10 123

10_122

10_124

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 123's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 123 at Knotilus!
	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 X10,3,11,4 X12,6,13,5 X4,18,5,17 X18,11,19,12 X2,15,3,16 X16,10,17,9 X20,14,1,13 X14,7,15,8 X6,19,7,20
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*]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 3, Braid index is 3		[{3, 10}, {2, 8}, {9, 7}, {8, 11}, {10, 6}, {7, 12}, {11, 4}, {5, 3}, {4, 1}, {6, 2}, {12, 5}, {1, 9}]

[edit Notes on presentations of 10 123]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["10 123"];

In[4]:=

PD[K]

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

Out[4]=

X8291 X10,3,11,4 X12,6,13,5 X4,18,5,17 X18,11,19,12 X2,15,3,16 X16,10,17,9 X20,14,1,13 X14,7,15,8 X6,19,7,20

In[5]:=

GaussCode[K]

Out[5]=

1, -6, 2, -4, 3, -10, 9, -1, 7, -2, 5, -3, 8, -9, 6, -7, 4, -5, 10, -8

In[6]:=

DTCode[K]

Out[6]=

8 10 12 14 16 18 20 2 4 6

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[10*]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

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 \textrm{BR}(3,\{-1,2,-1,2,-1,2,-1,2,-1,2\})}

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 3, 10, 3 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{3, 10}, {2, 8}, {9, 7}, {8, 11}, {10, 6}, {7, 12}, {11, 4}, {5, 3}, {4, 1}, {6, 2}, {12, 5}, {1, 9}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

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	$-q^{5}+5q^{4}-10q^{3}+15q^{2}-19q+21-19q^{-1}+15q^{-2}-10q^{-3}+5q^{-4}-q^{-5}$
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^8-a^2 z^6-z^6 a^{-2} +4 z^6-2 a^2 z^4-2 z^4 a^{-2} +3 z^4+a^2 z^2+z^2 a^{-2} -4 z^2+2 a^2+2 a^{-2} -3}
Kauffman polynomial (db, data sources)	$4az^{9}+4z^{9}a^{-1}+10a^{2}z^{8}+10z^{8}a^{-2}+20z^{8}+10a^{3}z^{7}+14az^{7}+14z^{7}a^{-1}+10z^{7}a^{-3}+5a^{4}z^{6}-11a^{2}z^{6}-11z^{6}a^{-2}+5z^{6}a^{-4}-32z^{6}+a^{5}z^{5}-15a^{3}z^{5}-38az^{5}-38z^{5}a^{-1}-15z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}-3a^{2}z^{4}-3z^{4}a^{-2}-5z^{4}a^{-4}+4z^{4}+5a^{3}z^{3}+21az^{3}+21z^{3}a^{-1}+5z^{3}a^{-3}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3$
The A2 invariant	$-q^{14}+3q^{12}-2q^{10}+3q^{8}-3q^{4}+4q^{2}-5+4q^{-2}-3q^{-4}+3q^{-8}-2q^{-10}+3q^{-12}-q^{-14}$
The G2 invariant	$q^{80}-4q^{78}+10q^{76}-20q^{74}+26q^{72}-25q^{70}+10q^{68}+30q^{66}-80q^{64}+140q^{62}-180q^{60}+158q^{58}-71q^{56}-100q^{54}+308q^{52}-473q^{50}+528q^{48}-391q^{46}+69q^{44}+343q^{42}-693q^{40}+822q^{38}-656q^{36}+239q^{34}+267q^{32}-647q^{30}+750q^{28}-495q^{26}+29q^{24}+435q^{22}-675q^{20}+551q^{18}-133q^{16}-414q^{14}+836q^{12}-944q^{10}+710q^{8}-173q^{6}-472q^{4}+970q^{2}-1165+970q^{-2}-472q^{-4}-173q^{-6}+710q^{-8}-944q^{-10}+836q^{-12}-414q^{-14}-133q^{-16}+551q^{-18}-675q^{-20}+435q^{-22}+29q^{-24}-495q^{-26}+750q^{-28}-647q^{-30}+267q^{-32}+239q^{-34}-656q^{-36}+822q^{-38}-693q^{-40}+343q^{-42}+69q^{-44}-391q^{-46}+528q^{-48}-473q^{-50}+308q^{-52}-100q^{-54}-71q^{-56}+158q^{-58}-180q^{-60}+140q^{-62}-80q^{-64}+30q^{-66}+10q^{-68}-25q^{-70}+26q^{-72}-20q^{-74}+10q^{-76}-4q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_123.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+4q^{9}-5q^{7}+5q^{5}-4q^{3}+2q+2q^{-1}-4q^{-3}+5q^{-5}-5q^{-7}+4q^{-9}-q^{-11}$
2	$q^{32}-4q^{30}+q^{28}+15q^{26}-21q^{24}-12q^{22}+53q^{20}-27q^{18}-51q^{16}+75q^{14}-q^{12}-76q^{10}+49q^{8}+31q^{6}-53q^{4}-q^{2}+45-q^{-2}-53q^{-4}+31q^{-6}+49q^{-8}-76q^{-10}-q^{-12}+75q^{-14}-51q^{-16}-27q^{-18}+53q^{-20}-12q^{-22}-21q^{-24}+15q^{-26}+q^{-28}-4q^{-30}+q^{-32}$
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	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} }
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	$q^{56}-8q^{54}+28q^{52}-52q^{50}+38q^{48}+65q^{46}-253q^{44}+409q^{42}-326q^{40}-151q^{38}+912q^{36}-1524q^{34}+1436q^{32}-334q^{30}-1493q^{28}+3199q^{26}-3741q^{24}+2539q^{22}+148q^{20}-3147q^{18}+5022q^{16}-4816q^{14}+2595q^{12}+368q^{10}-2626q^{8}+3081q^{6}-1935q^{4}+375q^{2}+395+375q^{-2}-1935q^{-4}+3081q^{-6}-2626q^{-8}+368q^{-10}+2595q^{-12}-4816q^{-14}+5022q^{-16}-3147q^{-18}+148q^{-20}+2539q^{-22}-3741q^{-24}+3199q^{-26}-1493q^{-28}-334q^{-30}+1436q^{-32}-1524q^{-34}+912q^{-36}-151q^{-38}-326q^{-40}+409q^{-42}-253q^{-44}+65q^{-46}+38q^{-48}-52q^{-50}+28q^{-52}-8q^{-54}+q^{-56}$

