9 16

9_15

9_17

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 16's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 16 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X16,6,17,5 X18,8,1,7 X6,18,7,17 X10,16,11,15 X14,10,15,9 X8,14,9,13 X2,12,3,11
Gauss code	1, -9, 2, -1, 3, -5, 4, -8, 7, -6, 9, -2, 8, -7, 6, -3, 5, -4
Dowker-Thistlethwaite code	4 12 16 18 14 2 8 10 6
Conway Notation	[3,3,2+]

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

[edit Notes on presentations of 9 16]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 16"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X12,4,13,3 X16,6,17,5 X18,8,1,7 X6,18,7,17 X10,16,11,15 X14,10,15,9 X8,14,9,13 X2,12,3,11

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 16 18 14 2 8 10 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}(3,\{1,1,1,1,2,2,-1,2,2,2\})}

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 3, 10, 3 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,7\}} Nakanishi index 1 Maximal Thurston-Bennequin number [5][-16] Hyperbolic Volume 9.88301 A-Polynomial See Data:9 16/A-polynomial

[edit Notes for 9 16's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} 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 3} Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for 9 16's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-5t^{2}+8t-9+8t^{-1}-5t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+7z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 39, 6 }
Jones polynomial	$-q^{12}+3q^{11}-5q^{10}+6q^{9}-7q^{8}+6q^{7}-5q^{6}+4q^{5}-q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+z^{6}a^{-8}+5z^{4}a^{-6}+3z^{4}a^{-8}-z^{4}a^{-10}+8z^{2}a^{-6}-2z^{2}a^{-10}+4a^{-6}-3a^{-8}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-7}+4z^{7}a^{-9}+3z^{7}a^{-11}+z^{6}a^{-6}-z^{6}a^{-8}+3z^{6}a^{-10}+5z^{6}a^{-12}-2z^{5}a^{-7}-8z^{5}a^{-9}-z^{5}a^{-11}+5z^{5}a^{-13}-5z^{4}a^{-6}-4z^{4}a^{-8}-8z^{4}a^{-10}-6z^{4}a^{-12}+3z^{4}a^{-14}-2z^{3}a^{-7}-z^{3}a^{-9}-5z^{3}a^{-11}-5z^{3}a^{-13}+z^{3}a^{-15}+8z^{2}a^{-6}+6z^{2}a^{-8}+z^{2}a^{-10}+2z^{2}a^{-12}-z^{2}a^{-14}+4za^{-7}+4za^{-9}+2za^{-11}+2za^{-13}-4a^{-6}-3a^{-8}$
The A2 invariant	$q^{-10}+3q^{-14}+q^{-16}+2q^{-18}+q^{-20}-2q^{-22}-3q^{-26}+q^{-34}-q^{-36}$
The G2 invariant	$q^{-50}+3q^{-54}-2q^{-56}+3q^{-58}-q^{-60}+8q^{-64}-11q^{-66}+16q^{-68}-11q^{-70}+6q^{-72}+11q^{-74}-21q^{-76}+32q^{-78}-26q^{-80}+18q^{-82}+q^{-84}-23q^{-86}+33q^{-88}-31q^{-90}+19q^{-92}-18q^{-96}+22q^{-98}-18q^{-100}+11q^{-104}-24q^{-106}+22q^{-108}-14q^{-110}-8q^{-112}+27q^{-114}-40q^{-116}+41q^{-118}-27q^{-120}+2q^{-122}+22q^{-124}-40q^{-126}+46q^{-128}-36q^{-130}+16q^{-132}+10q^{-134}-26q^{-136}+30q^{-138}-20q^{-140}+2q^{-142}+13q^{-144}-19q^{-146}+13q^{-148}-q^{-150}-15q^{-152}+25q^{-154}-26q^{-156}+19q^{-158}-4q^{-160}-14q^{-162}+23q^{-164}-27q^{-166}+24q^{-168}-14q^{-170}+3q^{-172}+6q^{-174}-13q^{-176}+15q^{-178}-12q^{-180}+9q^{-182}-2q^{-184}-q^{-186}+2q^{-188}-4q^{-190}+3q^{-192}-2q^{-194}+q^{-196}$

Further Quantum Invariants

Further quantum knot invariants for 9_16.

A1 Invariants.

