10 59

10_58

10_60

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 59's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 59 at Knotilus!

Two figure 8 knots on a loop, interlinked.

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13
Gauss code	1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7
Dowker-Thistlethwaite code	4 8 10 14 2 18 6 20 12 16
Conway Notation	[22,211,2]

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

[edit Notes on presentations of 10 59]

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 59"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 14 2 18 6 20 12 16

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22,211,2]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 13.3899 A-Polynomial See Data:10 59/A-polynomial

[edit Notes for 10 59's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ 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 1} Rasmussen s-Invariant -2

[edit Notes for 10 59'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^3-7 t^2+18 t-23+18 t^{-1} -7 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-z^4-z^2+1}
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 75, 2 }
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-10q^{4}+12q^{3}-12q^{2}+12q-9+6q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+3z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+5z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-4z^{2}+a^{2}+4a^{-2}-3a^{-4}+a^{-6}-2$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+7z^{8}a^{-2}+4z^{8}a^{-4}+3z^{8}+3az^{7}+8z^{7}a^{-1}+11z^{7}a^{-3}+6z^{7}a^{-5}+a^{2}z^{6}-9z^{6}a^{-2}+z^{6}a^{-4}+5z^{6}a^{-6}-4z^{6}-9az^{5}-27z^{5}a^{-1}-28z^{5}a^{-3}-7z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}-8z^{4}a^{-2}-11z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}-6z^{4}+8az^{3}+21z^{3}a^{-1}+20z^{3}a^{-3}+4z^{3}a^{-5}-3z^{3}a^{-7}+3a^{2}z^{2}+11z^{2}a^{-2}+10z^{2}a^{-4}+3z^{2}a^{-6}-z^{2}a^{-8}+8z^{2}-2az-5za^{-1}-4za^{-3}+za^{-7}-a^{2}-4a^{-2}-3a^{-4}-a^{-6}-2$
The A2 invariant	$q^{10}-q^{6}+2q^{4}-2q^{2}+2q^{-2}-q^{-4}+4q^{-6}-q^{-8}+q^{-10}-q^{-12}-3q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
The G2 invariant	$q^{46}-2q^{44}+6q^{42}-10q^{40}+12q^{38}-10q^{36}-q^{34}+23q^{32}-44q^{30}+64q^{28}-63q^{26}+33q^{24}+18q^{22}-83q^{20}+135q^{18}-149q^{16}+112q^{14}-28q^{12}-76q^{10}+158q^{8}-183q^{6}+145q^{4}-55q^{2}-52+123q^{-2}-140q^{-4}+86q^{-6}+9q^{-8}-96q^{-10}+143q^{-12}-111q^{-14}+24q^{-16}+88q^{-18}-182q^{-20}+220q^{-22}-177q^{-24}+68q^{-26}+74q^{-28}-192q^{-30}+256q^{-32}-225q^{-34}+126q^{-36}+6q^{-38}-123q^{-40}+179q^{-42}-167q^{-44}+85q^{-46}+20q^{-48}-97q^{-50}+117q^{-52}-74q^{-54}-17q^{-56}+101q^{-58}-146q^{-60}+127q^{-62}-63q^{-64}-31q^{-66}+113q^{-68}-154q^{-70}+149q^{-72}-93q^{-74}+23q^{-76}+41q^{-78}-84q^{-80}+92q^{-82}-79q^{-84}+52q^{-86}-16q^{-88}-10q^{-90}+27q^{-92}-32q^{-94}+28q^{-96}-19q^{-98}+11q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 10_59.

A1 Invariants.

