8 16

8_15

8_17

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 16's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 16 at Knotilus!

Square depiction.

Knot presentations

Planar diagram presentation	X6271 X14,6,15,5 X16,11,1,12 X12,7,13,8 X8394 X4,9,5,10 X10,15,11,16 X2,14,3,13
Gauss code	1, -8, 5, -6, 2, -1, 4, -5, 6, -7, 3, -4, 8, -2, 7, -3
Dowker-Thistlethwaite code	6 8 14 12 4 16 2 10
Conway Notation	[.2.20]

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

[edit Notes on presentations of 8 16]

Knot 8_16.

A graph, knot 8_16.

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["8 16"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X14,6,15,5 X16,11,1,12 X12,7,13,8 X8394 X4,9,5,10 X10,15,11,16 X2,14,3,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 8 14 12 4 16 2 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[.2.20]

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,-1,2,-1,-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, 8, 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, 6}, {8, 11}, {9, 7}, {4, 8}, {6, 9}, {5, 3}, {10, 4}, {1, 5}, {11, 2}, {7, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-2] Hyperbolic Volume 10.579 A-Polynomial See Data:8 16/A-polynomial

[edit Notes for 8 16'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 1} 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 1} 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 3} Rasmussen s-Invariant 2

[edit Notes for 8 16'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^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }
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^6+2 z^4+z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 35, -2 }
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^2+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }
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 a^2 z^6-a^4 z^4+4 a^2 z^4-z^4-2 a^4 z^2+5 a^2 z^2-2 z^2-a^4+2 a^2}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^3 a^7+3 z^4 a^6-z^2 a^6+5 z^5 a^5-5 z^3 a^5+2 z a^5+5 z^6 a^4-7 z^4 a^4+4 z^2 a^4-a^4+2 z^7 a^3+3 z^5 a^3-10 z^3 a^3+4 z a^3+8 z^6 a^2-18 z^4 a^2+10 z^2 a^2-2 a^2+2 z^7 a-z^5 a-6 z^3 a+3 z a+3 z^6-8 z^4+5 z^2+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} }
The A2 invariant	$-q^{18}+q^{16}-q^{14}+q^{10}-q^{8}+2q^{6}-q^{4}+2q^{2}+1+q^{-4}-q^{-6}$
The G2 invariant	$q^{100}-2q^{98}+3q^{96}-4q^{94}+2q^{92}-q^{90}-2q^{88}+9q^{86}-12q^{84}+15q^{82}-14q^{80}+7q^{78}+2q^{76}-16q^{74}+28q^{72}-31q^{70}+24q^{68}-10q^{66}-11q^{64}+26q^{62}-30q^{60}+21q^{58}-5q^{56}-15q^{54}+23q^{52}-19q^{50}+2q^{48}+22q^{46}-36q^{44}+36q^{42}-20q^{40}-4q^{38}+30q^{36}-45q^{34}+46q^{32}-33q^{30}+12q^{28}+14q^{26}-32q^{24}+39q^{22}-30q^{20}+14q^{18}+5q^{16}-20q^{14}+24q^{12}-14q^{10}-q^{8}+21q^{6}-29q^{4}+26q^{2}-6-17q^{-2}+35q^{-4}-37q^{-6}+28q^{-8}-10q^{-10}-12q^{-12}+24q^{-14}-25q^{-16}+20q^{-18}-9q^{-20}-2q^{-22}+6q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 8_16.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+2q^{11}-2q^{9}+q^{7}+2q-q^{-1}+2q^{-3}-q^{-5}$