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

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]=

$-q^{5}+5q^{4}-10q^{3}+15q^{2}-19q+21-19q^{-1}+15q^{-2}-10q^{-3}+5q^{-4}-q^{-5}$

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^8-a^2 z^6-z^6 a^{-2} +4 z^6-2 a^2 z^4-2 z^4 a^{-2} +3 z^4+a^2 z^2+z^2 a^{-2} -4 z^2+2 a^2+2 a^{-2} -3}

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a28,}

Same Jones Polynomial (up to mirroring, 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\leftrightarrow q^{-1}} ): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 123"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

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} } , $-q^{5}+5q^{4}-10q^{3}+15q^{2}-19q+21-19q^{-1}+15q^{-2}-10q^{-3}+5q^{-4}-q^{-5}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11a28,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-2, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 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 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 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 -\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}} 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{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 $j$, alternation over 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} ). 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   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                   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}} $i=-1$ $i=1$ 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} 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^{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}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the 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 n+1} -dimensional representation of 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 sl(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 n}	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_n}
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^{15}-5 q^{14}+5 q^{13}+15 q^{12}-41 q^{11}+14 q^{10}+80 q^9-121 q^8-10 q^7+206 q^6-197 q^5-85 q^4+331 q^3-215 q^2-169 q+383-169 q^{-1} -215 q^{-2} +331 q^{-3} -85 q^{-4} -197 q^{-5} +206 q^{-6} -10 q^{-7} -121 q^{-8} +80 q^{-9} +14 q^{-10} -41 q^{-11} +15 q^{-12} +5 q^{-13} -5 q^{-14} + q^{-15} }
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^{30}+5 q^{29}-5 q^{28}-10 q^{27}+11 q^{26}+31 q^{25}-20 q^{24}-95 q^{23}+46 q^{22}+200 q^{21}-25 q^{20}-405 q^{19}-59 q^{18}+674 q^{17}+283 q^{16}-980 q^{15}-659 q^{14}+1229 q^{13}+1193 q^{12}-1376 q^{11}-1800 q^{10}+1364 q^9+2413 q^8-1205 q^7-2948 q^6+930 q^5+3353 q^4-584 q^3-3597 q^2+192 q+3691+192 q^{-1} -3597 q^{-2} -584 q^{-3} +3353 q^{-4} +930 q^{-5} -2948 q^{-6} -1205 q^{-7} +2413 q^{-8} +1364 q^{-9} -1800 q^{-10} -1376 q^{-11} +1193 q^{-12} +1229 q^{-13} -659 q^{-14} -980 q^{-15} +283 q^{-16} +674 q^{-17} -59 q^{-18} -405 q^{-19} -25 q^{-20} +200 q^{-21} +46 q^{-22} -95 q^{-23} -20 q^{-24} +31 q^{-25} +11 q^{-26} -10 q^{-27} -5 q^{-28} +5 q^{-29} - q^{-30} }
