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]=

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

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 $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 8 16's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-4t^{2}+8t-9+8t^{-1}-4t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, -2 }
Jones polynomial	$-q^{2}+3q-4+6q^{-1}-6q^{-2}+6q^{-3}-5q^{-4}+3q^{-5}-q^{-6}$
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)	$z^{3}a^{7}+3z^{4}a^{6}-z^{2}a^{6}+5z^{5}a^{5}-5z^{3}a^{5}+2za^{5}+5z^{6}a^{4}-7z^{4}a^{4}+4z^{2}a^{4}-a^{4}+2z^{7}a^{3}+3z^{5}a^{3}-10z^{3}a^{3}+4za^{3}+8z^{6}a^{2}-18z^{4}a^{2}+10z^{2}a^{2}-2a^{2}+2z^{7}a-z^{5}a-6z^{3}a+3za+3z^{6}-8z^{4}+5z^{2}+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$
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	Failed to parse (SVG (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} }

Further Quantum Invariants

Further quantum knot invariants for 8_16.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{13}+2 q^{11}-2 q^9+q^7+2 q- q^{-1} +2 q^{-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	Failed to parse (SVG (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^{69}+2 q^{67}-3 q^{63}+q^{61}+6 q^{59}-16 q^{55}+26 q^{51}+6 q^{49}-33 q^{47}-19 q^{45}+35 q^{43}+28 q^{41}-30 q^{39}-32 q^{37}+19 q^{35}+34 q^{33}-5 q^{31}-28 q^{29}-8 q^{27}+20 q^{25}+15 q^{23}-12 q^{21}-24 q^{19}+5 q^{17}+29 q^{15}+2 q^{13}-32 q^{11}-6 q^9+34 q^7+16 q^5-32 q^3-25 q+28 q^{-1} +31 q^{-3} -16 q^{-5} -35 q^{-7} +5 q^{-9} +33 q^{-11} +7 q^{-13} -24 q^{-15} -13 q^{-17} +12 q^{-19} +15 q^{-21} -4 q^{-23} -9 q^{-25} -2 q^{-27} +3 q^{-29} +2 q^{-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	Failed to parse (SVG (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^{228}-2 q^{226}+3 q^{222}-3 q^{220}-2 q^{218}+10 q^{214}+q^{212}-8 q^{210}-19 q^{206}-10 q^{204}+21 q^{202}+57 q^{200}+33 q^{198}-31 q^{196}-73 q^{194}-135 q^{192}-75 q^{190}+106 q^{188}+304 q^{186}+281 q^{184}+25 q^{182}-308 q^{180}-659 q^{178}-562 q^{176}+42 q^{174}+883 q^{172}+1266 q^{170}+813 q^{168}-284 q^{166}-1689 q^{164}-2162 q^{162}-1163 q^{160}+975 q^{158}+2814 q^{156}+3056 q^{154}+1343 q^{152}-1837 q^{150}-4170 q^{148}-3998 q^{146}-992 q^{144}+2915 q^{142}+5291 q^{140}+4527 q^{138}+469 q^{136}-4039 q^{134}-6182 q^{132}-4287 q^{130}+289 q^{128}+4762 q^{126}+6374 q^{124}+3726 q^{122}-1108 q^{120}-5157 q^{118}-5719 q^{116}-2863 q^{114}+1612 q^{112}+4916 q^{110}+4828 q^{108}+1928 q^{106}-1918 q^{104}-4121 q^{102}-3794 q^{100}-1232 q^{98}+1821 q^{96}+3366 q^{94}+2838 q^{92}+684 q^{90}-1514 q^{88}-2714 q^{86}-2150 q^{84}-380 q^{82}+1434 q^{80}+2265 q^{78}+1615 q^{76}+57 q^{74}-1565 q^{72}-2025 q^{70}-1146 q^{68}+573 q^{66}+1878 q^{64}+1816 q^{62}+420 q^{60}-1428 q^{58}-2213 q^{56}-1420 q^{54}+671 q^{52}+2378 q^{50}+2383 q^{48}+523 q^{46}-1971 q^{44}-3147 q^{42}-2159 q^{40}+730 q^{38}+3282 q^{36}+3646 q^{34}+1333 q^{32}-2119 q^{30}-4283 q^{28}-3695 q^{26}-273 q^{24}+3444 q^{22}+5009 q^{20}+3166 q^{18}-818 q^{16}-4378 q^{14}-5287 q^{12}-2546 q^{10}+1786 q^8+5046 q^6+5012 q^4+1936 q^2-2346-5245 q^{-2} -4675 q^{-4} -1385 q^{-6} +2685 q^{-8} +4881 q^{-10} +4252 q^{-12} +1110 q^{-14} -2616 q^{-16} -4443 q^{-18} -3709 q^{-20} -876 q^{-22} +2111 q^{-24} +3844 q^{-26} +3229 q^{-28} +835 q^{-30} -1658 q^{-32} -3060 q^{-34} -2633 q^{-36} -978 q^{-38} +1160 q^{-40} +2332 q^{-42} +2106 q^{-44} +900 q^{-46} -623 q^{-48} -1572 q^{-50} -1682 q^{-52} -795 q^{-54} +263 q^{-56} +992 q^{-58} +1131 q^{-60} +696 q^{-62} +16 q^{-64} -602 q^{-66} -702 q^{-68} -501 q^{-70} -106 q^{-72} +244 q^{-74} +412 q^{-76} +347 q^{-78} +92 q^{-80} -78 q^{-82} -190 q^{-84} -179 q^{-86} -97 q^{-88} +17 q^{-90} +83 q^{-92} +70 q^{-94} +51 q^{-96} +7 q^{-98} -20 q^{-100} -34 q^{-102} -16 q^{-104} + q^{-108} +8 q^{-110} +5 q^{-112} +2 q^{-114} -3 q^{-116} -2 q^{-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	$-q^{23}+q^{21}-2q^{19}+q^{17}-q^{15}+q^{13}+q^{9}+q^{7}+2q^{3}+2q^{-1}-q^{-3}+q^{-5}-q^{-7}$
1,0,1	$q^{68}-4q^{66}+6q^{64}-2q^{62}-7q^{60}+19q^{58}-24q^{56}+9q^{54}+19q^{52}-46q^{50}+54q^{48}-25q^{46}-26q^{44}+76q^{42}-97q^{40}+71q^{38}-10q^{36}-66q^{34}+109q^{32}-114q^{30}+64q^{28}-2q^{26}-41q^{24}+53q^{22}-21q^{20}+3q^{18}+3q^{16}+30q^{14}-75q^{12}+94q^{10}-80q^{8}+18q^{6}+56q^{4}-104q^{2}+124-82q^{-2}+36q^{-4}+28q^{-6}-59q^{-8}+60q^{-10}-43q^{-12}+10q^{-14}+7q^{-16}-14q^{-18}+10q^{-20}-4q^{-22}+q^{-24}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{52}-q^{50}-2q^{48}+3q^{46}+q^{44}-5q^{42}+2q^{40}+6q^{38}-q^{36}-4q^{34}+3q^{32}-9q^{28}-5q^{26}+3q^{24}-4q^{22}-5q^{20}+10q^{18}+4q^{16}-2q^{14}+7q^{12}+9q^{10}-2q^{8}-q^{6}+5q^{4}+2q^{2}-5-q^{-2}+3q^{-4}-2q^{-6}-2q^{-8}+2q^{-10}-q^{-14}+q^{-16}$
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	$q^{68}-2q^{64}-2q^{62}+q^{60}+5q^{58}+2q^{56}-5q^{54}-5q^{52}+2q^{50}+8q^{48}+q^{46}-8q^{44}-4q^{42}+5q^{40}+5q^{38}-4q^{36}-6q^{34}+q^{32}+6q^{30}-6q^{26}-q^{24}+5q^{22}+3q^{20}-3q^{18}-3q^{16}+4q^{14}+5q^{12}-2q^{10}-6q^{8}+2q^{6}+8q^{4}+4q^{2}-6-6q^{-2}+4q^{-4}+8q^{-6}-q^{-8}-6q^{-10}-3q^{-12}+4q^{-14}+3q^{-16}-2q^{-18}-2q^{-20}+q^{-24}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{58}-2q^{56}+q^{54}-2q^{52}+5q^{50}-5q^{48}+4q^{46}-5q^{44}+8q^{42}-5q^{40}+3q^{38}-4q^{36}+q^{32}-5q^{30}+2q^{28}-9q^{26}+9q^{24}-9q^{22}+11q^{20}-9q^{18}+12q^{16}-5q^{14}+10q^{12}-4q^{10}+4q^{8}+q^{6}+2q^{2}-4+6q^{-2}-6q^{-4}+5q^{-6}-6q^{-8}+5q^{-10}-4q^{-12}+3q^{-14}-2q^{-16}+q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

