10 76: Difference between revisions

Revision as of 17:53, 31 August 2005

10_75

10_77

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 76's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 76 at Knotilus!

[edit 10 76 Quick Notes]

[edit 10 76 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_14,10,15,9 X_12,3,13,4 X_2,13,3,14 X_18,6,19,5 X_20,8,1,7 X_6,20,7,19 X_8,18,9,17 X_16,12,17,11 X_10,16,11,15
Gauss code	1, -4, 3, -1, 5, -7, 6, -8, 2, -10, 9, -3, 4, -2, 10, -9, 8, -5, 7, -6
Dowker-Thistlethwaite code	4 12 18 20 14 16 2 10 8 6
Conway Notation	[3,3,2++]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 4,

Braid index is 4

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

[edit Notes on presentations of 10 76]

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 76"];

In[4]:= PD[K]

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

Out[4]= X₄₂₅₁ X_14,10,15,9 X_12,3,13,4 X_2,13,3,14 X_18,6,19,5 X_20,8,1,7 X_6,20,7,19 X_8,18,9,17 X_16,12,17,11 X_10,16,11,15

In[5]:= GaussCode[K]

Out[5]= 1, -4, 3, -1, 5, -7, 6, -8, 2, -10, 9, -3, 4, -2, 10, -9, 8, -5, 7, -6

In[6]:= DTCode[K]

Out[6]= 4 12 18 20 14 16 2 10 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [3,3,2++]

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}(4,\{1,1,1,1,2,-1,-3,2,2,2,-3\})}

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]= { 4, 11, 4 }

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

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{2,3\}}
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[1][-13]
Hyperbolic Volume	11.5129
A-Polynomial	See Data:10 76/A-polynomial

[edit Notes for 10 76's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$2$
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 2}
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	-4

[edit Notes for 10 76'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 -2 t^3+7 t^2-12 t+15-12 t^{-1} +7 t^{-2} -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 -2 z^6-5 z^4-2 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 57, 4 }
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^{10}-3 q^9+6 q^8-8 q^7+9 q^6-10 q^5+8 q^4-6 q^3+4 q^2-q+1}
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 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -4 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +4 z^2 a^{-2} -6 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2 a^{-8} +4 a^{-2} -4 a^{-4} + a^{-8} }
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}a^{-8}+z^{7}a^{-3}-2z^{7}a^{-5}+2z^{7}a^{-7}+5z^{7}a^{-9}+z^{6}a^{-2}-9z^{6}a^{-6}-3z^{6}a^{-8}+5z^{6}a^{-10}-2z^{5}a^{-3}+7z^{5}a^{-5}-2z^{5}a^{-7}-8z^{5}a^{-9}+3z^{5}a^{-11}-5z^{4}a^{-2}-7z^{4}a^{-4}+10z^{4}a^{-6}+4z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}-2z^{3}a^{-3}-15z^{3}a^{-5}-3z^{3}a^{-7}+7z^{3}a^{-9}-3z^{3}a^{-11}+8z^{2}a^{-2}+9z^{2}a^{-4}-7z^{2}a^{-6}-4z^{2}a^{-8}+3z^{2}a^{-10}-z^{2}a^{-12}+4za^{-3}+8za^{-5}+2za^{-7}-2za^{-9}-4a^{-2}-4a^{-4}+a^{-8}$
The A2 invariant	$1+q^{-2}+2q^{-4}+3q^{-6}-q^{-8}+q^{-10}-3q^{-12}-2q^{-14}-2q^{-18}+2q^{-20}-q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-30}$
The G2 invariant	$q^{-2}+3q^{-6}-2q^{-8}+3q^{-10}-q^{-12}+7q^{-16}-9q^{-18}+14q^{-20}-12q^{-22}+12q^{-24}+q^{-26}-12q^{-28}+31q^{-30}-40q^{-32}+46q^{-34}-33q^{-36}+q^{-38}+33q^{-40}-65q^{-42}+80q^{-44}-67q^{-46}+28q^{-48}+17q^{-50}-59q^{-52}+71q^{-54}-63q^{-56}+24q^{-58}+17q^{-60}-50q^{-62}+46q^{-64}-24q^{-66}-19q^{-68}+58q^{-70}-75q^{-72}+60q^{-74}-21q^{-76}-35q^{-78}+87q^{-80}-115q^{-82}+106q^{-84}-59q^{-86}-3q^{-88}+66q^{-90}-101q^{-92}+104q^{-94}-68q^{-96}+17q^{-98}+32q^{-100}-60q^{-102}+55q^{-104}-22q^{-106}-19q^{-108}+51q^{-110}-52q^{-112}+28q^{-114}+11q^{-116}-53q^{-118}+76q^{-120}-74q^{-122}+48q^{-124}-8q^{-126}-33q^{-128}+60q^{-130}-64q^{-132}+54q^{-134}-28q^{-136}+3q^{-138}+15q^{-140}-29q^{-142}+28q^{-144}-21q^{-146}+12q^{-148}-2q^{-150}-3q^{-152}+5q^{-154}-6q^{-156}+4q^{-158}-2q^{-160}+q^{-162}$

