10 88

10_87

10_89

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 88's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 88 at Knotilus!

Square quasi-symmetrical depiction.

Knot presentations

Planar diagram presentation	X4251 X20,14,1,13 X8394 X2,9,3,10 X14,7,15,8 X18,15,19,16 X12,6,13,5 X10,18,11,17 X16,12,17,11 X6,19,7,20
Gauss code	1, -4, 3, -1, 7, -10, 5, -3, 4, -8, 9, -7, 2, -5, 6, -9, 8, -6, 10, -2
Dowker-Thistlethwaite code	4 8 12 14 2 16 20 18 10 6
Conway Notation	[.21.21]

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

[edit Notes on presentations of 10 88]

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 88"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X20,14,1,13 X8394 X2,9,3,10 X14,7,15,8 X18,15,19,16 X12,6,13,5 X10,18,11,17 X16,12,17,11 X6,19,7,20

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 12 14 2 16 20 18 10 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[.21.21]

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]=

${\textrm {BR}}(5,\{-1,2,-1,-3,2,-3,2,4,-3,4\})$

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]=

{ 5, 10, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Negative amphicheiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 15.6466 A-Polynomial See Data:10 88/A-polynomial

[edit Notes for 10 88'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 0

[edit Notes for 10 88'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+8 t^2-24 t+35-24 t^{-1} +8 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)	$\{1\}$
Determinant and Signature	{ 101, 0 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^5+4 q^4-8 q^3+13 q^2-16 q+17-16 q^{-1} +13 q^{-2} -8 q^{-3} +4 q^{-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^6+2 a^2 z^4+2 z^4 a^{-2} -2 z^4-a^4 z^2+2 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} -3 z^2+a^2+ a^{-2} -1}
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 2 a z^9+2 z^9 a^{-1} +6 a^2 z^8+6 z^8 a^{-2} +12 z^8+7 a^3 z^7+14 a z^7+14 z^7 a^{-1} +7 z^7 a^{-3} +4 a^4 z^6-2 a^2 z^6-2 z^6 a^{-2} +4 z^6 a^{-4} -12 z^6+a^5 z^5-11 a^3 z^5-32 a z^5-32 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-10 a^2 z^4-10 z^4 a^{-2} -6 z^4 a^{-4} -8 z^4-a^5 z^3+6 a^3 z^3+19 a z^3+19 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +3 a^4 z^2+7 a^2 z^2+7 z^2 a^{-2} +3 z^2 a^{-4} +8 z^2-a^3 z-4 a z-4 z a^{-1} -z a^{-3} -a^2- a^{-2} -1}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{16}+q^{14}+2 q^{12}-3 q^{10}+3 q^8-2 q^4+3 q^2-3+3 q^{-2} -2 q^{-4} +3 q^{-8} -3 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+15q^{72}-15q^{70}+4q^{68}+21q^{66}-53q^{64}+91q^{62}-111q^{60}+94q^{58}-33q^{56}-75q^{54}+204q^{52}-299q^{50}+314q^{48}-218q^{46}+18q^{44}+223q^{42}-417q^{40}+487q^{38}-379q^{36}+134q^{34}+154q^{32}-373q^{30}+418q^{28}-277q^{26}+21q^{24}+235q^{22}-365q^{20}+303q^{18}-66q^{16}-243q^{14}+490q^{12}-562q^{10}+415q^{8}-96q^{6}-286q^{4}+588q^{2}-699+588q^{-2}-286q^{-4}-96q^{-6}+415q^{-8}-562q^{-10}+490q^{-12}-243q^{-14}-66q^{-16}+303q^{-18}-365q^{-20}+235q^{-22}+21q^{-24}-277q^{-26}+418q^{-28}-373q^{-30}+154q^{-32}+134q^{-34}-379q^{-36}+487q^{-38}-417q^{-40}+223q^{-42}+18q^{-44}-218q^{-46}+314q^{-48}-299q^{-50}+204q^{-52}-75q^{-54}-33q^{-56}+94q^{-58}-111q^{-60}+91q^{-62}-53q^{-64}+21q^{-66}+4q^{-68}-15q^{-70}+15q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_88.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{11}+3 q^9-4 q^7+5 q^5-3 q^3+q+ q^{-1} -3 q^{-3} +5 q^{-5} -4 q^{-7} +3 q^{-9} - q^{-11} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}-3 q^{30}+11 q^{26}-14 q^{24}-10 q^{22}+37 q^{20}-18 q^{18}-37 q^{16}+54 q^{14}-53 q^{10}+36 q^8+21 q^6-37 q^4-q^2+29- q^{-2} -37 q^{-4} +21 q^{-6} +36 q^{-8} -53 q^{-10} +54 q^{-14} -37 q^{-16} -18 q^{-18} +37 q^{-20} -10 q^{-22} -14 q^{-24} +11 q^{-26} -3 q^{-30} + q^{-32} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{63}+3 q^{61}-7 q^{57}-2 q^{55}+18 q^{53}+13 q^{51}-44 q^{49}-36 q^{47}+70 