Weight	Invariant
1	$q^{-5}+3q^{-9}-q^{-11}+q^{-13}-q^{-15}-q^{-17}+q^{-19}-2q^{-21}+2q^{-23}-q^{-25}$
2	$q^{-10}+4q^{-16}+q^{-18}-4q^{-20}+6q^{-22}+3q^{-24}-9q^{-26}+3q^{-28}+6q^{-30}-9q^{-32}-q^{-34}+6q^{-36}-4q^{-38}-4q^{-40}+3q^{-42}+3q^{-44}-4q^{-46}-2q^{-48}+9q^{-50}-3q^{-52}-8q^{-54}+9q^{-56}-q^{-58}-6q^{-60}+5q^{-62}-2q^{-66}+q^{-68}$
3	$q^{-15}+q^{-21}+4q^{-23}+q^{-25}-3q^{-27}-2q^{-29}+9q^{-31}+8q^{-33}-7q^{-35}-16q^{-37}+6q^{-39}+20q^{-41}+2q^{-43}-26q^{-45}-12q^{-47}+25q^{-49}+18q^{-51}-21q^{-53}-27q^{-55}+14q^{-57}+28q^{-59}-7q^{-61}-32q^{-63}+q^{-65}+27q^{-67}+8q^{-69}-26q^{-71}-9q^{-73}+21q^{-75}+16q^{-77}-13q^{-79}-19q^{-81}+3q^{-83}+21q^{-85}+8q^{-87}-22q^{-89}-19q^{-91}+18q^{-93}+29q^{-95}-14q^{-97}-32q^{-99}+8q^{-101}+31q^{-103}-3q^{-105}-23q^{-107}-q^{-109}+16q^{-111}-11q^{-115}+2q^{-117}+5q^{-119}-3q^{-123}+2q^{-127}-q^{-129}$
4	$q^{-20}+q^{-26}+q^{-28}+4q^{-30}-3q^{-34}+q^{-38}+13q^{-40}+6q^{-42}-8q^{-44}-15q^{-46}-15q^{-48}+22q^{-50}+30q^{-52}+11q^{-54}-26q^{-56}-60q^{-58}-4q^{-60}+44q^{-62}+65q^{-64}+16q^{-66}-86q^{-68}-68q^{-70}-3q^{-72}+93q^{-74}+94q^{-76}-44q^{-78}-99q^{-80}-84q^{-82}+50q^{-84}+135q^{-86}+37q^{-88}-67q^{-90}-132q^{-92}-17q^{-94}+118q^{-96}+88q^{-98}-14q^{-100}-130q^{-102}-60q^{-104}+80q^{-106}+103q^{-108}+18q^{-110}-109q^{-112}-75q^{-114}+47q^{-116}+101q^{-118}+35q^{-120}-77q^{-122}-85q^{-124}+86q^{-128}+64q^{-130}-17q^{-132}-83q^{-134}-72q^{-136}+36q^{-138}+91q^{-140}+78q^{-142}-46q^{-144}-135q^{-146}-46q^{-148}+68q^{-150}+147q^{-152}+25q^{-154}-129q^{-156}-100q^{-158}+6q^{-160}+137q^{-162}+67q^{-164}-69q^{-166}-76q^{-168}-34q^{-170}+71q^{-172}+49q^{-174}-24q^{-176}-26q^{-178}-25q^{-180}+27q^{-182}+15q^{-184}-12q^{-186}-q^{-188}-8q^{-190}+10q^{-192}+3q^{-194}-6q^{-196}+q^{-198}-2q^{-200}+3q^{-202}-2q^{-206}+q^{-208}$
5	$q^{-25}+q^{-31}+q^{-33}+q^{-35}+3q^{-37}-3q^{-41}+4q^{-45}+5q^{-47}+10q^{-49}+3q^{-51}-11q^{-53}-17q^{-55}-9q^{-57}+7q^{-59}+33q^{-61}+35q^{-63}+4q^{-65}-40q^{-67}-67q^{-69}-48q^{-71}+27q^{-73}+98q^{-75}+102q^{-77}+28q^{-79}-97q^{-81}-177q^{-83}-115q^{-85}+48q^{-87}+201q^{-89}+229q^{-91}+69q^{-93}-185q^{-95}-320q^{-97}-211q^{-99}+71q^{-101}+343q^{-103}+366q^{-105}+91q^{-107}-288q^{-109}-462q^{-111}-280q^{-113}+140q^{-115}+481q^{-117}+454q^{-119}+46q^{-121}-418q^{-123}-555q^{-125}-241q^{-127}+281q^{-129}+592q^{-131}+400q^{-133}-129q^{-135}-548q^{-137}-500q^{-139}-21q^{-141}+476q^{-143}+542q^{-145}+129q^{-147}-374q^{-149}-537q^{-151}-202q^{-153}+311q^{-155}+495q^{-157}+221q^{-159}-238q^{