Further Quantum Invariants

Further quantum knot invariants for 10_76.

A1 Invariants.

Weight	Invariant
1	$q+3q^{-3}-2q^{-5}+2q^{-7}-2q^{-9}-q^{-11}+q^{-13}-2q^{-15}+3q^{-17}-2q^{-19}+q^{-21}$
2	$q^{6}+3-q^{-2}-3q^{-4}+8q^{-6}-2q^{-8}-11q^{-10}+12q^{-12}+3q^{-14}-18q^{-16}+9q^{-18}+9q^{-20}-14q^{-22}+2q^{-24}+10q^{-26}-3q^{-28}-7q^{-30}+4q^{-32}+10q^{-34}-12q^{-36}-4q^{-38}+18q^{-40}-11q^{-42}-9q^{-44}+15q^{-46}-4q^{-48}-7q^{-50}+6q^{-52}-2q^{-56}+q^{-58}$
3	$q^{15}+2q^{7}-q^{5}-q^{3}+q+6q^{-1}-3q^{-3}-10q^{-5}+21q^{-9}+3q^{-11}-29q^{-13}-18q^{-15}+36q^{-17}+37q^{-19}-36q^{-21}-54q^{-23}+25q^{-25}+73q^{-27}-9q^{-29}-78q^{-31}-12q^{-33}+79q^{-35}+25q^{-37}-67q^{-39}-43q^{-41}+58q^{-43}+47q^{-45}-37q^{-47}-50q^{-49}+17q^{-51}+50q^{-53}+4q^{-55}-46q^{-57}-31q^{-59}+38q^{-61}+52q^{-63}-23q^{-65}-74q^{-67}+10q^{-69}+83q^{-71}+9q^{-73}-82q^{-75}-22q^{-77}+70q^{-79}+33q^{-81}-53q^{-83}-34q^{-85}+34q^{-87}+26q^{-89}-17q^{-91}-20q^{-93}+9q^{-95}+13q^{-97}-5q^{-99}-6q^{-101}+q^{-103}+3q^{-105}-2q^{-109}+q^{-111}$
4	$q^{28}-q^{20}+2q^{18}-q^{16}+3q^{12}-2q^{10}+3q^{8}-6q^{6}-5q^{4}+8q^{2}+6+15q^{-2}-17q^{-4}-32q^{-6}-5q^{-8}+22q^{-10}+71q^{-12}+3q^{-14}-76q^{-16}-79q^{-18}-15q^{-20}+156q^{-22}+118q^{-24}-45q^{-26}-187q^{-28}-182q^{-30}+141q^{-32}+267q^{-34}+135q^{-36}-175q^{-38}-382q^{-40}-48q^{-42}+278q^{-44}+354q^{-46}+10q^{-48}-433q^{-50}-266q^{-52}+125q^{-54}+433q^{-56}+213q^{-58}-319q^{-60}-360q^{-62}-48q^{-64}+365q^{-66}+301q^{-68}-162q^{-70}-333q^{-72}-148q^{-74}+237q^{-76}+295q^{-78}-9q^{-80}-255q^{-82}-214q^{-84}+84q^{-86}+257q^{-88}+167q^{-90}-131q^{-92}-277q^{-94}-132q^{-96}+171q^{-98}+360q^{-100}+64q^{-102}-282q^{-104}-362q^{-106}-6q^{-108}+447q^{-110}+278q^{-112}-145q^{-114}-456q^{-116}-212q^{-118}+341q^{-120}+350q^{-122}+53q^{-124}-333q^{-126}-287q^{-128}+136q^{-130}+234q^{-132}+143q^{-134}-135q^{-136}-194q^{-138}+18q^{-140}+76q^{-142}+98q^{-144}-26q^{-146}-78q^{-148}+q^{-150}+8q^{-152}+39q^{-154}-5q^{-156}-25q^{-158}+5q^{-160}-3q^{-162}+12q^{-164}-7q^{-168}+2q^{-170}-2q^{-172}+3q^{-174}-2q^{-178}+q^{-180}$