q^{45}+93 q^{43}-88 q^{41}-184 q^{39}+77 q^{37}+291 q^{35}-15 q^{33}-384 q^{31}-101 q^{29}+441 q^{27}+232 q^{25}-426 q^{23}-355 q^{21}+345 q^{19}+441 q^{17}-229 q^{15}-461 q^{13}+93 q^{11}+432 q^9+41 q^7-362 q^5-164 q^3+271 q+271 q^{-1} -164 q^{-3} -362 q^{-5} +41 q^{-7} +432 q^{-9} +93 q^{-11} -461 q^{-13} -229 q^{-15} +441 q^{-17} +345 q^{-19} -355 q^{-21} -426 q^{-23} +232 q^{-25} +441 q^{-27} -101 q^{-29} -384 q^{-31} -15 q^{-33} +291 q^{-35} +77 q^{-37} -184 q^{-39} -88 q^{-41} +93 q^{-43} +70 q^{-45} -36 q^{-47} -44 q^{-49} +13 q^{-51} +18 q^{-53} -2 q^{-55} -7 q^{-57} +3 q^{-61} - q^{-63} }
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^{104}-3 q^{102}+7 q^{98}-2 q^{96}-2 q^{94}-21 q^{92}+4 q^{90}+54 q^{88}+13 q^{86}-29 q^{84}-144 q^{82}-46 q^{80}+234 q^{78}+234 q^{76}+24 q^{74}-554 q^{72}-518 q^{70}+359 q^{68}+966 q^{66}+786 q^{64}-922 q^{62}-1844 q^{60}-517 q^{58}+1658 q^{56}+2747 q^{54}+35 q^{52}-3124 q^{50}-2894 q^{48}+724 q^{46}+4612 q^{44}+2698 q^{42}-2489 q^{40}-5095 q^{38}-2009 q^{36}+4299 q^{34}+5074 q^{32}+92 q^{30}-5038 q^{28}-4396 q^{26}+1926 q^{24}+5220 q^{22}+2525 q^{20}-2979 q^{18}-4789 q^{16}-623 q^{14}+3557 q^{12}+3551 q^{10}-574 q^8-3759 q^6-2380 q^4+1504 q^2+3699+1504 q^{-2} -2380 q^{-4} -3759 q^{-6} -574 q^{-8} +3551 q^{-10} +3557 q^{-12} -623 q^{-14} -4789 q^{-16} -2979 q^{-18} +2525 q^{-20} +5220 q^{-22} +1926 q^{-24} -4396 q^{-26} -5038 q^{-28} +92 q^{-30} +5074 q^{-32} +4299 q^{-34} -2009 q^{-36} -5095 q^{-38} -2489 q^{-40} +2698 q^{-42} +4612 q^{-44} +724 q^{-46} -2894 q^{-48} -3124 q^{-50} +35 q^{-52} +2747 q^{-54} +1658 q^{-56} -517 q^{-58} -1844 q^{-60} -922 q^{-62} +786 q^{-64} +966 q^{-66} +359 q^{-68} -518 q^{-70} -554 q^{-72} +24 q^{-74} +234 q^{-76} +234 q^{-78} -46 q^{-80} -144 q^{-82} -29 q^{-84} +13 q^{-86} +54 q^{-88} +4 q^{-90} -21 q^{-92} -2 q^{-94} -2 q^{-96} +7 q^{-98} -3 q^{-102} + q^{-104} }
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}+3 q^{153}-7 q^{149}+2 q^{147}+6 q^{145}+5 q^{143}+4 q^{141}-14 q^{139}-41 q^{137}-10 q^{135}+70 q^{133}+105 q^{131}+43 q^{129}-135 q^{127}-298 q^{125}-225 q^{123}+222 q^{121}+732 q^{119}+682 q^{117}-159 q^{115}-1354 q^{113}-1787 q^{111}-496 q^{109}+2095 q^{107}+3755 q^{105}+2268 q^{103}-2201 q^{101}-6450 q^{99}-5989 q^{97}+635 q^{95}+9175 q^{93}+11773 q^{91}+3865 q^{89}-10226 q^{87}-18885 q^{85}-12188 q^{83}+7671 q^{81}+25415 q^{79}+23813 q^{77}-4 q^{75}-28588 q^{73}-36643 q^{71}-13038 q^{69}+26035 q^{67}+47593 q^{65}+29531 q^{63}-16857 q^{61}-53210 q^{59}-46175 q^{57}+1965 q^{55}+51702 q^{53}+59244 q^{51}+15546 q^{49}-43098 q^{47}-65778 q^{45}-32065 q^{43}+29307 q^{41}+65010 q^{39}+44420 q^{37}-13624 q^{35}-57872 q^{33}-50838 q^{31}-1066 q^{29}+46663 q^{27}+51692 q^{25}+12623 q^{23}-34147 q^{21}-48336 q^{19}-20550 q^{17}+22334 q^{15}+42967 q^{13}+25616 q^{11}-12308 q^9-37451 q^7-29220 q^5+3904 q^3+32846 q+32846 q^{-1} +3904 q^{-3} -29220 q^{-5} -37451 q^{-7} -12308 q^{-9} +25616 q^{-11} +42967 q^{-13} +22334 q^{-15} -20550 q^{-17} -48336 q^{-19} -34147 q^{-21} +12623 q^{-23} +51692 q^{-25} +46663 q^{-27} -1066 q^{-29} -50838 q^{-31} -57872 q^{-33} -13624 q^{-35} +44420 q^{-37} +65010 q^{-39} +29307 q^{-41} -32065 q^{-43} -65778 q^{-45} -43098 q^{-47} +15546 q^{-49} +59244 q^{-51} +51702 q^{-53} +1965 q^{-55} -46175 q^{-57} -53210 q^{-59} -16857 q^{-61} +29531 q^{-63} +47593 q^{-65} +26035 q^{-67} -13038 q^{-69} -36643 q^{-71} -28588 q^{-73} -4 q^{-75} +23813 q^{-77} +25415 q^{-79} +7671 q^{-81} -12188 q^{-83} -18885 q^{-85} -10226 q^{-87} +3865 q^{-89} +11773 q^{-91} +9175 q^{-93} +635 q^{-95} -5989 q^{-97} -6450 q^{-99} -2201 q^{-101} +2268 q^{-103} +3755 q^{-105} +2095 q^{-107} -496 q^{-109} -1787 q^{-111} -1354 q^{-113} -159 q^{-115} +682 q^{-117} +732 q^{-119} +222 q^{-121} -225 q^{-123} -298 q^{-125} -135 q^{-127} +43 q^{-129} +105 q^{-131} +70 q^{-133} -10 q^{-135} -41 q^{-137} -14 q^{-139} +4 q^{-141} +5 q^{-143} +6 q^{-145} +2 q^{-147} -7 q^{-149} +3 q^{-153} - q^{-155} }