2	$q^{36}-2q^{34}+5q^{30}-7q^{28}-2q^{26}+11q^{24}-6q^{22}-5q^{20}+8q^{18}-q^{16}-5q^{14}+q^{12}+5q^{10}-2q^{8}-5q^{6}+7q^{4}+2q^{2}-9+6q^{-2}+6q^{-4}-8q^{-6}+q^{-8}+6q^{-10}-3q^{-12}-2q^{-14}+q^{-16}$
3	$-q^{69}+2q^{67}-3q^{63}+q^{61}+6q^{59}-16q^{55}+26q^{51}+6q^{49}-33q^{47}-19q^{45}+35q^{43}+28q^{41}-30q^{39}-32q^{37}+19q^{35}+34q^{33}-5q^{31}-28q^{29}-8q^{27}+20q^{25}+15q^{23}-12q^{21}-24q^{19}+5q^{17}+29q^{15}+2q^{13}-32q^{11}-6q^{9}+34q^{7}+16q^{5}-32q^{3}-25q+28q^{-1}+31q^{-3}-16q^{-5}-35q^{-7}+5q^{-9}+33q^{-11}+7q^{-13}-24q^{-15}-13q^{-17}+12q^{-19}+15q^{-21}-4q^{-23}-9q^{-25}-2q^{-27}+3q^{-29}+2q^{-31}-q^{-33}$
4	$q^{112}-2q^{110}+3q^{106}-3q^{104}-4q^{100}+8q^{98}+13q^{96}-16q^{94}-17q^{92}-17q^{90}+36q^{88}+65q^{86}-17q^{84}-75q^{82}-91q^{80}+44q^{78}+160q^{76}+60q^{74}-91q^{72}-198q^{70}-38q^{68}+185q^{66}+163q^{64}-9q^{62}-209q^{60}-134q^{58}+86q^{56}+165q^{54}+89q^{52}-103q^{50}-139q^{48}-29q^{46}+81q^{44}+115q^{42}+12q^{40}-83q^{38}-95q^{36}+3q^{34}+104q^{32}+79q^{30}-45q^{28}-130q^{26}-38q^{24}+99q^{22}+130q^{20}-20q^{18}-163q^{16}-79q^{14}+87q^{12}+180q^{10}+32q^{8}-165q^{6}-139q^{4}+17q^{2}+193+119q^{-2}-89q^{-4}-164q^{-6}-94q^{-8}+114q^{-10}+158q^{-12}+35q^{-14}-91q^{-16}-144q^{-18}-9q^{-20}+90q^{-22}+89q^{-24}+18q^{-26}-80q^{-28}-56q^{-30}-4q^{-32}+41q^{-34}+45q^{-36}-6q^{-38}-20q^{-40}-20q^{-42}-3q^{-44}+12q^{-46}+5q^{-48}+2q^{-50}-3q^{-52}-2q^{-54}+q^{-56}$
5	$-q^{165}+2q^{163}-3q^{159}+3q^{157}+2q^{155}-2q^{153}-4q^{151}-5q^{149}+16q^{145}+23q^{143}-5q^{141}-47q^{139}-56q^{137}+q^{135}+88q^{133}+133q^{131}+54q^{129}-148q^{127}-270q^{125}-158q^{123}+161q^{121}+436q^{119}+366q^{117}-87q^{115}-593q^{113}-639q^{111}-92q^{109}+642q^{107}+901q^{105}+381q^{103}-545q^{101}-1081q^{99}-686q^{97}+319q^{95}+1079q^{93}+926q^{91}-9q^{89}-909q^{87}-1027q^{85}-283q^{83}+632q^{81}+953q^{79}+492q^{77}-306q^{75}-772q^{73}-588q^{71}+26q^{69}+532q^{67}+573q^{65}+188q^{63}-301q^{61}-515q^{59}-314q^{57}+121q^{55}+446q^{53}+394q^{51}-6q^{49}-406q^{47}-447q^{45}-64q^{43}+409q^{41}+512q^{39}+102q^{37}-444q^{35}-591q^{33}-159q^{31}+485q^{29}+704q^{27}+242q^{25}-502q^{23}-819q^{21}-375q^{19}+450q^{17}+918q^{15}+557q^{13}-325q^{11}-945q^{9}-746q^{7}+108q^{5}+868q^{3}+905q+178q^{-1}-685q^{-3}-956q^{-5}-454q^{-7}+383q^{-9}+878q^{-11}+668q^{-13}-46q^{-15}-663q^{-17}-733q^{-19}-257q^{-21}+361q^{-23}+642q^{-25}+439q^{-27}-53q^{-29}-440q^{-31}-465q^{-33}-162q^{-35}+194q^{-37}+354q^{-39}+259q^{-41}+4q^{-43}-201q^{-45}-224q^{-47}-99q^{-49}+52q^{-51}+134q^{-53}+115q^{-55}+21q^{-57}-52q^{-59}-68q^{-61}-39q^{-63}+26q^{-67}+28q^{-69}+8q^{-71}-5q^{-73}-8q^{-75}-5q^{-77}-2q^{-79}+3q^{-81}+2q^{-83}-q^{-85}$
