10, 9, -1, 7, -2, 5, -3, 8, -9, 6, -7, 4, -5, 10, -8]
In[5]:=	BR[Knot[10, 123]]
Out[5]=	BR[3, {-1, 2, -1, 2, -1, 2, -1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[10, 123]][t]
Out[6]=	-4 6 15 24 2 3 4 29 + t - -- + -- - -- - 24 t + 15 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 123]][z]
Out[7]=	2 4 6 8 1 - 2 z - z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 123], Knot[11, Alternating, 28]}
In[9]:=	{KnotDet[Knot[10, 123]], KnotSignature[Knot[10, 123]]}
Out[9]=	{121, 0}
In[10]:=	J=Jones[Knot[10, 123]][q]
Out[10]=	-5 5 10 15 19 2 3 4 5 21 - q + -- - -- + -- - -- - 19 q + 15 q - 10 q + 5 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 123]}
In[12]:=	A2Invariant[Knot[10, 123]][q]
Out[12]=	-14 3 2 3 3 4 2 4 8 10 -5 - q + --- - --- + -- - -- + -- + 4 q - 3 q + 3 q - 2 q + 12 10 8 4 2 q q q q q 12 14 3 q - q
In[13]:=	Kauffman[Knot[10, 123]][a, z]
Out[13]=	2 3 3 2 2 2 z 2 6 z 2 2 5 z 21 z -3 - -- - 2 a - --- - 2 a z + 12 z + ---- + 6 a z + ---- + ----- + 2 a 2 3 a a a a 4 4 5 3 3 3 4 5 z 3 z 2 4 4 4 z 21 a z + 5 a z + 4 z - ---- - ---- - 3 a z - 5 a z + -- - 4 2 5 a a a 5 5 6 6 15 z 38 z 5 3 5 5 5 6 5 z 11 z ----- - ----- - 38 a z - 15 a z + a z - 32 z + ---- - ----- - 3 a 4 2 a a a 7 7 2 6 4 6 10 z 14 z 7 3 7 8 11 a z + 5 a z + ----- + ----- + 14 a z + 10 a z + 20 z + 3 a a 8 9 10 z 2 8 4 z 9 ----- + 10 a z + ---- + 4 a z 2 a a
In[14]:=	{Vassiliev[2][Knot[10, 123]], Vassiliev[3][Knot[10, 123]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 123]][q, t]
Out[15]=	11 1 4 1 6 4 9 6 -- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 10 9 3 3 2 5 2 5 3 ---- + --- + 9 q t + 10 q t + 6 q t + 9 q t + 4 q t + 3 q t q t 7 3 7 4 9 4 11 5 6 q t + q t + 4 q t + q t

10 123

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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