-161}-462q^{-163}-233q^{-165}+212q^{-167}+424q^{-169}+221q^{-171}-174q^{-173}-401q^{-175}-242q^{-177}+139q^{-179}+380q^{-181}+275q^{-183}-64q^{-185}-342q^{-187}-333q^{-189}-54q^{-191}+274q^{-193}+392q^{-195}+209q^{-197}-154q^{-199}-423q^{-201}-384q^{-203}-23q^{-205}+406q^{-207}+536q^{-209}+229q^{-211}-307q^{-213}-634q^{-215}-432q^{-217}+153q^{-219}+637q^{-221}+589q^{-223}+24q^{-225}-555q^{-227}-648q^{-229}-184q^{-231}+404q^{-233}+617q^{-235}+289q^{-237}-251q^{-239}-502q^{-241}-307q^{-243}+105q^{-245}+359q^{-247}+273q^{-249}-21q^{-251}-229q^{-253}-193q^{-255}-15q^{-257}+122q^{-259}+119q^{-261}+24q^{-263}-62q^{-265}-68q^{-267}-6q^{-269}+28q^{-271}+24q^{-273}+6q^{-275}-13q^{-277}-11q^{-279}+q^{-281}+9q^{-283}+2q^{-285}-5q^{-287}-2q^{-289}+q^{-291}+q^{-295}+2q^{-297}-3q^{-299}+2q^{-303}-q^{-305}$
6	$q^{-30}+q^{-36}+q^{-38}+q^{-40}+3q^{-44}-3q^{-48}+q^{-50}+4q^{-52}+7q^{-54}+2q^{-56}+7q^{-58}-q^{-60}-15q^{-62}-14q^{-64}-4q^{-66}+14q^{-68}+18q^{-70}+39q^{-72}+23q^{-74}-21q^{-76}-57q^{-78}-70q^{-80}-38q^{-82}+2q^{-84}+103q^{-86}+142q^{-88}+98q^{-90}-16q^{-92}-149q^{-94}-226q^{-96}-226q^{-98}-12q^{-100}+216q^{-102}+378q^{-104}+341q^{-106}+108q^{-108}-250q^{-110}-594q^{-112}-542q^{-114}-230q^{-116}+304q^{-118}+733q^{-120}+844q^{-122}+446q^{-124}-359q^{-126}-939q^{-128}-1136q^{-130}-653q^{-132}+242q^{-134}+1195q^{-136}+1504q^{-138}+875q^{-140}-194q^{-142}-1363q^{-144}-1799q^{-146}-1273q^{-148}+193q^{-150}+1597q^{-152}+2082q^{-154}+1504q^{-156}-118q^{-158}-1745q^{-160}-2514q^{-162}-1636q^{-164}+195q^{-166}+1945q^{-168}+2712q^{-170}+1765q^{-172}-273q^{-174}-2326q^{-176}-2802q^{-178}-1647q^{-180}+531q^{-182}+2513q^{-184}+2866q^{-186}+1428q^{-188}-1036q^{-190}-2643q^{-192}-2650q^{-194}-948q^{-196}+1406q^{-198}+2752q^{-200}+2262q^{-202}+202q^{-204}-1745q^{-206}-2565q^{-208}-1602q^{-210}+424q^{-212}+2035q^{-214}+2152q^{-216}+701q^{-218}-1003q^{-220}-1991q^{-222}-1457q^{-224}+73q^{-226}+1462q^{-228}+1680q^{-230}+600q^{-232}-764q^{-234}-1576q^{-236}-1118q^{-238}+116q^{-240}+1277q^{-242}+1445q^{-244}+522q^{-246}-723q^{-248}-1509q^{-250}-1174q^{-252}-96q^{-254}+1125q^{-256}+1590q^{-258}+1040q^{-260}-200q^{-262}-1369q^{-264}-1675q^{-266}-1067q^{-268}+310q^{-270}+1560q^{-272}+2010q^{-274}+1189q^{-276}-404q^{-278}-1854q^{-280}-2408q^{-282}-1420q^{-284}+534q^{-286}+2418q^{-288}+2785q^{-290}+1487q^{-292}-840q^{-294}-2898q^{-296}-3129q^{-298}-1380q^{-300}+1432q^{-302}+3224q^{-304}+3100q^{-306}+995q^{-308}-1852q^{-310}-3415q^{-312}-2780q^{-314}-279q^{-316}+2092q^{-318}+3141