6	$q^{228}-2q^{226}+3q^{222}-3q^{220}-2q^{218}+10q^{214}+q^{212}-8q^{210}-19q^{206}-10q^{204}+21q^{202}+57q^{200}+33q^{198}-31q^{196}-73q^{194}-135q^{192}-75q^{190}+106q^{188}+304q^{186}+281q^{184}+25q^{182}-308q^{180}-659q^{178}-562q^{176}+42q^{174}+883q^{172}+1266q^{170}+813q^{168}-284q^{166}-1689q^{164}-2162q^{162}-1163q^{160}+975q^{158}+2814q^{156}+3056q^{154}+1343q^{152}-1837q^{150}-4170q^{148}-3998q^{146}-992q^{144}+2915q^{142}+5291q^{140}+4527q^{138}+469q^{136}-4039q^{134}-6182q^{132}-4287q^{130}+289q^{128}+4762q^{126}+6374q^{124}+3726q^{122}-1108q^{120}-5157q^{118}-5719q^{116}-2863q^{114}+1612q^{112}+4916q^{110}+4828q^{108}+1928q^{106}-1918q^{104}-4121q^{102}-3794q^{100}-1232q^{98}+1821q^{96}+3366q^{94}+2838q^{92}+684q^{90}-1514q^{88}-2714q^{86}-2150q^{84}-380q^{82}+1434q^{80}+2265q^{78}+1615q^{76}+57q^{74}-1565q^{72}-2025q^{70}-1146q^{68}+573q^{66}+1878q^{64}+1816q^{62}+420q^{60}-1428q^{58}-2213q^{56}-1420q^{54}+671q^{52}+2378q^{50}+2383q^{48}+523q^{46}-1971q^{44}-3147q^{42}-2159q^{40}+730q^{38}+3282q^{36}+3646q^{34}+1333q^{32}-2119q^{30}-4283q^{28}-3695q^{26}-273q^{24}+3444q^{22}+5009q^{20}+3166q^{18}-818q^{16}-4378q^{14}-5287q^{12}-2546q^{10}+1786q^{8}+5046q^{6}+5012q^{4}+1936q^{2}-2346-5245q^{-2}-4675q^{-4}-1385q^{-6}+2685q^{-8}+4881q^{-10}+4252q^{-12}+1110q^{-14}-2616q^{-16}-4443q^{-18}-3709q^{-20}-876q^{-22}+2111q^{-24}+3844q^{-26}+3229q^{-28}+835q^{-30}-1658q^{-32}-3060q^{-34}-2633q^{-36}-978q^{-38}+1160q^{-40}+2332q^{-42}+2106q^{-44}+900q^{-46}-623q^{-48}-1572q^{-50}-1682q^{-52}-795q^{-54}+263q^{-56}+992q^{-58}+1131q^{-60}+696q^{-62}+16q^{-64}-602q^{-66}-702q^{-68}-501q^{-70}-106q^{-72}+244q^{-74}+412q^{-76}+347q^{-78}+92q^{-80}-78q^{-82}-190q^{-84}-179q^{-86}-97q^{-88}+17q^{-90}+83q^{-92}+70q^{-94}+51q^{-96}+7q^{-98}-20q^{-100}-34q^{-102}-16q^{-104}+q^{-108}+8q^{-110}+5q^{-112}+2q^{-114}-3q^{-116}-2q^{-118}+q^{-120}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{18}+q^{16}-q^{14}+q^{10}-q^{8}+2q^{6}-q^{4}+2q^{2}+1+q^{-4}-q^{-6}$
1,1	$q^{52}-4q^{50}+8q^{48}-12q^{46}+22q^{44}-36q^{42}+46q^{40}-62q^{38}+82q^{36}-92q^{34}+96q^{32}-90q^{30}+67q^{28}-34q^{26}-18q^{24}+70q^{22}-120q^{20}+166q^{18}-188q^{16}+200q^{14}-190q^{12}+164q^{10}-124q^{8}+70q^{6}-20q^{4}-28q^{2}+76-100q^{-2}+115q^{-4}-106q^{-6}+94q^{-8}-70q^{-10}+44q^{-12}-28q^{-14}+12q^{-16}-4q^{-18}+q^{-20}$
2,0	$q^{46}-q^{44}+2q^{40}-2q^{38}-2q^{36}+3q^{32}-2q^{30}-4q^{28}+4q^{26}+2q^{24}-3q^{22}+4q^{18}-q^{16}-q^{14}+2q^{12}-3q^{8}-q^{6}+4q^{4}-2q^{2}+6q^{-2}+3q^{-4}-2q^{-6}+2q^{-10}-3q^{-14}-q^{-16}+q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{42}-2q^{40}-q^{38}+5q^{36}-3q^{34}-3q^{32}+8q^{30}-4q^{28}-5q^{26}+6q^{24}-4q^{22}-4q^{20}+2q^{18}+q^{16}-q^{12}+5q^{10}+4q^{8}-5q^{6}+4q^{4}+6q^{2}-6+2q^{-2}+4q^{-4}-6q^{-6}+2q^{-8}+q^{-10}-2q^{-12}+q^{-14}$
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^{23}+q^{21}-2 q^{19}+q^{17}-q^{15}+q^{13}+q^9+q^7+2 q^3+2 q^{-1} - q^{-3} + q^{-5} - q^{-7} }
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^{68}-4 q^{66}+6 q^{64}-2 q^{62}-7 q^{60}+19 q^{58}-24 q^{56}+9 q^{54}+19 q^{52}-46 q^{50}+54 q^{48}-25 q^{46}-26 q^{44}+76 q^{42}-97 q^{40}+71 q^{38}-10 q^{36}-66 q^{34}+109 q^{32}-114 q^{30}+64 q^{28}-2 q^{26}-41 q^{24}+53 q^{22}-21 q^{20}+3 q^{18}+3 q^{16}+30 q^{14}-75 q^{12}+94 q^{10}-80 q^8+18 q^6+56 q^4-104 q^2+124-82 q^{-2} +36 q^{-4} +28 q^{-6} -59 q^{-8} +60 q^{-10} -43 q^{-12} +10 q^{-14} +7 q^{-16} -14 q^{-18} +10 q^{-20} -4 q^{-22} + q^{-24} }