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^{16}+q^{14}+2 q^{12}-3 q^{10}+3 q^8-2 q^4+3 q^2-3+3 q^{-2} -2 q^{-4} +3 q^{-8} -3 q^{-10} +2 q^{-12} + q^{-14} - q^{-16} }
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^{42}-q^{40}-3 q^{38}+q^{36}+7 q^{34}+q^{32}-14 q^{30}-3 q^{28}+21 q^{26}+4 q^{24}-25 q^{22}-5 q^{20}+28 q^{18}+8 q^{16}-32 q^{14}+q^{12}+26 q^{10}-6 q^8-15 q^6+7 q^4+7 q^2-10+7 q^{-2} +7 q^{-4} -15 q^{-6} -6 q^{-8} +26 q^{-10} + q^{-12} -32 q^{-14} +8 q^{-16} +28 q^{-18} -5 q^{-20} -25 q^{-22} +4 q^{-24} +21 q^{-26} -3 q^{-28} -14 q^{-30} + q^{-32} +7 q^{-34} + q^{-36} -3 q^{-38} - q^{-40} + q^{-42} }

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}-3 q^{32}+q^{30}+7 q^{28}-14 q^{26}+4 q^{24}+22 q^{22}-30 q^{20}+5 q^{18}+35 q^{16}-41 q^{14}+2 q^{12}+35 q^{10}-30 q^8-6 q^6+22 q^4-2 q^2-10-2 q^{-2} +22 q^{-4} -6 q^{-6} -30 q^{-8} +35 q^{-10} +2 q^{-12} -41 q^{-14} +35 q^{-16} +5 q^{-18} -30 q^{-20} +22 q^{-22} +4 q^{-24} -14 q^{-26} +7 q^{-28} + q^{-30} -3 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^{21}+q^{19}+2 q^{15}-3 q^{13}+4 q^{11}-2 q^9+2 q^7-2 q^5+2 q^3-q- q^{-1} +2 q^{-3} -2 q^{-5} +2 q^{-7} -2 q^{-9} +4 q^{-11} -3 q^{-13} +2 q^{-15} + q^{-19} - q^{-21} }