Weight	Invariant
1	$q^{7}-2q^{5}+3q^{3}-3q+3q^{-1}+2q^{-7}-4q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$
2	$q^{22}-2q^{20}-2q^{18}+8q^{16}-3q^{14}-12q^{12}+17q^{10}+4q^{8}-26q^{6}+15q^{4}+17q^{2}-27+4q^{-2}+22q^{-4}-14q^{-6}-9q^{-8}+14q^{-10}+6q^{-12}-16q^{-14}-3q^{-16}+24q^{-18}-15q^{-20}-18q^{-22}+29q^{-24}-5q^{-26}-19q^{-28}+17q^{-30}+q^{-32}-9q^{-34}+5q^{-36}-2q^{-40}+q^{-42}$
3	$q^{45}-2q^{43}-2q^{41}+3q^{39}+8q^{37}-3q^{35}-19q^{33}-q^{31}+32q^{29}+16q^{27}-45q^{25}-44q^{23}+50q^{21}+79q^{19}-35q^{17}-115q^{15}+3q^{13}+142q^{11}+42q^{9}-147q^{7}-91q^{5}+129q^{3}+134q-102q^{-1}-154q^{-3}+61q^{-5}+165q^{-7}-19q^{-9}-156q^{-11}-18q^{-13}+137q^{-15}+54q^{-17}-106q^{-19}-87q^{-21}+62q^{-23}+119q^{-25}-17q^{-27}-137q^{-29}-42q^{-31}+145q^{-33}+91q^{-35}-128q^{-37}-134q^{-39}+103q^{-41}+152q^{-43}-64q^{-45}-143q^{-47}+23q^{-49}+121q^{-51}-q^{-53}-85q^{-55}-11q^{-57}+51q^{-59}+12q^{-61}-27q^{-63}-8q^{-65}+14q^{-67}+2q^{-69}-5q^{-71}+2q^{-75}-2q^{-79}+q^{-81}$
4	$q^{76}-2q^{74}-2q^{72}+3q^{70}+3q^{68}+8q^{66}-10q^{64}-19q^{62}-q^{60}+14q^{58}+53q^{56}-q^{54}-67q^{52}-64q^{50}-13q^{48}+157q^{46}+117q^{44}-58q^{42}-213q^{40}-229q^{38}+174q^{36}+366q^{34}+219q^{32}-227q^{30}-624q^{28}-168q^{26}+432q^{24}+724q^{22}+204q^{20}-792q^{18}-775q^{16}-15q^{14}+977q^{12}+909q^{10}-399q^{8}-1123q^{6}-743q^{4}+676q^{2}+1331+282q^{-2}-940q^{-4}-1202q^{-6}+105q^{-8}+1243q^{-10}+766q^{-12}-499q^{-14}-1218q^{-16}-332q^{-18}+871q^{-20}+921q^{-22}-84q^{-24}-976q^{-26}-609q^{-28}+410q^{-30}+928q^{-32}+334q^{-34}-601q^{-36}-855q^{-38}-183q^{-40}+800q^{-42}+823q^{-44}+3q^{-46}-971q^{-48}-901q^{-50}+374q^{-52}+1141q^{-54}+764q^{-56}-682q^{-58}-1373q^{-60}-296q^{-62}+941q^{-64}+1238q^{-66}-68q^{-68}-1213q^{-70}-735q^{-72}+345q^{-74}+1079q^{-76}+379q^{-78}-622q^{-80}-646q^{-82}-105q^{-84}+558q^{-86}+371q^{-88}-157q^{-90}-293q^{-92}-175q^{-94}+169q^{-96}+168q^{-98}-8q^{-100}-65q^{-102}-83q^{-104}+34q^{-106}+40q^{-108}-2q^{-110}-21q^{-114}+7q^{-116}+5q^{-118}-4q^{-120}+4q^{-122}-3q^{-124}+2q^{-126}-2q^{-130}+q^{-132}$
5	$q^{115}-2q^{113}-2q^{111}+3q^{109}+3q^{107}+3q^{105}+q^{103}-10q^{101}-19q^{99}-q^{97}+23q^{95}+35q^{93}+27q^{91}-23q^{89}-85q^{87}-89q^{85}+9q^{83}+138q^{81}+196q^{79}+96q^{77}-158q^{75}-378q^{73}-322q^{71}+72q^{69}+543q^{67}+695q^{65}+268q^{63}-568q^{61}-1177q^{59}-914q^{57}+266q^{55}+1540q^{53}+1828q^{51}+587q^{49}-1513q^{47}-2818q^{45}-1968q^{43}+793q^{41}+3448q^{39}+3691q^{37}+794q^{35}-3334q^{33}-5328q^{31}-3066q^{29}+2184q^{27}+6311q^{25}+5617q^{23}+15q^{21}-6259q^{19}-7859q^{17}-2875q^{15}+5033q^{13}+9235q^{11}+5829q^{9}-2791q^{7}-9491q^{5}-8329q^{3}+86q+8648q^{-1}+9869q^{-3}+2601q^{-5}-6974q^{-7}-10438q^{-9}-4742q^{-11}+4983q^{-13}+10058q^{-15}+6127q^{-17}-3010q^{-19}-9090q^{-21}-6793q^{-23}+1407q^{-25}+7849q^{-27}+6891q^{-29}-169q^{-31}-6607q^{-33}-6724q^{-35}-809q^{-37}+5491q^{-39}+6538q^{-41}+1725q^{-43}-4415q^{-45}-6478q^{-47}-2842q^{-49}+3226q^{-51}+6503q^{-53}+4267q^{-55}-1664q^{-57}-6445q^{-59}-5982q^{-61}-366q^{-63}+5954q^{-65}+7714q^{-67}+2969q^{-69}-4836q^{-71}-9102q^{-73}-5758q^{-75}+2879q^{-77}+9654q^{-79}+8445q^{-81}-279q^{-83}-9174q^{-85}-10342q^{-87}-2584q^{-89}+7509q^{-91}+11130q^{-93}+5151q^{-95}-5068