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

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^{42}+2 q^{40}-3 q^{38}+5 q^{36}-7 q^{34}+7 q^{32}-8 q^{30}+6 q^{28}-3 q^{26}+4 q^{22}-6 q^{20}+10 q^{18}-13 q^{16}+14 q^{14}-13 q^{12}+11 q^{10}-8 q^8+5 q^6-2 q^2+6-6 q^{-2} +8 q^{-4} -6 q^{-6} +6 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-14} }
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^{68}-2 q^{64}-2 q^{62}+q^{60}+5 q^{58}+2 q^{56}-5 q^{54}-5 q^{52}+2 q^{50}+8 q^{48}+q^{46}-8 q^{44}-4 q^{42}+5 q^{40}+5 q^{38}-4 q^{36}-6 q^{34}+q^{32}+6 q^{30}-6 q^{26}-q^{24}+5 q^{22}+3 q^{20}-3 q^{18}-3 q^{16}+4 q^{14}+5 q^{12}-2 q^{10}-6 q^8+2 q^6+8 q^4+4 q^2-6-6 q^{-2} +4 q^{-4} +8 q^{-6} - q^{-8} -6 q^{-10} -3 q^{-12} +4 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-24} }

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^{58}-2 q^{56}+q^{54}-2 q^{52}+5 q^{50}-5 q^{48}+4 q^{46}-5 q^{44}+8 q^{42}-5 q^{40}+3 q^{38}-4 q^{36}+q^{32}-5 q^{30}+2 q^{28}-9 q^{26}+9 q^{24}-9 q^{22}+11 q^{20}-9 q^{18}+12 q^{16}-5 q^{14}+10 q^{12}-4 q^{10}+4 q^8+q^6+2 q^2-4+6 q^{-2} -6 q^{-4} +5 q^{-6} -6 q^{-8} +5 q^{-10} -4 q^{-12} +3 q^{-14} -2 q^{-16} + q^{-18} }