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 q^{50}-5 q^{49}+5 q^{48}+10 q^{47}-16 q^{46}-q^{45}-25 q^{44}+50 q^{43}+75 q^{42}-111 q^{41}-76 q^{40}-144 q^{39}+300 q^{38}+521 q^{37}-300 q^{36}-645 q^{35}-1029 q^{34}+795 q^{33}+2396 q^{32}+536 q^{31}-1785 q^{30}-4555 q^{29}-371 q^{28}+5976 q^{27}+5172 q^{26}-707 q^{25}-11055 q^{24}-6857 q^{23}+7535 q^{22}+13845 q^{21}+6944 q^{20}-15956 q^{19}-18552 q^{18}+2209 q^{17}+21368 q^{16}+20543 q^{15}-14191 q^{14}-29641 q^{13}-9205 q^{12}+22728 q^{11}+33968 q^{10}-6460 q^9-35045 q^8-21175 q^7+18340 q^6+42330 q^5+2887 q^4-34554 q^3-29818 q^2+11268 q+44955+11268 q^{-1} -29818 q^{-2} -34554 q^{-3} +2887 q^{-4} +42330 q^{-5} +18340 q^{-6} -21175 q^{-7} -35045 q^{-8} -6460 q^{-9} +33968 q^{-10} +22728 q^{-11} -9205 q^{-12} -29641 q^{-13} -14191 q^{-14} +20543 q^{-15} +21368 q^{-16} +2209 q^{-17} -18552 q^{-18} -15956 q^{-19} +6944 q^{-20} +13845 q^{-21} +7535 q^{-22} -6857 q^{-23} -11055 q^{-24} -707 q^{-25} +5172 q^{-26} +5976 q^{-27} -371 q^{-28} -4555 q^{-29} -1785 q^{-30} +536 q^{-31} +2396 q^{-32} +795 q^{-33} -1029 q^{-34} -645 q^{-35} -300 q^{-36} +521 q^{-37} +300 q^{-38} -144 q^{-39} -76 q^{-40} -111 q^{-41} +75 q^{-42} +50 q^{-43} -25 q^{-44} - q^{-45} -16 q^{-46} +10 q^{-47} +5 q^{-48} -5 q^{-49} + q^{-50} }
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^{75}+5 q^{74}-5 q^{73}-10 q^{72}+16 q^{71}+6 q^{70}-5 q^{69}-5 q^{68}-30 q^{67}-25 q^{66}+82 q^{65}+120 q^{64}-26 q^{63}-216 q^{62}-316 q^{61}-60 q^{60}+551 q^{59}+1025 q^{58}+464 q^{57}-1237 q^{56}-2571 q^{55}-1825 q^{54}+1562 q^{53}+5586 q^{52}+5808 q^{51}-618 q^{50}-9956 q^{49}-13478 q^{48}-4712 q^{47}+13601 q^{46}+26496 q^{45}+17652 q^{44}-12935 q^{43}-42692 q^{42}-41331 q^{41}+1497 q^{40}+57823 q^{39}+75942 q^{38}+25990 q^{37}-63660 q^{36}-117173 q^{35}-72692 q^{34}+51927 q^{33}+156391 q^{32}+136191 q^{31}-16419 q^{30}-183351 q^{29}-208746 q^{28}-43200 q^{27}+188890 q^{26}+279271 q^{25}+121855 q^{24}-169188 q^{23}-337053 q^{22}-209298 q^{21}+125956 q^{20}+374479 q^{19}+294696 q^{18}-65918 q^{17}-389525 q^{16}-368820 q^{15}-1823 q^{14}+384561 q^{13}+426473 q^{12}+69028 q^{11}-364895 q^{10}-466752 q^9-130155 q^8+336393 q^7+491875 q^6+182697 q^5-303191 q^4-504874 q^3-227767 q^2+267093 q+509105+267093 q^{-1} -227767 q^{-2} -504874 q^{-3} -303191 q^{-4} +182697 q^{-5} +491875 q^{-6} +336393 q^{-7} -130155 q^{-8} -466752 q^{-9} -364895 q^{-10} +69028 q^{-11} +426473 q^{-12} +384561 