5	$q^{45}-q^{37}-q^{35}+2q^{33}+3q^{27}-5q^{23}-2q^{19}-q^{17}+10q^{15}+10q^{13}-4q^{11}-9q^{9}-20q^{7}-18q^{5}+15q^{3}+43q+39q^{-1}+6q^{-3}-60q^{-5}-103q^{-7}-50q^{-9}+64q^{-11}+165q^{-13}+168q^{-15}-4q^{-17}-237q^{-19}-315q^{-21}-143q^{-23}+213q^{-25}+505q^{-27}+420q^{-29}-80q^{-31}-615q^{-33}-751q^{-35}-264q^{-37}+569q^{-39}+1085q^{-41}+751q^{-43}-298q^{-45}-1258q^{-47}-1298q^{-49}-243q^{-51}+1179q^{-53}+1780q^{-55}+920q^{-57}-819q^{-59}-2018q^{-61}-1629q^{-63}+196q^{-65}+1996q^{-67}+2188q^{-69}+519q^{-71}-1668q^{-73}-2508q^{-75}-1203q^{-77}+1184q^{-79}+2542q^{-81}+1713q^{-83}-602q^{-85}-2384q^{-87}-2002q^{-89}+130q^{-91}+2039q^{-93}+2073q^{-95}+269q^{-97}-1690q^{-99}-2005q^{-101}-479q^{-103}+1337q^{-105}+1828q^{-107}+644q^{-109}-1045q^{-111}-1666q^{-113}-744q^{-115}+779q^{-117}+1507q^{-119}+879q^{-121}-482q^{-123}-1387q^{-125}-1092q^{-127}+127q^{-129}+1251q^{-131}+1354q^{-133}+348q^{-135}-1024q^{-137}-1665q^{-139}-924q^{-141}+686q^{-143}+1876q^{-145}+1573q^{-147}-168q^{-149}-1950q^{-151}-2164q^{-153}-479q^{-155}+1770q^{-157}+2602q^{-159}+1171q^{-161}-1359q^{-163}-2742q^{-165}-1762q^{-167}+739q^{-169}+2579q^{-171}+2132q^{-173}-104q^{-175}-2104q^{-177}-2191q^{-179}-454q^{-181}+1477q^{-183}+1971q^{-185}+780q^{-187}-854q^{-189}-1530q^{-191}-867q^{-193}+345q^{-195}+1034q^{-197}+772q^{-199}-47q^{-201}-607q^{-203}-545q^{-205}-93q^{-207}+295q^{-209}+339q^{-211}+114q^{-213}-131q^{-215}-179q^{-217}-69q^{-219}+45q^{-221}+78q^{-223}+38q^{-225}-13q^{-227}-40q^{-229}-14q^{-231}+13q^{-233}+13q^{-235}+3q^{-237}-4q^{-239}-4q^{-241}-7q^{-243}+3q^{-245}+7q^{-247}-q^{-249}-2q^{-251}+q^{-253}-q^{-255}-2q^{-257}+3q^{-259}-2q^{-263}+q^{-265}$

A2 Invariants.