q^{-97}-10600q^{-99}-6856q^{-101}+2367q^{-103}+8890q^{-105}+7465q^{-107}+29q^{-109}-6579q^{-111}-6955q^{-113}-1623q^{-115}+4154q^{-117}+5651q^{-119}+2375q^{-121}-2159q^{-123}-4050q^{-125}-2356q^{-127}+802q^{-129}+2556q^{-131}+1882q^{-133}-61q^{-135}-1413q^{-137}-1291q^{-139}-223q^{-141}+697q^{-143}+767q^{-145}+230q^{-147}-284q^{-149}-397q^{-151}-173q^{-153}+101q^{-155}+194q^{-157}+89q^{-159}-34q^{-161}-73q^{-163}-41q^{-165}+4q^{-167}+27q^{-169}+22q^{-171}-5q^{-173}-12q^{-175}-2q^{-179}-q^{-181}+5q^{-183}+q^{-185}-3q^{-187}+2q^{-189}-2q^{-193}+q^{-195}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}-q^{6}+2q^{4}-2q^{2}+2q^{-2}-q^{-4}+4q^{-6}-q^{-8}+q^{-10}-q^{-12}-3q^{-14}+2q^{-16}-q^{-18}+q^{-22}$
2,0	$q^{28}-2q^{24}-2q^{22}+3q^{20}+5q^{18}-4q^{16}-6q^{14}+6q^{12}+9q^{10}-6q^{8}-11q^{6}+7q^{4}+12q^{2}-9-9q^{-2}+11q^{-4}+3q^{-6}-7q^{-8}+7q^{-12}-2q^{-14}+10q^{-18}-3q^{-20}-12q^{-22}+5q^{-24}+8q^{-26}-15q^{-28}-8q^{-30}+13q^{-32}+8q^{-34}-8q^{-36}-4q^{-38}+10q^{-40}+3q^{-42}-6q^{-44}-2q^{-46}+q^{-48}-q^{-52}+q^{-56}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-2q^{18}+2q^{16}+3q^{14}-8q^{12}+7q^{10}+5q^{8}-16q^{6}+13q^{4}+6q^{2}-21+14q^{-2}+9q^{-4}-18q^{-6}+8q^{-8}+11q^{-10}-5q^{-12}-4q^{-14}+3q^{-16}+7q^{-18}-13q^{-20}-7q^{-22}+20q^{-24}-13q^{-26}-10q^{-28}+24q^{-30}-9q^{-32}-10q^{-34}+15q^{-36}-3q^{-38}-7q^{-40}+5q^{-42}-2q^{-46}+q^{-48}$
1,0,0	$q^{13}+q^{9}-q^{7}+2q^{5}-3q^{3}+q-2q^{-1}+2q^{-3}+3q^{-7}+3q^{-9}+q^{-13}-3q^{-15}-4q^{-19}+2q^{-21}-q^{-23}+q^{-25}+q^{-29}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-2q^{18}+6q^{16}-9q^{14}+14q^{12}-19q^{10}+23q^{8}-26q^{6}+25q^{4}-22q^{2}+13-4q^{-2}-7q^{-4}+22q^{-6}-32q^{-8}+45q^{-10}-47q^{-12}+50q^{-14}-45q^{-16}+37q^{-18}-27q^{-20}+13q^{-22}-2q^{-24}-11q^{-26}+18q^{-28}-24q^{-30}+25q^{-32}-24q^{-34}+21q^{-36}-15q^{-38}+11q^{-40}-7q^{-42}+4q^{-44}-2q^{-46}+q^{-48}$
1,0	$q^{34}-2q^{30}-2q^{28}+4q^{26}+6q^{24}-3q^{22}-11q^{20}-3q^{18}+15q^{16}+13q^{14}-12q^{12}-22q^{10}+2q^{8}+27q^{6}+12q^{4}-23q^{2}-22+11q^{-2}+26q^{-4}+2q^{-6}-23q^{-8}-8q^{-10}+17q^{-12}+13q^{-14}-12q^{-16}-11q^{-18}+10q^{-20}+15q^{-22}-7q^{-24}-18q^{-26}+2q^{-28}+19q^{-30}+2q^{-32}-22q^{-34}-11q^{-36}+19q^{-38}+19q^{-40}-13q^{-42}-25q^{-44}+2q^{-46}+27q^{-48}+11q^{-50}-17q^{-52}-19q^{-54}+5q^{-56}+18q^{-58}+5q^{-60}-9q^{-62}-9q^{-64}+q^{-66}+6q^{-68}+2q^{-70}-2q^{-72}-2q^{-74}+q^{-78}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-2q^{44}+6q^{42}-10q^{40}+12q^{38}-10q^{36}-q^{34}+23q^{32}-44q^{30}+64q^{28}-63q^{26}+33q^{24}+18q^{22}-83q^{20}+135q^{18}-149q^{16}+112q^{14}-28q^{12}-76q^{10}+158q^{8}-183q^{6}+145q^{4}-55q^{2}-52+123q^{-2}-140q^{-4}+86q^{-6}+9q^{-8}-96q^{-10}+143q^{-12}-111q^{-14}+24q^{-16}+88q^{-18}-182q^{-20}+220q^{-22}-177q^{-24}+68q^{-26}+74q^{-28}-192q^{-30}+256q^{-32}-225q^{-34}+126q^{-36}+6q^{-38}-123q^{-40}+179q^{-42}-167q^{-44}+85q^{-46}+20q^{-48}-97q^{-50}+117q^{-52}-74q^{-54}-17q^{-56}+101q^{-58}-146q^{-60}+127q^{-62}-63q^{-64}-31q^{-66}+113q^{-68}-154q^{-70}+149q^{-72}-93q^{-74}+23q^{-76}+41q^{-78}-84q^{-80}+92q^{-82}-79q^{-84}+52q^{-86}-16q^{-88}-10q^{-90}+27q^{-92}-32q^{-94}+28q^{-96}-19q^{-98}+11q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