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^{100}-2 q^{98}+3 q^{96}-4 q^{94}+2 q^{92}-q^{90}-2 q^{88}+9 q^{86}-12 q^{84}+15 q^{82}-14 q^{80}+7 q^{78}+2 q^{76}-16 q^{74}+28 q^{72}-31 q^{70}+24 q^{68}-10 q^{66}-11 q^{64}+26 q^{62}-30 q^{60}+21 q^{58}-5 q^{56}-15 q^{54}+23 q^{52}-19 q^{50}+2 q^{48}+22 q^{46}-36 q^{44}+36 q^{42}-20 q^{40}-4 q^{38}+30 q^{36}-45 q^{34}+46 q^{32}-33 q^{30}+12 q^{28}+14 q^{26}-32 q^{24}+39 q^{22}-30 q^{20}+14 q^{18}+5 q^{16}-20 q^{14}+24 q^{12}-14 q^{10}-q^8+21 q^6-29 q^4+26 q^2-6-17 q^{-2} +35 q^{-4} -37 q^{-6} +28 q^{-8} -10 q^{-10} -12 q^{-12} +24 q^{-14} -25 q^{-16} +20 q^{-18} -9 q^{-20} -2 q^{-22} +6 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32} }

.

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["8 16"];

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^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }

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^6+2 z^4+z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 35, -2 }

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

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 a^2 z^6-a^4 z^4+4 a^2 z^4-z^4-2 a^4 z^2+5 a^2 z^2-2 z^2-a^4+2 a^2}

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^3 a^7+3 z^4 a^6-z^2 a^6+5 z^5 a^5-5 z^3 a^5+2 z a^5+5 z^6 a^4-7 z^4 a^4+4 z^2 a^4-a^4+2 z^7 a^3+3 z^5 a^3-10 z^3 a^3+4 z a^3+8 z^6 a^2-18 z^4 a^2+10 z^2 a^2-2 a^2+2 z^7 a-z^5 a-6 z^3 a+3 z a+3 z^6-8 z^4+5 z^2+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} }

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_156, K11n15, K11n56, K11n58,}

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}} ): {10_156,}

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["8 16"];

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^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-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^2+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} } }

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]=

{10_156, K11n15, K11n56, K11n58,}

In[6]:=

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

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

Out[6]=

{10_156,}

Vassiliev invariants

V2 and V3:

(1, -1)

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 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 -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 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 \frac{14}{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{38}{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 -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{80}{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{64}{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 -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 \frac{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 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{56}{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{152}{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{1951}{30}} 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{114}{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 -\frac{298}{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}{18}} 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{449}{30}}