q^{-320}+2142q^{-322}-230q^{-324}-2201q^{-326}-2606q^{-328}-1226q^{-330}+569q^{-332}+1893q^{-334}+1871q^{-336}+601q^{-338}-806q^{-340}-1463q^{-342}-1008q^{-344}-171q^{-346}+688q^{-348}+960q^{-350}+510q^{-352}-133q^{-354}-525q^{-356}-424q^{-358}-203q^{-360}+144q^{-362}+323q^{-364}+193q^{-366}-3q^{-368}-131q^{-370}-87q^{-372}-73q^{-374}+13q^{-376}+80q^{-378}+36q^{-380}-9q^{-382}-27q^{-384}+4q^{-386}-14q^{-388}+16q^{-392}-7q^{-396}-7q^{-398}+11q^{-400}-3q^{-402}-2q^{-404}+5q^{-406}-2q^{-408}-q^{-410}-2q^{-412}+3q^{-414}-2q^{-418}+q^{-420}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-10}+3q^{-14}+q^{-16}+2q^{-18}+q^{-20}-2q^{-22}-3q^{-26}+q^{-34}-q^{-36}$
1,1	$q^{-20}+6q^{-24}-4q^{-26}+20q^{-28}-22q^{-30}+50q^{-32}-58q^{-34}+83q^{-36}-98q^{-38}+102q^{-40}-104q^{-42}+75q^{-44}-60q^{-46}+4q^{-48}+34q^{-50}-85q^{-52}+128q^{-54}-162q^{-56}+192q^{-58}-189q^{-60}+184q^{-62}-156q^{-64}+118q^{-66}-73q^{-68}+24q^{-70}+16q^{-72}-52q^{-74}+73q^{-76}-84q^{-78}+88q^{-80}-84q^{-82}+74q^{-84}-58q^{-86}+46q^{-88}-36q^{-90}+22q^{-92}-12q^{-94}+8q^{-96}-4q^{-98}+q^{-100}$
2,0	$q^{-20}+3q^{-26}+4q^{-28}+3q^{-32}+6q^{-34}+4q^{-36}-q^{-38}+2q^{-40}+3q^{-42}-4q^{-44}-6q^{-46}-q^{-48}-5q^{-50}-8q^{-52}-2q^{-54}-q^{-56}-3q^{-58}+7q^{-62}+4q^{-64}+3q^{-68}+3q^{-70}-4q^{-72}-4q^{-74}+q^{-76}+q^{-84}-q^{-88}+q^{-90}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-20}+3q^{-24}+4q^{-26}+2q^{-28}+7q^{-30}+6q^{-32}-4q^{-34}+4q^{-36}-3q^{-38}-12q^{-40}-q^{-44}-7q^{-46}+3q^{-48}+5q^{-50}-q^{-52}-q^{-54}+3q^{-58}-5q^{-60}-2q^{-62}+7q^{-64}-5q^{-66}-3q^{-68}+8q^{-70}-3q^{-72}-3q^{-74}+5q^{-76}-q^{-78}-2q^{-80}+q^{-82}$
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^{-15} +3 q^{-19} + q^{-21} +4 q^{-23} + q^{-25} +2 q^{-27} - q^{-29} -2 q^{-31} -2 q^{-33} -3 q^{-35} - q^{-39} +2 q^{-41} - q^{-43} + q^{-45} - q^{-47} }
1,0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-30} +6 q^{-34} +2 q^{-36} +11 q^{-38} +15 q^{-40} - q^{-42} +38 q^{-44} -18 q^{-46} +27 q^{-48} +10 q^{-50} -48 q^{-52} +63 q^{-54} -97 q^{-56} +36 q^{-58} -17 q^{-60} -72 q^{-62} +84 q^{-64} -102 q^{-66} +61 q^{-68} -3 q^{-70} -27 q^{-72} +56 q^{-74} -21 q^{-76} +6 q^{-78} +21 q^{-80} -35 q^{-84} +64 q^{-86} -71 q^{-88} +26 q^{-90} +30 q^{-92} -84 q^{-94} +97 q^{-96} -61 q^{-98} +2 q^{-100} +53 q^{-102} -76 q^{-104} +61 q^{-106} -21 q^{-108} -24 q^{-110} +48 q^{-112} -42 q^{-114} +19 q^{-116} +9 q^{-118} -24 q^{-120} +19 q^{-122} -7 q^{-124} -2 q^{-126} +6 q^{-128} -4 q^{-130} + q^{-132} }