Weight	Invariant
1,0	$1+q^{-2}+2q^{-4}+3q^{-6}-q^{-8}+q^{-10}-3q^{-12}-2q^{-14}-2q^{-18}+2q^{-20}-q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-30}$
1,1	$q^{4}+6-4q^{-2}+18q^{-4}-20q^{-6}+42q^{-8}-62q^{-10}+93q^{-12}-140q^{-14}+182q^{-16}-240q^{-18}+272q^{-20}-302q^{-22}+288q^{-24}-244q^{-26}+164q^{-28}-46q^{-30}-92q^{-32}+246q^{-34}-378q^{-36}+502q^{-38}-576q^{-40}+612q^{-42}-587q^{-44}+512q^{-46}-402q^{-48}+254q^{-50}-107q^{-52}-40q^{-54}+160q^{-56}-244q^{-58}+292q^{-60}-300q^{-62}+282q^{-64}-246q^{-66}+195q^{-68}-144q^{-70}+102q^{-72}-66q^{-74}+38q^{-76}-20q^{-78}+10q^{-80}-4q^{-82}+q^{-84}$
2,0	$q^{4}+q^{2}+1+2q^{-2}+5q^{-4}+3q^{-6}+q^{-10}+5q^{-12}-4q^{-14}-10q^{-16}-2q^{-18}-9q^{-22}-6q^{-24}+6q^{-26}+5q^{-28}-2q^{-30}+4q^{-32}+9q^{-34}-3q^{-36}-q^{-38}+7q^{-40}-6q^{-44}+2q^{-46}+4q^{-48}-6q^{-50}-5q^{-52}+5q^{-54}+3q^{-56}-6q^{-58}-q^{-60}+4q^{-62}-q^{-64}-2q^{-66}+2q^{-70}-q^{-74}+q^{-76}$

A3 Invariants.

Weight	Invariant
0,1,0	$1+3q^{-4}+4q^{-6}+2q^{-8}+6q^{-10}+7q^{-12}-7q^{-14}+2q^{-16}-20q^{-20}+4q^{-24}-14q^{-26}+5q^{-28}+12q^{-30}-4q^{-32}+q^{-34}+5q^{-36}+3q^{-38}-5q^{-40}-3q^{-42}+11q^{-44}-5q^{-46}-9q^{-48}+14q^{-50}-4q^{-52}-11q^{-54}+11q^{-56}-q^{-58}-6q^{-60}+5q^{-62}-2q^{-66}+q^{-68}$
1,0,0	$q^{-1}+q^{-3}+3q^{-5}+2q^{-7}+4q^{-9}-q^{-11}+q^{-13}-4q^{-15}-2q^{-17}-3q^{-19}-q^{-21}-q^{-25}+2q^{-27}-q^{-29}+2q^{-31}-q^{-33}+2q^{-35}-q^{-37}+q^{-39}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}+q^{-4}+3q^{-6}+5q^{-8}+7q^{-10}+7q^{-12}+9q^{-14}+8q^{-16}+4q^{-18}-4q^{-20}-4q^{-22}-10q^{-24}-21q^{-26}-16q^{-28}-4q^{-30}-10q^{-32}-9q^{-34}+12q^{-36}+15q^{-38}+3q^{-40}+4q^{-42}+17q^{-44}+3q^{-46}-9q^{-48}+4q^{-50}+6q^{-52}-11q^{-54}-2q^{-56}+10q^{-58}-5q^{-60}-8q^{-62}+5q^{-64}+4q^{-66}-9q^{-68}-3q^{-70}+8q^{-72}+q^{-74}-6q^{-76}+2q^{-78}+4q^{-80}-2q^{-82}-q^{-84}+q^{-86}$
1,0,0,0	$q^{-2}+q^{-4}+3q^{-6}+3q^{-8}+3q^{-10}+4q^{-12}-q^{-14}+q^{-16}-4q^{-18}-3q^{-20}-3q^{-22}-3q^{-24}-q^{-26}-q^{-28}+q^{-30}-q^{-32}+2q^{-34}-q^{-36}+2q^{-38}+2q^{-44}-q^{-46}+q^{-48}$

B2 Invariants.