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}+3 q^{32}-7 q^{30}+13 q^{28}-22 q^{26}+32 q^{24}-40 q^{22}+48 q^{20}-49 q^{18}+45 q^{16}-33 q^{14}+16 q^{12}+7 q^{10}-32 q^8+56 q^6-76 q^4+90 q^2-96+90 q^{-2} -76 q^{-4} +56 q^{-6} -32 q^{-8} +7 q^{-10} +16 q^{-12} -33 q^{-14} +45 q^{-16} -49 q^{-18} +48 q^{-20} -40 q^{-22} +32 q^{-24} -22 q^{-26} +13 q^{-28} -7 q^{-30} +3 q^{-32} - q^{-34} }
1,0	$q^{56}-3q^{52}-3q^{50}+4q^{48}+10q^{46}-18q^{42}-13q^{40}+19q^{38}+31q^{36}-4q^{34}-43q^{32}-19q^{30}+40q^{28}+42q^{26}-21q^{24}-53q^{22}-4q^{20}+49q^{18}+22q^{16}-36q^{14}-31q^{12}+22q^{10}+33q^{8}-11q^{6}-33q^{4}+4q^{2}+35+4q^{-2}-33q^{-4}-11q^{-6}+33q^{-8}+22q^{-10}-31q^{-12}-36q^{-14}+22q^{-16}+49q^{-18}-4q^{-20}-53q^{-22}-21q^{-24}+42q^{-26}+40q^{-28}-19q^{-30}-43q^{-32}-4q^{-34}+31q^{-36}+19q^{-38}-13q^{-40}-18q^{-42}+10q^{-46}+4q^{-48}-3q^{-50}-3q^{-52}+q^{-56}$

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}-3 q^{78}+7 q^{76}-13 q^{74}+15 q^{72}-15 q^{70}+4 q^{68}+21 q^{66}-53 q^{64}+91 q^{62}-111 q^{60}+94 q^{58}-33 q^{56}-75 q^{54}+204 q^{52}-299 q^{50}+314 q^{48}-218 q^{46}+18 q^{44}+223 q^{42}-417 q^{40}+487 q^{38}-379 q^{36}+134 q^{34}+154 q^{32}-373 q^{30}+418 q^{28}-277 q^{26}+21 q^{24}+235 q^{22}-365 q^{20}+303 q^{18}-66 q^{16}-243 q^{14}+490 q^{12}-562 q^{10}+415 q^8-96 q^6-286 q^4+588 q^2-699+588 q^{-2} -286 q^{-4} -96 q^{-6} +415 q^{-8} -562 q^{-10} +490 q^{-12} -243 q^{-14} -66 q^{-16} +303 q^{-18} -365 q^{-20} +235 q^{-22} +21 q^{-24} -277 q^{-26} +418 q^{-28} -373 q^{-30} +154 q^{-32} +134 q^{-34} -379 q^{-36} +487 q^{-38} -417 q^{-40} +223 q^{-42} +18 q^{-44} -218 q^{-46} +314 q^{-48} -299 q^{-50} +204 q^{-52} -75 q^{-54} -33 q^{-56} +94 q^{-58} -111 q^{-60} +91 q^{-62} -53 q^{-64} +21 q^{-66} +4 q^{-68} -15 q^{-70} +15 q^{-72} -13 q^{-74} +7 q^{-76} -3 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 88"];

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+8 t^2-24 t+35-24 t^{-1} +8 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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 101, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^5+4 q^4-8 q^3+13 q^2-16 q+17-16 q^{-1} +13 q^{-2} -8 q^{-3} +4 q^{-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^6+2 a^2 z^4+2 z^4 a^{-2} -2 z^4-a^4 z^2+2 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} -3 z^2+a^2+ a^{-2} -1}

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 2 a z^9+2 z^9 a^{-1} +6 a^2 z^8+6 z^8 a^{-2} +12 z^8+7 a^3 z^7+14 a z^7+14 z^7 a^{-1} +7 z^7 a^{-3} +4 a^4 z^6-2 a^2 z^6-2 z^6 a^{-2} +4 z^6 a^{-4} -12 z^6+a^5 z^5-11 a^3 z^5-32 a z^5-32 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-10 a^2 z^4-10 z^4 a^{-2} -6 z^4 a^{-4} -8 z^4-a^5 z^3+6 a^3 z^3+19 a z^3+19 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +3 a^4 z^2+7 a^2 z^2+7 z^2 a^{-2} +3 z^2 a^{-4} +8 z^2-a^3 z-4 a z-4 z a^{-1} -z a^{-3} -a^2- a^{-2} -1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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 88"];

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+8 t^2-24 t+35-24 t^{-1} +8 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^5+4 q^4-8 q^3+13 q^2-16 q+17-16 q^{-1} +13 q^{-2} -8 q^{-3} +4 q^{-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]=

{}

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:

(-1, 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 $-4$ 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 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 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{34}{3}} $0$ 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} $0$ $0$ $-{\frac {32}{3}}$ $0$ ${\frac {56}{3}}$ ${\frac {136}{3}}$ ${\frac {449}{30}}$ $-{\frac {326}{5}}$ ${\frac {4978}{45}}$ $-{\frac {257}{18}}$ ${\frac {449}{30}}$