.

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 59"];

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^3-7 t^2+18 t-23+18 t^{-1} -7 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-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]=

{ 75, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-3q^{6}+6q^{5}-10q^{4}+12q^{3}-12q^{2}+12q-9+6q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+3z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+5z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-4z^{2}+a^{2}+4a^{-2}-3a^{-4}+a^{-6}-2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+7z^{8}a^{-2}+4z^{8}a^{-4}+3z^{8}+3az^{7}+8z^{7}a^{-1}+11z^{7}a^{-3}+6z^{7}a^{-5}+a^{2}z^{6}-9z^{6}a^{-2}+z^{6}a^{-4}+5z^{6}a^{-6}-4z^{6}-9az^{5}-27z^{5}a^{-1}-28z^{5}a^{-3}-7z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}-8z^{4}a^{-2}-11z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}-6z^{4}+8az^{3}+21z^{3}a^{-1}+20z^{3}a^{-3}+4z^{3}a^{-5}-3z^{3}a^{-7}+3a^{2}z^{2}+11z^{2}a^{-2}+10z^{2}a^{-4}+3z^{2}a^{-6}-z^{2}a^{-8}+8z^{2}-2az-5za^{-1}-4za^{-3}+za^{-7}-a^{2}-4a^{-2}-3a^{-4}-a^{-6}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_40, K11n66,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_106,}

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 59"];

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]=

{ $t^{3}-7t^{2}+18t-23+18t^{-1}-7t^{-2}+t^{-3}$, $q^{7}-3q^{6}+6q^{5}-10q^{4}+12q^{3}-12q^{2}+12q-9+6q^{-1}-3q^{-2}+q^{-3}$ }