Weight

Invariant

0,1

1+3q^{-4}-2q^{-6}+6q^{-8}-6q^{-10}+11q^{-12}-11q^{-14}+14q^{-16}-14q^{-18}+12q^{-20}-10q^{-22}+2q^{-24}+2q^{-26}-11q^{-28}+16q^{-30}-24q^{-32}+27q^{-34}-29q^{-36}+27q^{-38}-23q^{-40}+17q^{-42}-9q^{-44}+3q^{-46}+5q^{-48}-10q^{-50}+14q^{-52}-15q^{-54}+15q^{-56}-13q^{-58}+10q^{-60}-7q^{-62}+4q^{-64}-2q^{-66}+q^{-68}

1,0

q^{2}+3q^{-6}+3q^{-8}+q^{-10}-2q^{-12}+q^{-14}+6q^{-16}+8q^{-18}-3q^{-20}-10q^{-22}-3q^{-24}+11q^{-26}+7q^{-28}-12q^{-30}-18q^{-32}-2q^{-34}+14q^{-36}+4q^{-38}-13q^{-40}-11q^{-42}+7q^{-44}+12q^{-46}-9q^{-50}+2q^{-52}+11q^{-54}+3q^{-56}-9q^{-58}-3q^{-60}+9q^{-62}+7q^{-64}-8q^{-66}-10q^{-68}+6q^{-70}+12q^{-72}-2q^{-74}-15q^{-76}-4q^{-78}+14q^{-80}+11q^{-82}-9q^{-84}-15q^{-86}+13q^{-90}+6q^{-92}-7q^{-94}-8q^{-96}+q^{-98}+6q^{-100}+2q^{-102}-2q^{-104}-2q^{-106}+q^{-110}

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} +3 q^{-6} + q^{-8} +7 q^{-10} + q^{-12} +10 q^{-14} - q^{-16} +12 q^{-18} -9 q^{-20} +8 q^{-22} -15 q^{-24} +5 q^{-26} -18 q^{-28} -11 q^{-32} -6 q^{-38} +12 q^{-40} -8 q^{-42} +20 q^{-44} -19 q^{-46} +22 q^{-48} -20 q^{-50} +23 q^{-52} -20 q^{-54} +16 q^{-56} -14 q^{-58} +13 q^{-60} -4 q^{-62} + q^{-64} -5 q^{-68} +10 q^{-70} -11 q^{-72} +9 q^{-74} -13 q^{-76} +13 q^{-78} -9 q^{-80} +8 q^{-82} -8 q^{-84} +6 q^{-86} -3 q^{-88} +2 q^{-90} -2 q^{-92} + q^{-94} }

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^{-2} +3 q^{-6} -2 q^{-8} +3 q^{-10} - q^{-12} +7 q^{-16} -9 q^{-18} +14 q^{-20} -12 q^{-22} +12 q^{-24} + q^{-26} -12 q^{-28} +31 q^{-30} -40 q^{-32} +46 q^{-34} -33 q^{-36} + q^{-38} +33 q^{-40} -65 q^{-42} +80 q^{-44} -67 q^{-46} +28 q^{-48} +17 q^{-50} -59 q^{-52} +71 q^{-54} -63 q^{-56} +24 q^{-58} +17 q^{-60} -50 q^{-62} +46 q^{-64} -24 q^{-66} -19 q^{-68} +58 q^{-70} -75 q^{-72} +60 q^{-74} -21 q^{-76} -35 q^{-78} +87 q^{-80} -115 q^{-82} +106 q^{-84} -59 q^{-86} -3 q^{-88} +66 q^{-90} -101 q^{-92} +104 q^{-94} -68 q^{-96} +17 q^{-98} +32 q^{-100} -60 q^{-102} +55 q^{-104} -22 q^{-106} -19 q^{-108} +51 q^{-110} -52 q^{-112} +28 q^{-114} +11 q^{-116} -53 q^{-118} +76 q^{-120} -74 q^{-122} +48 q^{-124} -8 q^{-126} -33 q^{-128} +60 q^{-130} -64 q^{-132} +54 q^{-134} -28 q^{-136} +3 q^{-138} +15 q^{-140} -29 q^{-142} +28 q^{-144} -21 q^{-146} +12 q^{-148} -2 q^{-150} -3 q^{-152} +5 q^{-154} -6 q^{-156} +4 q^{-158} -2 q^{-160} + q^{-162} }

.