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 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=} -2 is the signature of 8 16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 χ 5                 1 -1 3               2   2 1             2 1   -1 -1           4 2     2 -3         3 3       0 -5       3 3         0 -7     2 3           1 -9   1 3             -2 -11   2               2 -13 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=-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 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}^{2}\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}^{3}\oplus{\mathbb Z}_2^{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}^{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 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}^{3}\oplus{\mathbb Z}_2^{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}^{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 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}^{3}\oplus{\mathbb Z}_2^{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}^{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 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}^{3}\oplus{\mathbb Z}_2^{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}} 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} ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ 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}^{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 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}\oplus{\mathbb Z}_2^{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}^{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 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}_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^7-3 q^6-q^5+10 q^4-8 q^3-10 q^2+24 q-8-25 q^{-1} +35 q^{-2} -3 q^{-3} -37 q^{-4} +38 q^{-5} +4 q^{-6} -41 q^{-7} +32 q^{-8} +8 q^{-9} -32 q^{-10} +19 q^{-11} +7 q^{-12} -15 q^{-13} +6 q^{-14} +2 q^{-15} -3 q^{-16} + q^{-17} }
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^{15}+3 q^{14}+q^{13}-5 q^{12}-8 q^{11}+8 q^{10}+20 q^9-8 q^8-33 q^7-3 q^6+51 q^5+18 q^4-61 q^3-43 q^2+70 q+65-64 q^{-1} -96 q^{-2} +63 q^{-3} +113 q^{-4} -46 q^{-5} -136 q^{-6} +37 q^{-7} +147 q^{-8} -19 q^{-9} -160 q^{-10} +8 q^{-11} +159 q^{-12} +8 q^{-13} -155 q^{-14} -20 q^{-15} +139 q^{-16} +31 q^{-17} -116 q^{-18} -35 q^{-19} +88 q^{-20} +33 q^{-21} -58 q^{-22} -28 q^{-23} +34 q^{-24} +19 q^{-25} -19 q^{-26} -8 q^{-27} +8 q^{-28} +3 q^{-29} -3 q^{-30} -2 q^{-31} +3 q^{-32} - q^{-33} }
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^{26}-3 q^{25}-q^{24}+5 q^{23}+3 q^{22}+8 q^{21}-18 q^{20}-18 q^{19}+5 q^{18}+17 q^{17}+59 q^{16}-22 q^{15}-63 q^{14}-47 q^{13}-7 q^{12}+157 q^{11}+49 q^{10}-62 q^9-146 q^8-142 q^7+210 q^6+175 q^5+61 q^4-190 q^3-350 q^2+140 q+250+269 q^{-1} -116 q^{-2} -526 q^{-3} -16 q^{-4} +224 q^{-5} +466 q^{-6} +32 q^{-7} -619 q^{-8} -182 q^{-9} +140 q^{-10} +609 q^{-11} +182 q^{-12} -650 q^{-13} -319 q^{-14} +48 q^{-15} +694 q^{-16} +306 q^{-17} -625 q^{-18} -420 q^{-19} -50 q^{-20} +706 q^{-21} +401 q^{-22} -522 q^{-23} -454 q^{-24} -160 q^{-25} +596 q^{-26} +437 q^{-27} -330 q^{-28} -378 q^{-29} -239 q^{-30} +376 q^{-31} +362 q^{-32} -130 q^{-33} -206 q^{-34} -217 q^{-35} +153 q^{-36} +202 q^{-37} -23 q^{-38} -55 q^{-39} -117 q^{-40} +37 q^{-41} +67 q^{-42} -7 q^{-43} +3 q^{-44} -35 q^{-45} +8 q^{-46} +14 q^{-47} -7 q^{-48} +4 q^{-49} -6 q^{-50} +3 q^{-51} +2 q^{-52} -3 q^{-53} + q^{-54} }