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]=

{9_40, K11n66,}

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_106,}

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 $-4$ $-8$ $8$ $-{\frac {14}{3}}$ ${\frac {38}{3}}$ $32$ ${\frac {112}{3}}$ 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 24} Failed to parse (SVG (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{4769}{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{194}{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{1738}{45}} ${\frac {319}{18}}$ 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{31}{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 10 59. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                     1 1 13                   2   -2 11                 4 1   3 9               6 2     -4 7             6 4       2 5           6 6         0 3         6 6           0 1       4 7             3 -1     2 5               -3 -3   1 4                 3 -5   2                   -2 -7 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=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=-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}} Failed to parse (SVG (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}\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=-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}^{4}\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=-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}^{5}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=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}^{7}\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}^{6}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\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=6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_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^{20}-3 q^{19}+2 q^{18}+6 q^{17}-17 q^{16}+12 q^{15}+22 q^{14}-53 q^{13}+26 q^{12}+56 q^{11}-100 q^{10}+29 q^9+95 q^8-127 q^7+16 q^6+117 q^5-119 q^4-7 q^3+112 q^2-83 q-25+81 q^{-1} -39 q^{-2} -27 q^{-3} +40 q^{-4} -9 q^{-5} -14 q^{-6} +11 q^{-7} -3 q^{-9} + q^{-10} }
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^{39}-3 q^{38}+2 q^{37}+2 q^{36}-q^{35}-8 q^{34}+9 q^{33}+14 q^{32}-23 q^{31}-27 q^{30}+48 q^{29}+53 q^{28}-85 q^{27}-101 q^{26}+132 q^{25}+175 q^{24}-183 q^{23}-267 q^{22}+211 q^{21}+391 q^{20}-232 q^{19}-504 q^{18}+217 q^{17}+610 q^{16}-178 q^{15}-691 q^{14}+122 q^{13}+730 q^{12}-42 q^{11}-748 q^{10}-27 q^9+711 q^8+118 q^7-665 q^6-182 q^5+573 q^4+255 q^3-481 q^2-286 q+358+307 q^{-1} -245 q^{-2} -291 q^{-3} +138 q^{-4} +251 q^{-5} -56 q^{-6} -191 q^{-7} - q^{-8} +133 q^{-9} +24 q^{-10} -77 q^{-11} -30 q^{-12} +39 q^{-13} +23 q^{-14} -16 q^{-15} -14 q^{-16} +6 q^{-17} +5 q^{-18} -3 q^{-20} + q^{-21} }