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.

In[3]:= K = Knot["10 76"];

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 -2 t^3+7 t^2-12 t+15-12 t^{-1} +7 t^{-2} -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 -2 z^6-5 z^4-2 z^2+1}

In[6]:= Alexander[K, 2][t]

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

Out[6]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 57, 4 }

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^{10}-3 q^9+6 q^8-8 q^7+9 q^6-10 q^5+8 q^4-6 q^3+4 q^2-q+1}

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

In[10]:= Kauffman[K][a, z]

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

Out[10]= $z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}a^{-8}+z^{7}a^{-3}-2z^{7}a^{-5}+2z^{7}a^{-7}+5z^{7}a^{-9}+z^{6}a^{-2}-9z^{6}a^{-6}-3z^{6}a^{-8}+5z^{6}a^{-10}-2z^{5}a^{-3}+7z^{5}a^{-5}-2z^{5}a^{-7}-8z^{5}a^{-9}+3z^{5}a^{-11}-5z^{4}a^{-2}-7z^{4}a^{-4}+10z^{4}a^{-6}+4z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}-2z^{3}a^{-3}-15z^{3}a^{-5}-3z^{3}a^{-7}+7z^{3}a^{-9}-3z^{3}a^{-11}+8z^{2}a^{-2}+9z^{2}a^{-4}-7z^{2}a^{-6}-4z^{2}a^{-8}+3z^{2}a^{-10}-z^{2}a^{-12}+4za^{-3}+8za^{-5}+2za^{-7}-2za^{-9}-4a^{-2}-4a^{-4}+a^{-8}$

"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.

In[3]:= K = Knot["10 76"];

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 -2 t^3+7 t^2-12 t+15-12 t^{-1} +7 t^{-2} -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^{10}-3 q^9+6 q^8-8 q^7+9 q^6-10 q^5+8 q^4-6 q^3+4 q^2-q+1} }

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:

(-2, -6)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -8}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -48}

Failed to parse (SVG (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}

-{\frac {124}{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{172}{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 384}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 672}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 256}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 272}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{256}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1152}

Failed to parse (SVG (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{992}{3}}

-{\frac {1376}{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{56369}{15}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2468}{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{53396}{45}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3535}{9}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1711}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (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

s=

4 is the signature of 10 76. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-2