q^{-13} -1823 q^{-14} -368820 q^{-15} -389525 q^{-16} -65918 q^{-17} +294696 q^{-18} +374479 q^{-19} +125956 q^{-20} -209298 q^{-21} -337053 q^{-22} -169188 q^{-23} +121855 q^{-24} +279271 q^{-25} +188890 q^{-26} -43200 q^{-27} -208746 q^{-28} -183351 q^{-29} -16419 q^{-30} +136191 q^{-31} +156391 q^{-32} +51927 q^{-33} -72692 q^{-34} -117173 q^{-35} -63660 q^{-36} +25990 q^{-37} +75942 q^{-38} +57823 q^{-39} +1497 q^{-40} -41331 q^{-41} -42692 q^{-42} -12935 q^{-43} +17652 q^{-44} +26496 q^{-45} +13601 q^{-46} -4712 q^{-47} -13478 q^{-48} -9956 q^{-49} -618 q^{-50} +5808 q^{-51} +5586 q^{-52} +1562 q^{-53} -1825 q^{-54} -2571 q^{-55} -1237 q^{-56} +464 q^{-57} +1025 q^{-58} +551 q^{-59} -60 q^{-60} -316 q^{-61} -216 q^{-62} -26 q^{-63} +120 q^{-64} +82 q^{-65} -25 q^{-66} -30 q^{-67} -5 q^{-68} -5 q^{-69} +6 q^{-70} +16 q^{-71} -10 q^{-72} -5 q^{-73} +5 q^{-74} - q^{-75} }
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^{105}-5 q^{104}+5 q^{103}+10 q^{102}-16 q^{101}-6 q^{100}+35 q^{98}-15 q^{97}-20 q^{96}+54 q^{95}-111 q^{94}-45 q^{93}+67 q^{92}+286 q^{91}+100 q^{90}-200 q^{89}-160 q^{88}-876 q^{87}-519 q^{86}+547 q^{85}+2266 q^{84}+2135 q^{83}+369 q^{82}-1755 q^{81}-6510 q^{80}-6759 q^{79}-1634 q^{78}+9549 q^{77}+16670 q^{76}+15089 q^{75}+3490 q^{74}-23671 q^{73}-42311 q^{72}-38074 q^{71}+2177 q^{70}+53656 q^{69}+89894 q^{68}+80429 q^{67}-5927 q^{66}-116971 q^{65}-188750 q^{64}-139516 q^{63}+17510 q^{62}+223702 q^{61}+350846 q^{60}+248163 q^{59}-55084 q^{58}-421057 q^{57}-579766 q^{56}-398132 q^{55}+125462 q^{54}+718160 q^{53}+933692 q^{52}+566398 q^{51}-296113 q^{50}-1120591 q^{49}-1390524 q^{48}-737424 q^{47}+576892 q^{46}+1714502 q^{45}+1904108 q^{44}+799778 q^{43}-994356 q^{42}-2457541 q^{41}-2432667 q^{40}-718282 q^{39}+1687441 q^{38}+3280016 q^{37}+2803073 q^{36}+415859 q^{35}-2605860 q^{34}-4118660 q^{33}-2944138 q^{32}+316119 q^{31}+3666270 q^{30}+4743125 q^{29}+2723860 q^{28}-1414996 q^{27}-4788145 q^{26}-5050456 q^{25}-1878123 q^{24}+2771269 q^{23}+5680763 q^{22}+4861446 q^{21}+506904 q^{20}-4269721 q^{19}-6201858 q^{18}-3876461 q^{17}+1233253 q^{16}+5545016 q^{15}+6124552 q^{14}+2246899 q^{13}-3178939 q^{12}-6406680 q^{11}-5126450 q^{10}-178345 q^9+4894067 q^8+6587383 q^7+3410011 q^6-2125705 q^5-6152435 q^4-5762576 q^3-1219025 q^2+4184939 q+6671309+4184939 q^{-1} -1219025 q^{-2} -5762576 q^{-3} -6152435 q^{-4} -2125705 q^{-5} +3410011 q^{-6} +6587383 q^{-7} +4894067 q^{-8} -178345 q^{-9} -5126450 q^{-10} -6406680 q^{-11} -3178939 q^{-12} +2246899 q^{-13} +6124552 q^{-14} +5545016 q^{-15} +1233253 q^{-16} -3876461 q^{-17} -6201858 q^{-18} -4269721 q^{-19} +506904 q^{-20} +4861446 q^{-21} +5680763 q^{-22} +2771269 q^{-23} -1878123 q^{-24} -5050456 q^{-25} -4788145 q^{-26} -1414996 q^{-27} +2723860 q^{-28} +4743125 