5	$-q^{40}+3q^{39}+q^{38}-5q^{37}-3q^{36}-3q^{35}+2q^{34}+16q^{33}+21q^{32}-7q^{31}-29q^{30}-42q^{29}-27q^{28}+32q^{27}+94q^{26}+87q^{25}-10q^{24}-124q^{23}-178q^{22}-93q^{21}+117q^{20}+292q^{19}+245q^{18}-29q^{17}-338q^{16}-449q^{15}-186q^{14}+317q^{13}+632q^{12}+463q^{11}-135q^{10}-730q^{9}-804q^{8}-159q^{7}+702q^{6}+1080q^{5}+579q^{4}-520q^{3}-1299q^{2}-996q+200+1351q^{-1}+1442q^{-2}+207q^{-3}-1336q^{-4}-1756q^{-5}-654q^{-6}+1152q^{-7}+2062q^{-8}+1089q^{-9}-975q^{-10}-2224q^{-11}-1479q^{-12}+708q^{-13}+2379q^{-14}+1833q^{-15}-513q^{-16}-2443q^{-17}-2123q^{-18}+276q^{-19}+2526q^{-20}+2379q^{-21}-103q^{-22}-2546q^{-23}-2596q^{-24}-107q^{-25}+2567q^{-26}+2779q^{-27}+297q^{-28}-2494q^{-29}-2921q^{-30}-542q^{-31}+2366q^{-32}+2993q^{-33}+786q^{-34}-2109q^{-35}-2962q^{-36}-1048q^{-37}+1752q^{-38}+2809q^{-39}+1252q^{-40}-1311q^{-41}-2501q^{-42}-1369q^{-43}+837q^{-44}+2065q^{-45}+1370q^{-46}-411q^{-47}-1566q^{-48}-1216q^{-49}+77q^{-50}+1060q^{-51}+975q^{-52}+125q^{-53}-640q^{-54}-696q^{-55}-182q^{-56}+326q^{-57}+428q^{-58}+171q^{-59}-134q^{-60}-243q^{-61}-112q^{-62}+51q^{-63}+109q^{-64}+59q^{-65}-12q^{-66}-41q^{-67}-33q^{-68}+6q^{-69}+22q^{-70}+2q^{-71}-3q^{-72}+q^{-73}-5q^{-74}-q^{-75}+6q^{-76}-3q^{-77}-2q^{-78}+3q^{-79}-q^{-80}$
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^{57}-3 q^{56}-q^{55}+5 q^{54}+3 q^{53}+3 q^{52}-7 q^{51}-19 q^{49}-19 q^{48}+19 q^{47}+30 q^{46}+47 q^{45}+12 q^{44}+13 q^{43}-85 q^{42}-133 q^{41}-63 q^{40}+19 q^{39}+159 q^{38}+182 q^{37}+268 q^{36}-20 q^{35}-301 q^{34}-413 q^{33}-376 q^{32}-42 q^{31}+282 q^{30}+886 q^{29}+660 q^{28}+134 q^{27}-552 q^{26}-1105 q^{25}-1100 q^{24}-605 q^{23}+996 q^{22}+1609 q^{21}+1657 q^{20}+654 q^{19}-879 q^{18}-2272 q^{17}-2743 q^{16}-659 q^{15}+1182 q^{14}+3059 q^{13}+3147 q^{12}+1515 q^{11}-1657 q^{10}-4476 q^9-3646 q^8-1651 q^7+2325 q^6+4974 q^5+5241 q^4+1485 q^3-3847 q^2-5842 q-5721-965 q^{-1} +4404 q^{-2} +8140 q^{-3} +5767 q^{-4} -771 q^{-5} -5808 q^{-6} -8981 q^{-7} -5297 q^{-8} +1663 q^{-9} +9049 q^{-10} +9327 q^{-11} +3213 q^{-12} -3965 q^{-13} -10546 q^{-14} -9014 q^{-15} -1759 q^{-16} +8461 q^{-17} +11491 q^{-18} +6665 q^{-19} -1652 q^{-20} -10910 q^{-21} -11566 q^{-22} -4648 q^{-23} +7473 q^{-24} +12667 q^{-25} +9159 q^{-26} +208 q^{-27} -10915 q^{-28} -13273 q^{-29} -6739 q^{-30} +6680 q^{-31} +13452 q^{-32} +11007 q^{-33} +1604 q^{-34} -10853 q^{-35} -14578 q^{-36} -8458 q^{-37} +5801 q^{-38} +13912 q^{-39} +12629 q^{-40} +3162 q^{-41} -10203 q^{-42} -15409 q^{-43} -10272 q^{-44} +4031 q^{-45} +13348 q^{-46} +13829 q^{-47} +5360 q^{-48} -8049 q^{-49} -14881 q^{-50} -11817 q^{-51} +978 q^{-52} +10786 q^{-53} +13502 q^{-54} +7563 q^{-55} -4203 q^{-56} -11981 q^{-57} -11729 q^{-58} -2326 q^{-59} +6311 q^{-60} +10646 q^{-61} +8125 q^{-62} -154 q^{-63} -7147 q^{-64} -9081 q^{-65} -3938 q^{-66} +1838 q^{-67} +6070 q^{-68} +6230 q^{-69} +1989 q^{-70} -2639 q^{-71} -5023 q^{-72} -3174 q^{-73} -538 q^{-74} +2163 q^{-75} +3224 q^{-76} +1817 q^{-77} -306 q^{-78} -1843 q^{-79} -1461 q^{-80} -780 q^{-81} +324 q^{-82} +1086 q^{-83} +818 q^{-84} +167 q^{-85} -438 q^{-86} -364 q^{-87} -327 q^{-88} -59 q^{-89} +245 q^{-90} +214 q^{-91} +70 q^{-92} -87 q^{-93} -31 q^{-94} -71 q^{-95} -36 q^{-96} +47 q^{-97} +33 q^{-98} +10 q^{-99} -25 q^{-100} +11 q^{-101} -7 q^{-102} -12 q^{-103} +11 q^{-104} +2 q^{-105} + q^{-106} -6 q^{-107} +3 q^{-108} +2 q^{-109} -3 q^{-110} + q^{-111} }