-1

0

1

2

3

4

5

6

7

8

χ

21

1

19

2

-2

17

4

1

3

15

4

2

-2

13

5

4

1

11

5

4

-1

9

3

5

-2

7

3

5

2

5

1

3

-2

3

1

4

3

1

0

-1

1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=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=-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=-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}_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=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}^{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=1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=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^{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=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^{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}^{4}\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=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}^{4}\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=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}\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=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 {\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=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}_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^{28}-3 q^{27}+2 q^{26}+7 q^{25}-16 q^{24}+5 q^{23}+26 q^{22}-40 q^{21}+3 q^{20}+55 q^{19}-62 q^{18}-5 q^{17}+77 q^{16}-68 q^{15}-16 q^{14}+81 q^{13}-55 q^{12}-24 q^{11}+65 q^{10}-32 q^9-24 q^8+38 q^7-11 q^6-15 q^5+15 q^4-2 q^3-5 q^2+4 q- q^{-1} + q^{-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 q^{54}-3 q^{53}+2 q^{52}+3 q^{51}-q^{50}-10 q^{49}+3 q^{48}+21 q^{47}-5 q^{46}-39 q^{45}+6 q^{44}+64 q^{43}+3 q^{42}-107 q^{41}-13 q^{40}+150 q^{39}+40 q^{38}-199 q^{37}-73 q^{36}+241 q^{35}+114 q^{34}-272 q^{33}-157 q^{32}+292 q^{31}+189 q^{30}-286 q^{29}-226 q^{28}+277 q^{27}+239 q^{26}-240 q^{25}-259 q^{24}+210 q^{23}+252 q^{22}-156 q^{21}-248 q^{20}+109 q^{19}+228 q^{18}-64 q^{17}-194 q^{16}+18 q^{15}+162 q^{14}+5 q^{13}-112 q^{12}-30 q^{11}+83 q^{10}+23 q^9-39 q^8-31 q^7+29 q^6+12 q^5-7 q^4-13 q^3+8 q^2+2 q-4 q^{-1} +3 q^{-2} - q^{-5} + q^{-6} }
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^{88}-3 q^{87}+2 q^{86}+3 q^{85}-5 q^{84}+5 q^{83}-12 q^{82}+9 q^{81}+15 q^{80}-20 q^{79}+13 q^{78}-42 q^{77}+29 q^{76}+59 q^{75}-51 q^{74}+6 q^{73}-121 q^{72}+81 q^{71}+183 q^{70}-73 q^{69}-52 q^{68}-333 q^{67}+140 q^{66}+461 q^{65}+18 q^{64}-150 q^{63}-756 q^{62}+94 q^{61}+847 q^{60}+315 q^{59}-159 q^{58}-1309 q^{57}-150 q^{56}+1158 q^{55}+738 q^{54}+10 q^{53}-1762 q^{52}-506 q^{51}+1238 q^{50}+1084 q^{49}+306 q^{48}-1951 q^{47}-809 q^{46}+1093 q^{45}+1230 q^{44}+604 q^{43}-1861 q^{42}-982 q^{41}+795 q^{40}+1189 q^{39}+850 q^{38}-1557 q^{37}-1040 q^{36}+410 q^{35}+1004 q^{34}+1021 q^{33}-1094 q^{32}-976 q^{31}-3 q^{30}+692 q^{29}+1062 q^{28}-562 q^{27}-756 q^{26}-311 q^{25}+301 q^{24}+895 q^{23}-119 q^{22}-412 q^{21}-387 q^{20}-25 q^{19}+561 q^{18}+88 q^{17}-102 q^{16}-255 q^{15}-151 q^{14}+238 q^{13}+83 q^{12}+40 q^{11}-92 q^{10}-113 q^9+67 q^8+19 q^7+43 q^6-13 q^5-45 q^4+18 q^3-8 q^2+16 q+2-13 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} + q^{-5} -4 q^{-6} +4 q^{-7} - q^{-8} - q^{-11} + q^{-12} }
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 \textrm{NotAvailable}(q)}
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 \textrm{NotAvailable}(q)}
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 \textrm{NotAvailable}(q)}

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.