q^{-29} +3666270 q^{-30} +316119 q^{-31} -2944138 q^{-32} -4118660 q^{-33} -2605860 q^{-34} +415859 q^{-35} +2803073 q^{-36} +3280016 q^{-37} +1687441 q^{-38} -718282 q^{-39} -2432667 q^{-40} -2457541 q^{-41} -994356 q^{-42} +799778 q^{-43} +1904108 q^{-44} +1714502 q^{-45} +576892 q^{-46} -737424 q^{-47} -1390524 q^{-48} -1120591 q^{-49} -296113 q^{-50} +566398 q^{-51} +933692 q^{-52} +718160 q^{-53} +125462 q^{-54} -398132 q^{-55} -579766 q^{-56} -421057 q^{-57} -55084 q^{-58} +248163 q^{-59} +350846 q^{-60} +223702 q^{-61} +17510 q^{-62} -139516 q^{-63} -188750 q^{-64} -116971 q^{-65} -5927 q^{-66} +80429 q^{-67} +89894 q^{-68} +53656 q^{-69} +2177 q^{-70} -38074 q^{-71} -42311 q^{-72} -23671 q^{-73} +3490 q^{-74} +15089 q^{-75} +16670 q^{-76} +9549 q^{-77} -1634 q^{-78} -6759 q^{-79} -6510 q^{-80} -1755 q^{-81} +369 q^{-82} +2135 q^{-83} +2266 q^{-84} +547 q^{-85} -519 q^{-86} -876 q^{-87} -160 q^{-88} -200 q^{-89} +100 q^{-90} +286 q^{-91} +67 q^{-92} -45 q^{-93} -111 q^{-94} +54 q^{-95} -20 q^{-96} -15 q^{-97} +35 q^{-98} -6 q^{-100} -16 q^{-101} +10 q^{-102} +5 q^{-103} -5 q^{-104} + q^{-105} }
7	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^{140}+5 q^{139}-5 q^{138}-10 q^{137}+16 q^{136}+6 q^{135}-30 q^{133}-15 q^{132}+65 q^{131}-9 q^{130}-25 q^{129}+36 q^{128}-11 q^{127}-42 q^{126}-195 q^{125}-115 q^{124}+385 q^{123}+395 q^{122}+319 q^{121}+136 q^{120}-646 q^{119}-1137 q^{118}-1890 q^{117}-1295 q^{116}+1720 q^{115}+4250 q^{114}+5950 q^{113}+4577 q^{112}-1660 q^{111}-9312 q^{110}-17220 q^{109}-18195 q^{108}-4883 q^{107}+17046 q^{106}+41839 q^{105}+52772 q^{104}+33340 q^{103}-13156 q^{102}-78969 q^{101}-129393 q^{100}-120485 q^{99}-37709 q^{98}+110481 q^{97}+258889 q^{96}+312390 q^{95}+212678 q^{94}-58409 q^{93}-408671 q^{92}-655018 q^{91}-629387 q^{90}-225618 q^{89}+455543 q^{88}+1111993 q^{87}+1386952 q^{86}+974072 q^{85}-120295 q^{84}-1490425 q^{83}-2486737 q^{82}-2411772 q^{81}-992130 q^{80}+1359420 q^{79}+3658056 q^{78}+4600767 q^{77}+3312410 q^{76}-69631 q^{75}-4295721 q^{74}-7250018 q^{73}-7038624 q^{72}-3064728 q^{71}+3458772 q^{70}+9564329 q^{69}+11904596 q^{68}+8481876 q^{67}-127110 q^{66}-10320484 q^{65}-16996224 q^{64}-16013247 q^{63}-6427765 q^{62}+8135986 q^{61}+20830152 q^{60}+24710309 q^{59}+16239649 q^{58}-1944871 q^{57}-21701623 q^{56}-32924207 q^{55}-28421781 q^{54}-8547351 q^{53}+18203416 q^{52}+38688427 q^{51}+41268842 q^{50}+22629174 q^{49}-9727922 q^{48}-40298558 q^{47}-52668841 q^{46}-38661673 q^{45}-3265789 q^{44}+36818791 q^{43}+60675114 q^{42}+54508287 q^{41}+19350236 q^{40}-28369637 q^{39}-64052090 q^{38}-68100082 q^{37}-36505322 q^{36}+16059829 q^{35}+62539208 q^{34}+77949831 q^{33}+52680978 q^{32}-1622508 q^{31}-56831245 q^{30}-83452910 q^{29}-66273208 q^{28}-13063190 q^{27}+48269134 