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^{64}-3 q^{63}+2 q^{62}+2 q^{61}-5 q^{60}+8 q^{59}-11 q^{58}+11 q^{57}+4 q^{56}-33 q^{55}+29 q^{54}-13 q^{53}+53 q^{52}-2 q^{51}-150 q^{50}+47 q^{49}+44 q^{48}+229 q^{47}-q^{46}-494 q^{45}-71 q^{44}+180 q^{43}+757 q^{42}+186 q^{41}-1157 q^{40}-612 q^{39}+204 q^{38}+1758 q^{37}+886 q^{36}-1891 q^{35}-1692 q^{34}-274 q^{33}+2903 q^{32}+2192 q^{31}-2188 q^{30}-2929 q^{29}-1351 q^{28}+3594 q^{27}+3638 q^{26}-1811 q^{25}-3696 q^{24}-2626 q^{23}+3524 q^{22}+4612 q^{21}-991 q^{20}-3719 q^{19}-3609 q^{18}+2852 q^{17}+4866 q^{16}-56 q^{15}-3125 q^{14}-4127 q^{13}+1833 q^{12}+4499 q^{11}+836 q^{10}-2120 q^9-4177 q^8+630 q^7+3613 q^6+1555 q^5-855 q^4-3700 q^3-508 q^2+2306 q+1817+367 q^{-1} -2651 q^{-2} -1163 q^{-3} +887 q^{-4} +1437 q^{-5} +1091 q^{-6} -1343 q^{-7} -1095 q^{-8} -105 q^{-9} +677 q^{-10} +1074 q^{-11} -347 q^{-12} -575 q^{-13} -397 q^{-14} +77 q^{-15} +618 q^{-16} +50 q^{-17} -129 q^{-18} -250 q^{-19} -115 q^{-20} +215 q^{-21} +66 q^{-22} +26 q^{-23} -75 q^{-24} -75 q^{-25} +45 q^{-26} +15 q^{-27} +23 q^{-28} -9 q^{-29} -21 q^{-30} +6 q^{-31} +5 q^{-33} -3 q^{-35} + q^{-36} }
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^{95}-3 q^{94}+2 q^{93}+2 q^{92}-5 q^{91}+4 q^{90}+5 q^{89}-9 q^{88}+q^{87}+4 q^{86}-17 q^{85}+11 q^{84}+32 q^{83}-4 q^{82}-22 q^{81}-41 q^{80}-49 q^{79}+50 q^{78}+155 q^{77}+101 q^{76}-115 q^{75}-315 q^{74}-273 q^{73}+163 q^{72}+669 q^{71}+638 q^{70}-185 q^{69}-1235 q^{68}-1341 q^{67}+41 q^{66}+2021 q^{65}+2581 q^{64}+489 q^{63}-2989 q^{62}-4499 q^{61}-1653 q^{60}+3912 q^{59}+7115 q^{58}+3765 q^{57}-4486 q^{56}-10276 q^{55}-6985 q^{54}+4288 q^{53}+13723 q^{52}+11201 q^{51}-3061 q^{50}-16799 q^{49}-16208 q^{48}+544 q^{47}+19255 q^{46}+21420 q^{45}+2918 q^{44}-20420 q^{43}-26301 q^{42}-7214 q^{41}+20423 q^{40}+30315 q^{39}+11642 q^{38}-19211 q^{37}-33076 q^{36}-15851 q^{35}+17079 q^{34}+34581 q^{33}+19447 q^{32}-14466 q^{31}-34836 q^{30}-22171 q^{29}+11463 q^{28}+34118 q^{27}+24228 q^{26}-8535 q^{25}-32600 q^{24}-25448 q^{23}+5395 q^{22}+30482 q^{21}+26291 q^{20}-2395 q^{19}-27787 q^{18}-26495 q^{17}-905 q^{16}+24567 q^{15}+26408 q^{14}+4043 q^{13}-20727 q^{12}-25537 q^{11}-7347 q^{10}+16367 q^9+24111 q^8+10123 q^7-11590 q^6-21606 q^5-12422 q^4+6642 q^3+18415 q^2+13587 q-2015-14338 q^{-1} -13643 q^{-2} -1920 q^{-3} +10000 q^{-4} +12425 q^{-5} +4685 q^{-6} -5718 q^{-7} -10237 q^{-8} -6122 q^{-9} +2092 q^{-10} +7441 q^{-11} +6285 q^{-12} +556 q^{-13} -4635 q^{-14} -5428 q^{-15} -2035 q^{-16} +2191 q^{-17} +4023 q^{-18} +2550 q^{-19} -507 q^{-20} -2531 q^{-21} -2278 q^{-22} -464 q^{-23} +1262 q^{-24} +1700 q^{-25} +798 q^{-26} -431 q^{-27} -1037 q^{-28} -752 q^{-29} -12 q^{-30} +520 q^{-31} +535 q^{-32} +178 q^{-33} -201 q^{-34} -325 q^{-35} -164 q^{-36} +49 q^{-37} +141 q^{-38} +122 q^{-39} +19 q^{-40} -71 q^{-41} -64 q^{-42} -9 q^{-43} +12 q^{-44} +24 q^{-45} +23 q^{-46} -9 q^{-47} -14 q^{-48} - q^{-49} +5 q^{-52} -3 q^{-54} + q^{-55} }