A4 Invariants.

Weight	Invariant
0,1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-30} +3 q^{-34} +4 q^{-36} +5 q^{-38} +7 q^{-40} +12 q^{-42} +6 q^{-44} +6 q^{-46} +6 q^{-48} -3 q^{-50} -10 q^{-52} -8 q^{-54} -8 q^{-56} -15 q^{-58} -8 q^{-60} +3 q^{-62} +3 q^{-64} -3 q^{-66} +8 q^{-68} +8 q^{-70} -5 q^{-72} -2 q^{-74} +3 q^{-76} -5 q^{-78} -6 q^{-80} +3 q^{-82} +2 q^{-84} -3 q^{-86} + q^{-88} +6 q^{-90} -3 q^{-94} +3 q^{-96} +2 q^{-98} -3 q^{-100} - q^{-102} + q^{-104} }
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^{-20} +3 q^{-24} + q^{-26} +4 q^{-28} +3 q^{-30} +2 q^{-32} +2 q^{-34} - q^{-36} - q^{-38} -4 q^{-40} -2 q^{-42} -3 q^{-44} - q^{-48} + q^{-50} + q^{-52} - q^{-54} + q^{-56} - q^{-58} }

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^{-20} +3 q^{-24} -2 q^{-26} +6 q^{-28} -5 q^{-30} +8 q^{-32} -6 q^{-34} +8 q^{-36} -5 q^{-38} +2 q^{-40} -5 q^{-44} +7 q^{-46} -13 q^{-48} +13 q^{-50} -15 q^{-52} +13 q^{-54} -12 q^{-56} +9 q^{-58} -5 q^{-60} +2 q^{-62} +3 q^{-64} -5 q^{-66} +7 q^{-68} -8 q^{-70} +7 q^{-72} -7 q^{-74} +5 q^{-76} -3 q^{-78} +2 q^{-80} - q^{-82} }
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^{-30} +3 q^{-38} +3 q^{-40} + q^{-42} -2 q^{-44} +2 q^{-46} +7 q^{-48} +6 q^{-50} -4 q^{-52} -5 q^{-54} + q^{-56} +8 q^{-58} + q^{-60} -10 q^{-62} -8 q^{-64} +2 q^{-66} +5 q^{-68} -3 q^{-70} -7 q^{-72} -2 q^{-74} +5 q^{-76} +2 q^{-78} -3 q^{-80} -2 q^{-82} +5 q^{-84} +4 q^{-86} -3 q^{-88} -6 q^{-90} +2 q^{-92} +6 q^{-94} - q^{-96} -7 q^{-98} -2 q^{-100} +7 q^{-102} +4 q^{-104} -5 q^{-106} -7 q^{-108} +2 q^{-110} +8 q^{-112} +2 q^{-114} -5 q^{-116} -5 q^{-118} +2 q^{-120} +5 q^{-122} + q^{-124} -2 q^{-126} -2 q^{-128} + q^{-132} }

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^{-30} +3 q^{-34} + q^{-36} +7 q^{-38} + q^{-40} +10 q^{-42} +10 q^{-46} -4 q^{-48} +4 q^{-50} -8 q^{-52} -8 q^{-56} -6 q^{-58} -2 q^{-60} -6 q^{-62} +5 q^{-64} -8 q^{-66} +11 q^{-68} -8 q^{-70} +12 q^{-72} -11 q^{-74} +10 q^{-76} -9 q^{-78} +8 q^{-80} -7 q^{-82} +2 q^{-84} -2 q^{-86} +3 q^{-90} -5 q^{-92} +4 q^{-94} -5 q^{-96} +8 q^{-98} -5 q^{-100} +4 q^{-102} -5 q^{-104} +5 q^{-106} -2 q^{-108} + q^{-110} -2 q^{-112} + q^{-114} }