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^{77}+3 q^{76}+q^{75}-5 q^{74}-3 q^{73}-3 q^{72}+7 q^{71}+5 q^{70}+3 q^{69}+17 q^{68}+7 q^{67}-20 q^{66}-35 q^{65}-50 q^{64}-13 q^{63}+24 q^{62}+39 q^{61}+119 q^{60}+117 q^{59}+50 q^{58}-63 q^{57}-236 q^{56}-265 q^{55}-206 q^{54}-102 q^{53}+231 q^{52}+496 q^{51}+605 q^{50}+507 q^{49}-59 q^{48}-577 q^{47}-990 q^{46}-1207 q^{45}-684 q^{44}+181 q^{43}+1236 q^{42}+2115 q^{41}+1858 q^{40}+879 q^{39}-694 q^{38}-2595 q^{37}-3326 q^{36}-2899 q^{35}-954 q^{34}+2197 q^{33}+4399 q^{32}+5232 q^{31}+3816 q^{30}-72 q^{29}-4116 q^{28}-7274 q^{27}-7578 q^{26}-3711 q^{25}+1847 q^{24}+7752 q^{23}+10998 q^{22}+8840 q^{21}+2904 q^{20}-5758 q^{19}-13028 q^{18}-14181 q^{17}-9460 q^{16}+851 q^{15}+12286 q^{14}+18243 q^{13}+16969 q^{12}+6719 q^{11}-8340 q^{10}-19818 q^9-23784 q^8-15878 q^7+1107 q^6+18085 q^5+28705 q^4+25303 q^3+8356 q^2-12909 q-30600-33687 q^{-1} -19117 q^{-2} +4988 q^{-3} +29521 q^{-4} +39865 q^{-5} +29561 q^{-6} +4823 q^{-7} -25363 q^{-8} -43600 q^{-9} -39112 q^{-10} -15197 q^{-11} +19379 q^{-12} +44807 q^{-13} +46617 q^{-14} +25329 q^{-15} -11993 q^{-16} -44047 q^{-17} -52503 q^{-18} -34421 q^{-19} +4638 q^{-20} +41971 q^{-21} +56398 q^{-22} +42097 q^{-23} +2550 q^{-24} -39239 q^{-25} -59199 q^{-26} -48383 q^{-27} -8627 q^{-28} +36527 q^{-29} +60871 q^{-30} +53343 q^{-31} +13846 q^{-32} -34098 q^{-33} -62288 q^{-34} -57325 q^{-35} -17878 q^{-36} +32218 q^{-37} +63392 q^{-38} +60622 q^{-39} +21301 q^{-40} -30840 q^{-41} -64672 q^{-42} -63577 q^{-43} -24173 q^{-44} +29684 q^{-45} +65879 q^{-46} +66514 q^{-47} +27181 q^{-48} -28413 q^{-49} -67044 q^{-50} -69481 q^{-51} -30530 q^{-52} +26397 q^{-53} +67545 q^{-54} +72462 q^{-55} +34715 q^{-56} -23142 q^{-57} -66976 q^{-58} -75019 q^{-59} -39541 q^{-60} +18181 q^{-61} +64426 q^{-62} +76462 q^{-63} +44850 q^{-64} -11370 q^{-65} -59471 q^{-66} -75993 q^{-67} -49674 q^{-68} +3119 q^{-69} +51715 q^{-70} +72699 q^{-71} +53060 q^{-72} +5813 q^{-73} -41513 q^{-74} -66212 q^{-75} -53989 q^{-76} -14044 q^{-77} +29826 q^{-78} +56667 q^{-79} +51610 q^{-80} +20314 q^{-81} -17896 q^{-82} -44973 q^{-83} -46094 q^{-84} -23700 q^{-85} +7529 q^{-86} +32625 q^{-87} +37962 q^{-88} +23751 q^{-89} +298 q^{-90} -21075 q^{-91} -28709 q^{-92} -21093 q^{-93} -4955 q^{-94} +11823 q^{-95} +19723 q^{-96} +16633 q^{-97} +6638 q^{-98} -5311 q^{-99} -12093 q^{-100} -11767 q^{-101} -6321 q^{-102} +1507 q^{-103} +6674 q^{-104} +7423 q^{-105} +4806 q^{-106} +201 q^{-107} -3153 q^{-108} -4130 q^{-109} -3212 q^{-110} -698 q^{-111} +1305 q^{-112} +2093 q^{-113} +1843 q^{-114} +559 q^{-115} -456 q^{-116} -903 q^{-117} -921 q^{-118} -363 q^{-119} +107 q^{-120} +370 q^{-121} +465 q^{-122} +154 q^{-123} -63 q^{-124} -123 q^{-125} -162 q^{-126} -53 q^{-127} -15 q^{-128} +35 q^{-129} +99 q^{-130} +17 q^{-131} -25 q^{-132} -15 q^{-133} -16 q^{-134} +9 q^{-135} -8 q^{-136} -6 q^{-137} +22 q^{-138} -8 q^{-140} -2 q^{-141} - q^{-142} +6 q^{-143} -3 q^{-144} -2 q^{-145} +3 q^{-146} - q^{-147} }

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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