q^{26}+84870725 q^{25}+76415546 q^{24}+26435776 q^{23}-38413430 q^{22}-83109137 q^{21}-83022604 q^{20}-37518962 q^{19}+28658568 q^{18}+79366271 q^{17}+86611850 q^{16}+45992202 q^{15}-19960604 q^{14}-74793555 q^{13}-88046411 q^{12}-52101041 q^{11}+12750874 q^{10}+70271955 q^9+88260356 q^8+56446409 q^7-6976270 q^6-66276594 q^5-88039714 q^4-59795350 q^3+2203806 q^2+62882041 q+87910159+62882041 q^{-1} +2203806 q^{-2} -59795350 q^{-3} -88039714 q^{-4} -66276594 q^{-5} -6976270 q^{-6} +56446409 q^{-7} +88260356 q^{-8} +70271955 q^{-9} +12750874 q^{-10} -52101041 q^{-11} -88046411 q^{-12} -74793555 q^{-13} -19960604 q^{-14} +45992202 q^{-15} +86611850 q^{-16} +79366271 q^{-17} +28658568 q^{-18} -37518962 q^{-19} -83022604 q^{-20} -83109137 q^{-21} -38413430 q^{-22} +26435776 q^{-23} +76415546 q^{-24} +84870725 q^{-25} +48269134 q^{-26} -13063190 q^{-27} -66273208 q^{-28} -83452910 q^{-29} -56831245 q^{-30} -1622508 q^{-31} +52680978 q^{-32} +77949831 q^{-33} +62539208 q^{-34} +16059829 q^{-35} -36505322 q^{-36} -68100082 q^{-37} -64052090 q^{-38} -28369637 q^{-39} +19350236 q^{-40} +54508287 q^{-41} +60675114 q^{-42} +36818791 q^{-43} -3265789 q^{-44} -38661673 q^{-45} -52668841 q^{-46} -40298558 q^{-47} -9727922 q^{-48} +22629174 q^{-49} +41268842 q^{-50} +38688427 q^{-51} +18203416 q^{-52} -8547351 q^{-53} -28421781 q^{-54} -32924207 q^{-55} -21701623 q^{-56} -1944871 q^{-57} +16239649 q^{-58} +24710309 q^{-59} +20830152 q^{-60} +8135986 q^{-61} -6427765 q^{-62} -16013247 q^{-63} -16996224 q^{-64} -10320484 q^{-65} -127110 q^{-66} +8481876 q^{-67} +11904596 q^{-68} +9564329 q^{-69} +3458772 q^{-70} -3064728 q^{-71} -7038624 q^{-72} -7250018 q^{-73} -4295721 q^{-74} -69631 q^{-75} +3312410 q^{-76} +4600767 q^{-77} +3658056 q^{-78} +1359420 q^{-79} -992130 q^{-80} -2411772 q^{-81} -2486737 q^{-82} -1490425 q^{-83} -120295 q^{-84} +974072 q^{-85} +1386952 q^{-86} +1111993 q^{-87} +455543 q^{-88} -225618 q^{-89} -629387 q^{-90} -655018 q^{-91} -408671 q^{-92} -58409 q^{-93} +212678 q^{-94} +312390 q^{-95} +258889 q^{-96} +110481 q^{-97} -37709 q^{-98} -120485 q^{-99} -129393 q^{-100} -78969 q^{-101} -13156 q^{-102} +33340 q^{-103} +52772 q^{-104} +41839 q^{-105} +17046 q^{-106} -4883 q^{-107} -18195 q^{-108} -17220 q^{-109} -9312 q^{-110} -1660 q^{-111} +4577 q^{-112} +5950 q^{-113} +4250 q^{-114} +1720 q^{-115} -1295 q^{-116} -1890 q^{-117} -1137 q^{-118} -646 q^{-119} +136 q^{-120} +319 q^{-121} +395 q^{-122} +385 q^{-123} -115 q^{-124} -195 q^{-125} -42 q^{-126} -11 q^{-127} +36 q^{-128} -25 q^{-129} -9 q^{-130} +65 q^{-131} -15 q^{-132} -30 q^{-133} +6 q^{-135} +16 q^{-136} -10 q^{-137} -5 q^{-138} +5 q^{-139} - q^{-140} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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