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

.

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]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

In[6]:=

Alexander[K, 2][t]

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

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

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]=

$z^{3}a^{7}+3z^{4}a^{6}-z^{2}a^{6}+5z^{5}a^{5}-5z^{3}a^{5}+2za^{5}+5z^{6}a^{4}-7z^{4}a^{4}+4z^{2}a^{4}-a^{4}+2z^{7}a^{3}+3z^{5}a^{3}-10z^{3}a^{3}+4za^{3}+8z^{6}a^{2}-18z^{4}a^{2}+10z^{2}a^{2}-2a^{2}+2z^{7}a-z^{5}a-6z^{3}a+3za+3z^{6}-8z^{4}+5z^{2}+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$

"Similar" Knots (within the Atlas)

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

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {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} } , $-q^{2}+3q-4+6q^{-1}-6q^{-2}+6q^{-3}-5q^{-4}+3q^{-5}-q^{-6}$ }

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 $4$ $-8$ $8$ 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}} $-{\frac {38}{3}}$ $-32$ $-{\frac {80}{3}}$ ${\frac {64}{3}}$ 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} ${\frac {32}{3}}$ $32$ 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}} $-{\frac {152}{3}}$ ${\frac {1951}{30}}$ 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}} $-{\frac {31}{18}}$ $-{\frac {449}{30}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

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

8_15 8_17