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 Failed to parse (SVG (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=} 0 is the signature of 10 88. 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                   3   3 7                 5 1   -4 5               8 3     5 3             8 5       -3 1           9 8         1 -1         8 9           1 -3       5 8             -3 -5     3 8               5 -7   1 5                 -4 -9   3                   3 -11 1                     -1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\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}^{5}\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=-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}^{8}\oplus{\mathbb Z}_2^{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}^{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 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}^{8}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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}^{9}\oplus{\mathbb Z}_2^{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 {\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=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}^{8}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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}^{5}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 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^{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}^{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 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^{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=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 $sl(2)$)

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}-4 q^{14}+3 q^{13}+12 q^{12}-29 q^{11}+7 q^{10}+59 q^9-84 q^8-12 q^7+150 q^6-138 q^5-65 q^4+239 q^3-153 q^2-123 q+275-123 q^{-1} -153 q^{-2} +239 q^{-3} -65 q^{-4} -138 q^{-5} +150 q^{-6} -12 q^{-7} -84 q^{-8} +59 q^{-9} +7 q^{-10} -29 q^{-11} +12 q^{-12} +3 q^{-13} -4 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}+4 q^{29}-3 q^{28}-7 q^{27}+4 q^{26}+24 q^{25}-8 q^{24}-64 q^{23}+12 q^{22}+130 q^{21}+15 q^{20}-245 q^{19}-84 q^{18}+391 q^{17}+229 q^{16}-551 q^{15}-453 q^{14}+674 q^{13}+771 q^{12}-760 q^{11}-1111 q^{10}+745 q^9+1471 q^8-664 q^7-1781 q^6+513 q^5+2025 q^4-325 q^3-2172 q^2+110 q+2223+110 q^{-1} -2172 q^{-2} -325 q^{-3} +2025 q^{-4} +513 q^{-5} -1781 q^{-6} -664 q^{-7} +1471 q^{-8} +745 q^{-9} -1111 q^{-10} -760 q^{-11} +771 q^{-12} +674 q^{-13} -453 q^{-14} -551 q^{-15} +229 q^{-16} +391 q^{-17} -84 q^{-18} -245 q^{-19} +15 q^{-20} +130 q^{-21} +12 q^{-22} -64 q^{-23} -8 q^{-24} +24 q^{-25} +4 q^{-26} -7 q^{-27} -3 q^{-28} +4 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}-4 q^{49}+3 q^{48}+7 q^{47}-9 q^{46}+q^{45}-23 q^{44}+28 q^{43}+57 q^{42}-50 q^{41}-41 q^{40}-138 q^{39}+126 q^{38}+337 q^{37}-50 q^{36}-251 q^{35}-716 q^{34}+162 q^{33}+1214 q^{32}+557 q^{31}-431 q^{30}-2424 q^{29}-760 q^{28}+2541 q^{27}+2732 q^{26}+658 q^{25}-5136 q^{24}-3919 q^{23}+2771 q^{22}+6350 q^{21}+4546 q^{20}-7050 q^{19}-9106 q^{18}+165 q^{17}+9436 q^{16}+10854 q^{15}-6275 q^{14}-14088 q^{13}-4965 q^{12}+10078 q^{11}+17176 q^{10}-2981 q^9-16783 q^8-10469 q^7+8268 q^6+21342 