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^{132}-3 q^{131}+2 q^{130}+2 q^{129}-5 q^{128}+4 q^{127}+q^{126}+7 q^{125}-19 q^{124}+q^{123}+20 q^{122}-25 q^{121}+16 q^{120}+18 q^{119}+21 q^{118}-73 q^{117}-31 q^{116}+64 q^{115}-44 q^{114}+82 q^{113}+112 q^{112}+42 q^{111}-292 q^{110}-247 q^{109}+90 q^{108}+21 q^{107}+497 q^{106}+606 q^{105}+152 q^{104}-1059 q^{103}-1376 q^{102}-425 q^{101}+220 q^{100}+2204 q^{99}+2916 q^{98}+1258 q^{97}-2865 q^{96}-5443 q^{95}-3951 q^{94}-726 q^{93}+6525 q^{92}+10723 q^{91}+7429 q^{90}-4175 q^{89}-15120 q^{88}-16463 q^{87}-8829 q^{86}+11686 q^{85}+28398 q^{84}+27511 q^{83}+3240 q^{82}-28283 q^{81}-44153 q^{80}-35885 q^{79}+6963 q^{78}+53013 q^{77}+69038 q^{76}+33878 q^{75}-31299 q^{74}-82485 q^{73}-89733 q^{72}-24463 q^{71}+67566 q^{70}+124650 q^{69}+94827 q^{68}-5378 q^{67}-110736 q^{66}-159124 q^{65}-88087 q^{64}+51617 q^{63}+169514 q^{62}+170878 q^{61}+53259 q^{60}-107320 q^{59}-215770 q^{58}-165370 q^{57}+2969 q^{56}+181264 q^{55}+232043 q^{54}+124295 q^{53}-71352 q^{52}-237847 q^{51}-226639 q^{50}-57865 q^{49}+159895 q^{48}+258830 q^{47}+180526 q^{46}-22299 q^{45}-226671 q^{44}-255929 q^{43}-107466 q^{42}+122916 q^{41}+254329 q^{40}+210290 q^{39}+20838 q^{38}-197919 q^{37}-258084 q^{36}-138340 q^{35}+85001 q^{34}+232503 q^{33}+219810 q^{32}+54199 q^{31}-162683 q^{30}-245235 q^{29}-157373 q^{28}+47338 q^{27}+201336 q^{26}+218957 q^{25}+84517 q^{24}-120303 q^{23}-222151 q^{22}-171612 q^{21}+4097 q^{20}+158404 q^{19}+208727 q^{18}+115433 q^{17}-65976 q^{16}-183941 q^{15}-177651 q^{14}-44691 q^{13}+99222 q^{12}+180816 q^{11}+138738 q^{10}-2953 q^9-125398 q^8-163229 q^7-86074 q^6+29317 q^5+128622 q^4+138288 q^3+51737 q^2-53665 q-120386-100518 q^{-1} -31211 q^{-2} +60715 q^{-3} +105510 q^{-4} +76269 q^{-5} +8646 q^{-6} -59455 q^{-7} -79642 q^{-8} -59587 q^{-9} +1761 q^{-10} +53025 q^{-11} +63614 q^{-12} +38668 q^{-13} -6747 q^{-14} -38361 q^{-15} -51131 q^{-16} -26219 q^{-17} +8138 q^{-18} +31017 q^{-19} +34299 q^{-20} +17373 q^{-21} -3803 q^{-22} -24902 q^{-23} -23742 q^{-24} -11308 q^{-25} +4420 q^{-26} +15214 q^{-27} +15812 q^{-28} +9456 q^{-29} -4507 q^{-30} -9950 q^{-31} -10160 q^{-32} -5012 q^{-33} +1515 q^{-34} +6152 q^{-35} +7515 q^{-36} +2307 q^{-37} -919 q^{-38} -3667 q^{-39} -3731 q^{-40} -2189 q^{-41} +469 q^{-42} +2720 q^{-43} +1729 q^{-44} +1151 q^{-45} -279 q^{-46} -1025 q^{-47} -1360 q^{-48} -625 q^{-49} +458 q^{-50} +384 q^{-51} +612 q^{-52} +269 q^{-53} -387 q^{-55} -305 q^{-56} +13 q^{-57} -27 q^{-58} +134 q^{-59} +108 q^{-60} +88 q^{-61} -67 q^{-62} -71 q^{-63} +2 q^{-64} -33 q^{-65} +12 q^{-66} +15 q^{-67} +32 q^{-68} -9 q^{-69} -14 q^{-70} +6 q^{-71} -7 q^{-72} +5 q^{-75} -3 q^{-77} + q^{-78} }
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^{175}-3 q^{174}+2 q^{173}+2 q^{172}-5 q^{171}+4 q^{170}+q^{169}+3 q^{168}-3 q^{167}-19 q^{166}+17 q^{165}+12 q^{164}-20 q^{163}+12 q^{162}+3 q^{161}+13 q^{160}-18 q^{159}-78 q^{158}+56 q^{157}+67 q^{156}-14 q^{155}+39 q^{154}-35 q^{153}-21 q^{152}-111 q^{151}-247 q^{150}+152 q^{149}+325 q^{148}+266 q^{147}+262 q^{146}-232 q^{145}-506 q^{144}-785 q^{143}-920 q^{142}+363 q^{141}+1475 q^{140}+2037 q^{139}+1839 q^{138}-467 q^{137}-2745 q^{136}-4522 q^{135}-4567 q^{134}-245 q^{133}+5220 q^{132}+9665 q^{131}+10101 q^{130}+2561 q^{129}-8323 q^{128}-18334 q^{127}-21216 q^{126}-9528 q^{125}+10914 q^{124}+32221 q^{123}+41491 q^{122}+25211 q^{121}-10127 q^{120}-51064 q^{119}-74442 q^{118}-56212 q^{117}-608 q^{116}+72262 q^{115}+123499 q^{114}+110369 q^{113}+30341 q^{112}-89468 