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^{-50} +3 q^{-54} -2 q^{-56} +3 q^{-58} - q^{-60} +8 q^{-64} -11 q^{-66} +16 q^{-68} -11 q^{-70} +6 q^{-72} +11 q^{-74} -21 q^{-76} +32 q^{-78} -26 q^{-80} +18 q^{-82} + q^{-84} -23 q^{-86} +33 q^{-88} -31 q^{-90} +19 q^{-92} -18 q^{-96} +22 q^{-98} -18 q^{-100} +11 q^{-104} -24 q^{-106} +22 q^{-108} -14 q^{-110} -8 q^{-112} +27 q^{-114} -40 q^{-116} +41 q^{-118} -27 q^{-120} +2 q^{-122} +22 q^{-124} -40 q^{-126} +46 q^{-128} -36 q^{-130} +16 q^{-132} +10 q^{-134} -26 q^{-136} +30 q^{-138} -20 q^{-140} +2 q^{-142} +13 q^{-144} -19 q^{-146} +13 q^{-148} - q^{-150} -15 q^{-152} +25 q^{-154} -26 q^{-156} +19 q^{-158} -4 q^{-160} -14 q^{-162} +23 q^{-164} -27 q^{-166} +24 q^{-168} -14 q^{-170} +3 q^{-172} +6 q^{-174} -13 q^{-176} +15 q^{-178} -12 q^{-180} +9 q^{-182} -2 q^{-184} - q^{-186} +2 q^{-188} -4 q^{-190} +3 q^{-192} -2 q^{-194} + q^{-196} }

.

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["9 16"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-5t^{2}+8t-9+8t^{-1}-5t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+7z^{4}+6z^{2}+1$

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]=

{ 39, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{12}+3q^{11}-5q^{10}+6q^{9}-7q^{8}+6q^{7}-5q^{6}+4q^{5}-q^{4}+q^{3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-6}+z^{6}a^{-8}+5z^{4}a^{-6}+3z^{4}a^{-8}-z^{4}a^{-10}+8z^{2}a^{-6}-2z^{2}a^{-10}+4a^{-6}-3a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-7}+4z^{7}a^{-9}+3z^{7}a^{-11}+z^{6}a^{-6}-z^{6}a^{-8}+3z^{6}a^{-10}+5z^{6}a^{-12}-2z^{5}a^{-7}-8z^{5}a^{-9}-z^{5}a^{-11}+5z^{5}a^{-13}-5z^{4}a^{-6}-4z^{4}a^{-8}-8z^{4}a^{-10}-6z^{4}a^{-12}+3z^{4}a^{-14}-2z^{3}a^{-7}-z^{3}a^{-9}-5z^{3}a^{-11}-5z^{3}a^{-13}+z^{3}a^{-15}+8z^{2}a^{-6}+6z^{2}a^{-8}+z^{2}a^{-10}+2z^{2}a^{-12}-z^{2}a^{-14}+4za^{-7}+4za^{-9}+2za^{-11}+2za^{-13}-4a^{-6}-3a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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["9 16"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^3-5 t^2+8 t-9+8 t^{-1} -5 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^{12}+3 q^{11}-5 q^{10}+6 q^9-7 q^8+6 q^7-5 q^6+4 q^5-q^4+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]=

{}

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:

(6, 14)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 112} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 668} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 84} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2688} ${\frac {13408}{3}}$ 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{2176}{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 496} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2304} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6272} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16032} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2016} Failed to parse (SVG (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{154271}{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{23548}{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{153164}{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{769}{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{6111}{5}}