q^5+1199 q^4-16789 q^3-14594 q^2+5083 q+22721+5083 q^{-1} -14594 q^{-2} -16789 q^{-3} +1199 q^{-4} +21342 q^{-5} +8268 q^{-6} -10469 q^{-7} -16783 q^{-8} -2981 q^{-9} +17176 q^{-10} +10078 q^{-11} -4965 q^{-12} -14088 q^{-13} -6275 q^{-14} +10854 q^{-15} +9436 q^{-16} +165 q^{-17} -9106 q^{-18} -7050 q^{-19} +4546 q^{-20} +6350 q^{-21} +2771 q^{-22} -3919 q^{-23} -5136 q^{-24} +658 q^{-25} +2732 q^{-26} +2541 q^{-27} -760 q^{-28} -2424 q^{-29} -431 q^{-30} +557 q^{-31} +1214 q^{-32} +162 q^{-33} -716 q^{-34} -251 q^{-35} -50 q^{-36} +337 q^{-37} +126 q^{-38} -138 q^{-39} -41 q^{-40} -50 q^{-41} +57 q^{-42} +28 q^{-43} -23 q^{-44} + q^{-45} -9 q^{-46} +7 q^{-47} +3 q^{-48} -4 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}+4 q^{74}-3 q^{73}-7 q^{72}+9 q^{71}+4 q^{70}-2 q^{69}+3 q^{68}-21 q^{67}-34 q^{66}+40 q^{65}+84 q^{64}+33 q^{63}-59 q^{62}-199 q^{61}-197 q^{60}+113 q^{59}+531 q^{58}+543 q^{57}-109 q^{56}-1040 q^{55}-1392 q^{54}-320 q^{53}+1822 q^{52}+3134 q^{51}+1551 q^{50}-2527 q^{49}-5861 q^{48}-4569 q^{47}+2283 q^{46}+9758 q^{45}+10091 q^{44}+71 q^{43}-13769 q^{42}-18660 q^{41}-6376 q^{40}+16455 q^{39}+29950 q^{38}+17815 q^{37}-15371 q^{36}-42477 q^{35}-34960 q^{34}+8400 q^{33}+53555 q^{32}+56888 q^{31}+6187 q^{30}-60539 q^{29}-81348 q^{28}-27953 q^{27}+60590 q^{26}+105028 q^{25}+55924 q^{24}-52997 q^{23}-125046 q^{22}-86597 q^{21}+37910 q^{20}+138741 q^{19}+117296 q^{18}-17294 q^{17}-145636 q^{16}-144641 q^{15}-6338 q^{14}+145775 q^{13}+167068 q^{12}+30435 q^{11}-140607 q^{10}-183710 q^9-53108 q^8+131586 q^7+194854 q^6+73319 q^5-119974 q^4-201061 q^3-91032 q^2+106443 q+203085+106443 q^{-1} -91032 q^{-2} -201061 q^{-3} -119974 q^{-4} +73319 q^{-5} +194854 q^{-6} +131586 q^{-7} -53108 q^{-8} -183710 q^{-9} -140607 q^{-10} +30435 q^{-11} +167068 q^{-12} +145775 q^{-13} -6338 q^{-14} -144641 q^{-15} -145636 q^{-16} -17294 q^{-17} +117296 q^{-18} +138741 q^{-19} +37910 q^{-20} -86597 q^{-21} -125046 q^{-22} -52997 q^{-23} +55924 q^{-24} +105028 q^{-25} +60590 q^{-26} -27953 q^{-27} -81348 q^{-28} -60539 q^{-29} +6187 q^{-30} +56888 q^{-31} +53555 q^{-32} +8400 q^{-33} -34960 q^{-34} -42477 q^{-35} -15371 q^{-36} +17815 q^{-37} +29950 q^{-38} +16455 q^{-39} -6376 q^{-40} -18660 q^{-41} -13769 q^{-42} +71 q^{-43} +10091 q^{-44} +9758 q^{-45} +2283 q^{-46} -4569 q^{-47} -5861 q^{-48} -2527 q^{-49} +1551 q^{-50} +3134 q^{-51} +1822 q^{-52} -320 q^{-53} -1392 q^{-54} -1040 q^{-55} -109 q^{-56} +543 q^{-57} +531 q^{-58} +113 q^{-59} -197 q^{-60} -199 q^{-61} -59 q^{-62} +33 q^{-63} +84 q^{-64} +40 q^{-65} -34 q^{-66} -21 q^{-67} +3 q^{-68} -2 q^{-69} +4 q^{-70} +9 q^{-71} -7 q^{-72} -3 q^{-73} +4 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}-4 q^{104}+3 q^{103}+7 q^{102}-9 q^{101}-4 q^{100}-3 q^{99}+22 q^{98}-10 q^{97}-2 q^{96}+44 q^{95}-68 q^{94}-58 q^{93}-20 q^{92}+145 q^{91}+95 q^{90}+46 q^{89}+125 q^{88}-408 q^{87}-531 q^{86}-317 q^{85}+615 q^{84}+967 q^{83}+1024 q^{82}+876 q^{81}-1619 q^{80}-3217 q^{79}-3248 q^{78}+420 q^{77}+4033 q^{76}+6961 q^{75}+7385 q^{74}-1329 q^{73}-11203 q^{72}-17531 q^{71}-10354 q^{70}+4368 q^{69}+23577 q^{68}+35863 q^{67}+19355 q^{66}-14959 q^{65}-52800 q^{64}-59173 q^{63}-30996 q^{62}+33402 q^{61}+99926 q^{60}+104116 q^{59}+38455 q^{58}-79161 q^{57}-162940 q^{56}-165555 q^{55}-42543 