q^{111}-187950 q^{110}-194369 q^{109}-91202 q^{108}+90580 q^{107}+262071 q^{106}+312183 q^{105}+195053 q^{104}-60419 q^{103}-333639 q^{102}-459859 q^{101}-349449 q^{100}-17771 q^{99}+383829 q^{98}+625054 q^{97}+554478 q^{96}+156798 q^{95}-391708 q^{94}-787693 q^{93}-798981 q^{92}-360028 q^{91}+339456 q^{90}+921759 q^{89}+1060754 q^{88}+620370 q^{87}-216558 q^{86}-1004231 q^{85}-1313097 q^{84}-916642 q^{83}+27372 q^{82}+1017968 q^{81}+1525433 q^{80}+1221199 q^{79}+214803 q^{78}-959249 q^{77}-1677609 q^{76}-1503543 q^{75}-482798 q^{74}+836629 q^{73}+1756491 q^{72}+1737910 q^{71}+750054 q^{70}-668316 q^{69}-1765174 q^{68}-1909362 q^{67}-990367 q^{66}+478970 q^{65}+1715130 q^{64}+2013147 q^{63}+1186804 q^{62}-290062 q^{61}-1624327 q^{60}-2057064 q^{59}-1333116 q^{58}+119079 q^{57}+1512667 q^{56}+2053620 q^{55}+1430994 q^{54}+26504 q^{53}-1393841 q^{52}-2018872 q^{51}-1491657 q^{50}-146548 q^{49}+1278259 q^{48}+1966185 q^{47}+1525924 q^{46}+247682 q^{45}-1165838 q^{44}-1904384 q^{43}-1547603 q^{42}-339934 q^{41}+1054432 q^{40}+1836659 q^{39}+1563270 q^{38}+433405 q^{37}-934402 q^{36}-1760650 q^{35}-1578328 q^{34}-535662 q^{33}+799050 q^{32}+1669468 q^{31}+1588648 q^{30}+650127 q^{29}-638978 q^{28}-1554650 q^{27}-1589201 q^{26}-773601 q^{25}+452900 q^{24}+1406920 q^{23}+1566532 q^{22}+898023 q^{21}-240689 q^{20}-1220620 q^{19}-1510509 q^{18}-1009004 q^{17}+13846 q^{16}+994613 q^{15}+1407490 q^{14}+1089286 q^{13}+214501 q^{12}-735297 q^{11}-1253458 q^{10}-1122016 q^9-421402 q^8+458102 q^7+1047594 q^6+1093289 q^5+586349 q^4-183526 q^3-802844 q^2-999366 q-688534-61652 q^{-1} +538408 q^{-2} +844839 q^{-3} +717559 q^{-4} +254425 q^{-5} -281280 q^{-6} -648043 q^{-7} -673235 q^{-8} -376793 q^{-9} +58547 q^{-10} +434006 q^{-11} +568182 q^{-12} +423484 q^{-13} +108117 q^{-14} -231142 q^{-15} -425527 q^{-16} -402092 q^{-17} -207529 q^{-18} +64525 q^{-19} +272883 q^{-20} +330682 q^{-21} +241488 q^{-22} +51003 q^{-23} -135388 q^{-24} -234540 q^{-25} -223883 q^{-26} -111853 q^{-27} +31530 q^{-28} +137589 q^{-29} +174030 q^{-30} +126180 q^{-31} +32687 q^{-32} -57695 q^{-33} -113998 q^{-34} -109320 q^{-35} -59734 q^{-36} +4273 q^{-37} +59446 q^{-38} +77364 q^{-39} +60427 q^{-40} +23421 q^{-41} -19957 q^{-42} -44848 q^{-43} -46742 q^{-44} -30772 q^{-45} -2624 q^{-46} +19304 q^{-47} +29060 q^{-48} +26577 q^{-49} +11643 q^{-50} -3586 q^{-51} -14239 q^{-52} -18098 q^{-53} -12038 q^{-54} -3445 q^{-55} +4502 q^{-56} +9813 q^{-57} +8779 q^{-58} +5232 q^{-59} +428 q^{-60} -4316 q^{-61} -5140 q^{-62} -4126 q^{-63} -1847 q^{-64} +1118 q^{-65} +2202 q^{-66} +2606 q^{-67} +1934 q^{-68} +94 q^{-69} -822 q^{-70} -1286 q^{-71} -1165 q^{-72} -354 q^{-73} -12 q^{-74} +464 q^{-75} +739 q^{-76} +340 q^{-77} +92 q^{-78} -184 q^{-79} -290 q^{-80} -119 q^{-81} -142 q^{-82} -34 q^{-83} +142 q^{-84} +103 q^{-85} +74 q^{-86} -9 q^{-87} -56 q^{-88} +6 q^{-89} -33 q^{-90} -33 q^{-91} +12 q^{-92} +15 q^{-93} +23 q^{-94} -14 q^{-96} +6 q^{-97} -7 q^{-99} +5 q^{-102} -3 q^{-104} + q^{-105} }

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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