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 $j-2r=s+1$ 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=} 6 is the signature of 9 16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 25                   1 -1 23                 2   2 21               3 1   -2 19             3 2     1 17           4 3       -1 15         2 3         -1 13       3 4           1 11     1 2             -1 9     3               3 7 1 1                 0 5 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=5$ 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 i=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 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}} Failed to parse (SVG (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}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2} Failed to parse (SVG (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^{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=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^{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=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}^{3}\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}^{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=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}^{2}\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=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}\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=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 {\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^{33}-3 q^{32}+2 q^{31}+6 q^{30}-14 q^{29}+7 q^{28}+16 q^{27}-31 q^{26}+12 q^{25}+28 q^{24}-42 q^{23}+10 q^{22}+35 q^{21}-42 q^{20}+3 q^{19}+35 q^{18}-32 q^{17}-4 q^{16}+27 q^{15}-17 q^{14}-7 q^{13}+15 q^{12}-5 q^{11}-4 q^{10}+5 q^9-q^7+q^6}
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^{62}-2 q^{61}-3 q^{60}+2 q^{59}+8 q^{58}-5 q^{57}-16 q^{56}+13 q^{55}+24 q^{54}-22 q^{53}-38 q^{52}+33 q^{51}+58 q^{50}-45 q^{49}-78 q^{48}+51 q^{47}+101 q^{46}-56 q^{45}-115 q^{44}+48 q^{43}+131 q^{42}-43 q^{41}-133 q^{40}+26 q^{39}+137 q^{38}-14 q^{37}-128 q^{36}-4 q^{35}+120 q^{34}+20 q^{33}-109 q^{32}-30 q^{31}+87 q^{30}+45 q^{29}-74 q^{28}-44 q^{27}+46 q^{26}+51 q^{25}-35 q^{24}-37 q^{23}+9 q^{22}+37 q^{21}-7 q^{20}-19 q^{19}-5 q^{18}+15 q^{17}+2 q^{16}-4 q^{15}-4 q^{14}+4 q^{13}+q^{12}-q^{10}+q^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 q^{102}-3 q^{101}+2 q^{100}+3 q^{99}-5 q^{98}+4 q^{97}-10 q^{96}+11 q^{95}+10 q^{94}-23 q^{93}+11 q^{92}-21 q^{91}+38 q^{90}+22 q^{89}-75 q^{88}+10 q^{87}-19 q^{86}+111 q^{85}+44 q^{84}-180 q^{83}-32 q^{82}-12 q^{81}+247 q^{80}+114 q^{79}-311 q^{78}-138 q^{77}-41 q^{76}+401 q^{75}+236 q^{74}-390 q^{73}-252 q^{72}-130 q^{71}+490 q^{70}+360 q^{69}-377 q^{68}-307 q^{67}-238 q^{66}+479 q^{65}+426 q^{64}-296 q^{63}-285 q^{62}-324 q^{61}+394 q^{60}+434 q^{59}-184 q^{58}-219 q^{57}-378 q^{56}+272 q^{55}+400 q^{54}-57 q^{53}-134 q^{52}-401 q^{51}+132 q^{50}+330 q^{49}+59 q^{48}-32 q^{47}-371 q^{46}-3 q^{45}+215 q^{44}+124 q^{43}+72 q^{42}-273 q^{41}-88 q^{40}+81 q^{39}+109 q^{38}+127 q^{37}-135 q^{36}-89 q^{35}-15 q^{34}+44 q^{33}+109 q^{32}-33 q^{31}-40 q^{30}-36 q^{29}-4 q^{28}+53 q^{27}+q^{26}-3 q^{25}-17 q^{24}-12 q^{23}+16 q^{22}+q^{21}+4 q^{20}-3 q^{19}-5 q^{18}+4 q^{17}+q^{15}-q^{13}+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 -q^{150}+3 q^{149}-2 q^{148}-3 q^{147}+5 q^{146}-q^{145}-2 q^{144}+4 q^{143}-5 q^{142}-6 q^{141}+12 q^{140}+6 q^{139}-10 q^{138}-8 q^{137}-7 q^{136}+13 q^{135}+30 q^{134}+10 