10_75

10_77

@@ Line 1: / Line 1: @@
+<!--                       WARNING! WARNING! WARNING!
-<!-- This page was  generated from the splice template "Rolfsen_Splice_Template". Please do not edit! -->
+<!-- This page was generated from the splice template [[Rolfsen_Splice_Base]]. Please do not edit!
-<!--  --> <!--
+<!-- You probably want to edit the template referred to immediately below. (See [[Category:Knot Page Template]].)
- -->
+<!-- This page itself was created by running [[Media:KnotPageSpliceRobot.nb]] on [[Rolfsen_Splice_Base]]. -->
+<!-- <math>\text{Null}</math> -->
+<!-- <math>\text{Null}</math> -->
 {{Rolfsen Knot Page|
 n = 10 |
@@ Line 41: / Line 44: @@
 coloured_jones_3 = <math>q^{54}-3 q^{53}+2 q^{52}+3 q^{51}-q^{50}-10 q^{49}+3 q^{48}+21 q^{47}-5 q^{46}-39 q^{45}+6 q^{44}+64 q^{43}+3 q^{42}-107 q^{41}-13 q^{40}+150 q^{39}+40 q^{38}-199 q^{37}-73 q^{36}+241 q^{35}+114 q^{34}-272 q^{33}-157 q^{32}+292 q^{31}+189 q^{30}-286 q^{29}-226 q^{28}+277 q^{27}+239 q^{26}-240 q^{25}-259 q^{24}+210 q^{23}+252 q^{22}-156 q^{21}-248 q^{20}+109 q^{19}+228 q^{18}-64 q^{17}-194 q^{16}+18 q^{15}+162 q^{14}+5 q^{13}-112 q^{12}-30 q^{11}+83 q^{10}+23 q^9-39 q^8-31 q^7+29 q^6+12 q^5-7 q^4-13 q^3+8 q^2+2 q-4 q^{-1} +3 q^{-2} - q^{-5} + q^{-6} </math> |
 coloured_jones_4 = <math>q^{88}-3 q^{87}+2 q^{86}+3 q^{85}-5 q^{84}+5 q^{83}-12 q^{82}+9 q^{81}+15 q^{80}-20 q^{79}+13 q^{78}-42 q^{77}+29 q^{76}+59 q^{75}-51 q^{74}+6 q^{73}-121 q^{72}+81 q^{71}+183 q^{70}-73 q^{69}-52 q^{68}-333 q^{67}+140 q^{66}+461 q^{65}+18 q^{64}-150 q^{63}-756 q^{62}+94 q^{61}+847 q^{60}+315 q^{59}-159 q^{58}-1309 q^{57}-150 q^{56}+1158 q^{55}+738 q^{54}+10 q^{53}-1762 q^{52}-506 q^{51}+1238 q^{50}+1084 q^{49}+306 q^{48}-1951 q^{47}-809 q^{46}+1093 q^{45}+1230 q^{44}+604 q^{43}-1861 q^{42}-982 q^{41}+795 q^{40}+1189 q^{39}+850 q^{38}-1557 q^{37}-1040 q^{36}+410 q^{35}+1004 q^{34}+1021 q^{33}-1094 q^{32}-976 q^{31}-3 q^{30}+692 q^{29}+1062 q^{28}-562 q^{27}-756 q^{26}-311 q^{25}+301 q^{24}+895 q^{23}-119 q^{22}-412 q^{21}-387 q^{20}-25 q^{19}+561 q^{18}+88 q^{17}-102 q^{16}-255 q^{15}-151 q^{14}+238 q^{13}+83 q^{12}+40 q^{11}-92 q^{10}-113 q^9+67 q^8+19 q^7+43 q^6-13 q^5-45 q^4+18 q^3-8 q^2+16 q+2-13 q^{-1} +9 q^{-2} -6 q^{-3} +3 q^{-4} + q^{-5} -4 q^{-6} +4 q^{-7} - q^{-8} - q^{-11} + q^{-12} </math> |
-coloured_jones_5 =  |
+coloured_jones_5 = <math>\textrm{NotAvailable}(q)</math> |
-coloured_jones_6 =  |
+coloured_jones_6 = <math>\textrm{NotAvailable}(q)</math> |
-coloured_jones_7 =  |
+coloured_jones_7 = <math>\textrm{NotAvailable}(q)</math> |
 computer_talk =
          <table>
@@ Line 50: / Line 53: @@
          <td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
          </tr>
-         <tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</td></tr>
+         <tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:27:48)...</td></tr>
          <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 76]]</nowiki></pre></td></tr>
 <tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[4, 2, 5, 1], X[14, 10, 15, 9], X[12, 3, 13, 4], X[2, 13, 3, 14],
@@ Line 70: / Line 73: @@
 <tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>4</nowiki></pre></td></tr>
          <tr  valign=top><td><pre style="color: blue; border: 0px; padding:  0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red;  border: 0px; padding:  0em"><nowiki>Show[DrawMorseLink[Knot[10, 76]]]</nowiki></pre></td></tr><tr><td></td><td  align=left>[[Image:10_76_ML.gif]]</td></tr><tr valign=top><td><tt><font  color=blue>Out[8]=</font></tt><td><tt><font  color=black>-Graphics-</font></tt></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>(#[Knot[10, 76]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr>
+         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki> (#[Knot[10, 76]]&) /@ {
+                   SymmetryType, UnknottingNumber, ThreeGenus,
+                   BridgeIndex, SuperBridgeIndex, NakanishiIndex
+                  }</nowiki></pre></td></tr>
 <tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, {2, 3}, 3, 3, NotAvailable, 1}</nowiki></pre></td></tr>
          <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[10, 76]][t]</nowiki></pre></td></tr>

10 76: Difference between revisions

Revision as of 17:53, 31 August 2005

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