q^{54}+151254 q^{53}+271312 q^{52}+233769 q^{51}+13377 q^{50}-249520 q^{49}-414572 q^{48}-308643 q^{47}+46760 q^{46}+420480 q^{45}+576578 q^{44}+343984 q^{43}-145228 q^{42}-645910 q^{41}-753595 q^{40}-338824 q^{39}+352823 q^{38}+902403 q^{37}+878082 q^{36}+270412 q^{35}-643372 q^{34}-1182741 q^{33}-942447 q^{32}-39231 q^{31}+986653 q^{30}+1396470 q^{29}+904710 q^{28}-317470 q^{27}-1371425 q^{26}-1527787 q^{25}-639520 q^{24}+759338 q^{23}+1682298 q^{22}+1515191 q^{21}+201994 q^{20}-1266624 q^{19}-1892397 q^{18}-1214631 q^{17}+352347 q^{16}+1696603 q^{15}+1921584 q^{14}+704085 q^{13}-991256 q^{12}-2009256 q^{11}-1615471 q^{10}-55878 q^9+1547088 q^8+2110070 q^7+1076094 q^6-686020 q^5-1969547 q^4-1844533 q^3-390299 q^2+1337527 q+2158285+1337527 q^{-1} -390299 q^{-2} -1844533 q^{-3} -1969547 q^{-4} -686020 q^{-5} +1076094 q^{-6} +2110070 q^{-7} +1547088 q^{-8} -55878 q^{-9} -1615471 q^{-10} -2009256 q^{-11} -991256 q^{-12} +704085 q^{-13} +1921584 q^{-14} +1696603 q^{-15} +352347 q^{-16} -1214631 q^{-17} -1892397 q^{-18} -1266624 q^{-19} +201994 q^{-20} +1515191 q^{-21} +1682298 q^{-22} +759338 q^{-23} -639520 q^{-24} -1527787 q^{-25} -1371425 q^{-26} -317470 q^{-27} +904710 q^{-28} +1396470 q^{-29} +986653 q^{-30} -39231 q^{-31} -942447 q^{-32} -1182741 q^{-33} -643372 q^{-34} +270412 q^{-35} +878082 q^{-36} +902403 q^{-37} +352823 q^{-38} -338824 q^{-39} -753595 q^{-40} -645910 q^{-41} -145228 q^{-42} +343984 q^{-43} +576578 q^{-44} +420480 q^{-45} +46760 q^{-46} -308643 q^{-47} -414572 q^{-48} -249520 q^{-49} +13377 q^{-50} +233769 q^{-51} +271312 q^{-52} +151254 q^{-53} -42543 q^{-54} -165555 q^{-55} -162940 q^{-56} -79161 q^{-57} +38455 q^{-58} +104116 q^{-59} +99926 q^{-60} +33402 q^{-61} -30996 q^{-62} -59173 q^{-63} -52800 q^{-64} -14959 q^{-65} +19355 q^{-66} +35863 q^{-67} +23577 q^{-68} +4368 q^{-69} -10354 q^{-70} -17531 q^{-71} -11203 q^{-72} -1329 q^{-73} +7385 q^{-74} +6961 q^{-75} +4033 q^{-76} +420 q^{-77} -3248 q^{-78} -3217 q^{-79} -1619 q^{-80} +876 q^{-81} +1024 q^{-82} +967 q^{-83} +615 q^{-84} -317 q^{-85} -531 q^{-86} -408 q^{-87} +125 q^{-88} +46 q^{-89} +95 q^{-90} +145 q^{-91} -20 q^{-92} -58 q^{-93} -68 q^{-94} +44 q^{-95} -2 q^{-96} -10 q^{-97} +22 q^{-98} -3 q^{-99} -4 q^{-100} -9 q^{-101} +7 q^{-102} +3 q^{-103} -4 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}+4 q^{139}-3 q^{138}-7 q^{137}+9 q^{136}+4 q^{135}+3 q^{134}-17 q^{133}-15 q^{132}+33 q^{131}-8 q^{130}-16 q^{129}+42 q^{128}+30 q^{127}+13 q^{126}-121 q^{125}-168 q^{124}+67 q^{123}+72 q^{122}+153 q^{121}+345 q^{120}+206 q^{119}+28 q^{118}-744 q^{117}-1273 q^{116}-608 q^{115}+135 q^{114}+1493 q^{113}+2823 q^{112}+2540 q^{111}+1128 q^{110}-2911 q^{109}-7248 q^{108}-7552 q^{107}-4466 q^{106}+3929 q^{105}+14027 q^{104}+18818 q^{103}+16073 q^{102}-110 q^{101}-24199 q^{100}-41450 q^{99}-42981 q^{98}-16792 q^{97}+30828 q^{96}+75952 q^{95}+97722 q^{94}+66617 q^{93}-17353 q^{92}-118190 q^{91}-190921 q^{90}-173528 q^{89}-48185 q^{88}+141342 q^{87}+318844 q^{86}+366831 q^{85}+218101 q^{84}-93530 q^{83}-454180 q^{82}-657775 q^{81}-546567 q^{80}-108698 q^{79}+519225 q^{78}+1018197 q^{77}+1077733 q^{76}+567378 q^{75}-394097 q^{74}-1360104 q^{73}-1798579 q^{72}-1366682 q^{71}-78293 q^{70}+1520419 q^{69}+2618368 q^{68}+2535184 q^{67}+1042851 q^{66}-1289355 