q^{133}-44 q^{132}-70 q^{131}-q^{130}+99 q^{129}+125 q^{128}+13 q^{127}-181 q^{126}-248 q^{125}-37 q^{124}+307 q^{123}+419 q^{122}+99 q^{121}-435 q^{120}-660 q^{119}-232 q^{118}+558 q^{117}+959 q^{116}+427 q^{115}-648 q^{114}-1248 q^{113}-696 q^{112}+651 q^{111}+1538 q^{110}+992 q^{109}-600 q^{108}-1732 q^{107}-1281 q^{106}+449 q^{105}+1865 q^{104}+1528 q^{103}-293 q^{102}-1862 q^{101}-1710 q^{100}+88 q^{99}+1826 q^{98}+1797 q^{97}+70 q^{96}-1679 q^{95}-1828 q^{94}-240 q^{93}+1547 q^{92}+1788 q^{91}+348 q^{90}-1340 q^{89}-1723 q^{88}-481 q^{87}+1166 q^{86}+1629 q^{85}+575 q^{84}-945 q^{83}-1520 q^{82}-693 q^{81}+721 q^{80}+1400 q^{79}+799 q^{78}-486 q^{77}-1246 q^{76}-877 q^{75}+208 q^{74}+1065 q^{73}+962 q^{72}+17 q^{71}-833 q^{70}-943 q^{69}-289 q^{68}+586 q^{67}+914 q^{66}+436 q^{65}-304 q^{64}-751 q^{63}-600 q^{62}+64 q^{61}+600 q^{60}+573 q^{59}+160 q^{58}-343 q^{57}-573 q^{56}-277 q^{55}+180 q^{54}+391 q^{53}+334 q^{52}+36 q^{51}-298 q^{50}-300 q^{49}-92 q^{48}+109 q^{47}+225 q^{46}+171 q^{45}-44 q^{44}-140 q^{43}-120 q^{42}-44 q^{41}+62 q^{40}+109 q^{39}+36 q^{38}-15 q^{37}-46 q^{36}-48 q^{35}-9 q^{34}+34 q^{33}+17 q^{32}+12 q^{31}-2 q^{30}-17 q^{29}-11 q^{28}+8 q^{27}+q^{26}+4 q^{25}+4 q^{24}-3 q^{23}-4 q^{22}+3 q^{21}+q^{18}-q^{16}+q^{15}}
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^{207}-3 q^{206}+2 q^{205}+3 q^{204}-5 q^{203}+q^{202}-q^{201}+8 q^{200}-10 q^{199}+q^{198}+17 q^{197}-23 q^{196}+q^{195}+6 q^{194}+24 q^{193}-26 q^{192}-13 q^{191}+35 q^{190}-54 q^{189}+19 q^{188}+51 q^{187}+68 q^{186}-93 q^{185}-99 q^{184}+21 q^{183}-98 q^{182}+147 q^{181}+247 q^{180}+198 q^{179}-272 q^{178}-446 q^{177}-200 q^{176}-199 q^{175}+539 q^{174}+890 q^{173}+648 q^{172}-544 q^{171}-1305 q^{170}-1037 q^{169}-654 q^{168}+1196 q^{167}+2297 q^{166}+1918 q^{165}-522 q^{164}-2629 q^{163}-2832 q^{162}-2034 q^{161}+1601 q^{160}+4268 q^{159}+4290 q^{158}+477 q^{157}-3678 q^{156}-5203 q^{155}-4535 q^{154}+966 q^{153}+5831 q^{152}+7137 q^{151}+2582 q^{150}-3554 q^{149}-6995 q^{148}-7347 q^{147}-783 q^{146}+6062 q^{145}+9195 q^{144}+4909 q^{143}-2256 q^{142}-7362 q^{141}-9231 q^{140}-2737 q^{139}+5074 q^{138}+9749 q^{137}+6359 q^{136}-663 q^{135}-6541 q^{134}-9681 q^{133}-3987 q^{132}+3697 q^{131}+9141 q^{130}+6665 q^{129}+506 q^{128}-5301 q^{127}-9131 q^{126}-4452 q^{125}+2476 q^{124}+8063 q^{123}+6330 q^{122}+1292 q^{121}-4056 q^{120}-8208 q^{119}-4620 q^{118}+1315 q^{117}+6829 q^{116}+5872 q^{115}+2104 q^{114}-2692 q^{113}-7128 q^{112}-4838 q^{111}-74 q^{110}+5299 q^{109}+5338 q^{108}+3092 q^{107}-988 q^{106}-5677 q^{105}-4955 q^{104}-1685 q^{103}+3273 q^{102}+4375 q^{101}+3912 q^{100}+959 q^{99}-3617 q^{98}-4465 q^{97}-3031 q^{96}+919 q^{95}+2673 q^{94}+3919 q^{93}+2566 q^{92}-1153 q^{91}-3027 q^{90}-3384 q^{89}-1063 q^{88}+495 q^{87}+2764 q^{86}+3042 q^{85}+900 q^{84}-989 q^{83}-2437 q^{82}-1830 q^{81}-1255 q^{80}+933 q^{79}+2164 q^{78}+1669 q^{77}+638 q^{76}-815 q^{75}-1252 q^{74}-1740 q^{73}-471 q^{72}+698 q^{71}+1143 q^{70}+1074 q^{69}+354 q^{68}-183 q^{67}-1111 q^{66}-780 q^{65}-255 q^{64}+248 q^{63}+591 q^{62}+551 q^{61}+397 q^{60}-306 q^{59}-382 q^{58}-366 q^{57}-181 q^{56}+57 q^{55}+239 q^{54}+345 q^{53}+38 q^{52}-24 q^{51}-133 q^{50}-144 q^{49}-105 q^{48}+11 q^{47}+131 q^{46}+38 q^{45}+53 q^{44}-30 q^{42}-61 q^{41}-28 q^{40}+30 q^{39}-2 q^{38}+21 q^{37}+13 q^{36}+6 q^{35}-17 q^{34}-12 q^{33}+9 q^{32}-6 q^{31}+3 q^{30}+3 q^{29}+5 q^{28}-3 q^{27}-4 q^{26}+4 q^{25}-q^{24}+q^{21}-q^{19}+q^{18}}

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.

9_15 9_17