q^{65}-3349882 q^{64}-3995819 q^{63}-2580161 q^{62}+454017 q^{61}+3732829 q^{60}+5554146 q^{59}+4652596 q^{58}+1132131 q^{57}-3491647 q^{56}-6921359 q^{55}-7080887 q^{54}-3483098 q^{53}+2414596 q^{52}+7768501 q^{51}+9563059 q^{50}+6458604 q^{49}-417981 q^{48}-7822357 q^{47}-11746214 q^{46}-9770005 q^{45}-2404931 q^{44}+6931124 q^{43}+13296490 q^{42}+13053332 q^{41}+5814614 q^{40}-5113233 q^{39}-14002256 q^{38}-15949372 q^{37}-9459051 q^{36}+2549624 q^{35}+13795700 q^{34}+18181824 q^{33}+12981758 q^{32}+470475 q^{31}-12777028 q^{30}-19616672 q^{29}-16076808 q^{28}-3614657 q^{27}+11154661 q^{26}+20259777 q^{25}+18557545 q^{24}+6592876 q^{23}-9195152 q^{22}-20238112 q^{21}-20364308 q^{20}-9198979 q^{19}+7151404 q^{18}+19740523 q^{17}+21549728 q^{16}+11338004 q^{15}-5213041 q^{14}-18966439 q^{13}-22240226 q^{12}-13018279 q^{11}+3484032 q^{10}+18080389 q^9+22588382 q^8+14321596 q^7-1979218 q^6-17180052 q^5-22733021 q^4-15371342 q^3+639284 q^2+16291800 q+22770653+16291800 q^{-1} +639284 q^{-2} -15371342 q^{-3} -22733021 q^{-4} -17180052 q^{-5} -1979218 q^{-6} +14321596 q^{-7} +22588382 q^{-8} +18080389 q^{-9} +3484032 q^{-10} -13018279 q^{-11} -22240226 q^{-12} -18966439 q^{-13} -5213041 q^{-14} +11338004 q^{-15} +21549728 q^{-16} +19740523 q^{-17} +7151404 q^{-18} -9198979 q^{-19} -20364308 q^{-20} -20238112 q^{-21} -9195152 q^{-22} +6592876 q^{-23} +18557545 q^{-24} +20259777 q^{-25} +11154661 q^{-26} -3614657 q^{-27} -16076808 q^{-28} -19616672 q^{-29} -12777028 q^{-30} +470475 q^{-31} +12981758 q^{-32} +18181824 q^{-33} +13795700 q^{-34} +2549624 q^{-35} -9459051 q^{-36} -15949372 q^{-37} -14002256 q^{-38} -5113233 q^{-39} +5814614 q^{-40} +13053332 q^{-41} +13296490 q^{-42} +6931124 q^{-43} -2404931 q^{-44} -9770005 q^{-45} -11746214 q^{-46} -7822357 q^{-47} -417981 q^{-48} +6458604 q^{-49} +9563059 q^{-50} +7768501 q^{-51} +2414596 q^{-52} -3483098 q^{-53} -7080887 q^{-54} -6921359 q^{-55} -3491647 q^{-56} +1132131 q^{-57} +4652596 q^{-58} +5554146 q^{-59} +3732829 q^{-60} +454017 q^{-61} -2580161 q^{-62} -3995819 q^{-63} -3349882 q^{-64} -1289355 q^{-65} +1042851 q^{-66} +2535184 q^{-67} +2618368 q^{-68} +1520419 q^{-69} -78293 q^{-70} -1366682 q^{-71} -1798579 q^{-72} -1360104 q^{-73} -394097 q^{-74} +567378 q^{-75} +1077733 q^{-76} +1018197 q^{-77} +519225 q^{-78} -108698 q^{-79} -546567 q^{-80} -657775 q^{-81} -454180 q^{-82} -93530 q^{-83} +218101 q^{-84} +366831 q^{-85} +318844 q^{-86} +141342 q^{-87} -48185 q^{-88} -173528 q^{-89} -190921 q^{-90} -118190 q^{-91} -17353 q^{-92} +66617 q^{-93} +97722 q^{-94} +75952 q^{-95} +30828 q^{-96} -16792 q^{-97} -42981 q^{-98} -41450 q^{-99} -24199 q^{-100} -110 q^{-101} +16073 q^{-102} +18818 q^{-103} +14027 q^{-104} +3929 q^{-105} -4466 q^{-106} -7552 q^{-107} -7248 q^{-108} -2911 q^{-109} +1128 q^{-110} +2540 q^{-111} +2823 q^{-112} +1493 q^{-113} +135 q^{-114} -608 q^{-115} -1273 q^{-116} -744 q^{-117} +28 q^{-118} +206 q^{-119} +345 q^{-120} +153 q^{-121} +72 q^{-122} +67 q^{-123} -168 q^{-124} -121 q^{-125} +13 q^{-126} +30 q^{-127} +42 q^{-128} -16 q^{-129} -8 q^{-130} +33 q^{-131} -15 q^{-132} -17 q^{-133} +3 q^{-134} +4 q^{-135} +9 q^{-136} -7 q^{-137} -3 q^{-138} +4 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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