9 16

9_15

9_17

Visit 9 16's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 16's page at Knotilus! Visit 9 16's page at the original Knot Atlas! 9 16 Quick Notes

9 16 Further Notes and Views

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: Length is 10, width is 3. Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index $\{4,7\}$ 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 $3$ Topological 4 genus $3$ 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	$q^{-15}+3q^{-19}+q^{-21}+4q^{-23}+q^{-25}+2q^{-27}-q^{-29}-2q^{-31}-2q^{-33}-3q^{-35}-q^{-39}+2q^{-41}-q^{-43}+q^{-45}-q^{-47}$
1,0,1	$q^{-30}+6q^{-34}+2q^{-36}+11q^{-38}+15q^{-40}-q^{-42}+38q^{-44}-18q^{-46}+27q^{-48}+10q^{-50}-48q^{-52}+63q^{-54}-97q^{-56}+36q^{-58}-17q^{-60}-72q^{-62}+84q^{-64}-102q^{-66}+61q^{-68}-3q^{-70}-27q^{-72}+56q^{-74}-21q^{-76}+6q^{-78}+21q^{-80}-35q^{-84}+64q^{-86}-71q^{-88}+26q^{-90}+30q^{-92}-84q^{-94}+97q^{-96}-61q^{-98}+2q^{-100}+53q^{-102}-76q^{-104}+61q^{-106}-21q^{-108}-24q^{-110}+48q^{-112}-42q^{-114}+19q^{-116}+9q^{-118}-24q^{-120}+19q^{-122}-7q^{-124}-2q^{-126}+6q^{-128}-4q^{-130}+q^{-132}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-30}+3q^{-34}+4q^{-36}+5q^{-38}+7q^{-40}+12q^{-42}+6q^{-44}+6q^{-46}+6q^{-48}-3q^{-50}-10q^{-52}-8q^{-54}-8q^{-56}-15q^{-58}-8q^{-60}+3q^{-62}+3q^{-64}-3q^{-66}+8q^{-68}+8q^{-70}-5q^{-72}-2q^{-74}+3q^{-76}-5q^{-78}-6q^{-80}+3q^{-82}+2q^{-84}-3q^{-86}+q^{-88}+6q^{-90}-3q^{-94}+3q^{-96}+2q^{-98}-3q^{-100}-q^{-102}+q^{-104}$
1,0,0,0	$q^{-20}+3q^{-24}+q^{-26}+4q^{-28}+3q^{-30}+2q^{-32}+2q^{-34}-q^{-36}-q^{-38}-4q^{-40}-2q^{-42}-3q^{-44}-q^{-48}+q^{-50}+q^{-52}-q^{-54}+q^{-56}-q^{-58}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-20}+3q^{-24}-2q^{-26}+6q^{-28}-5q^{-30}+8q^{-32}-6q^{-34}+8q^{-36}-5q^{-38}+2q^{-40}-5q^{-44}+7q^{-46}-13q^{-48}+13q^{-50}-15q^{-52}+13q^{-54}-12q^{-56}+9q^{-58}-5q^{-60}+2q^{-62}+3q^{-64}-5q^{-66}+7q^{-68}-8q^{-70}+7q^{-72}-7q^{-74}+5q^{-76}-3q^{-78}+2q^{-80}-q^{-82}$
1,0	$q^{-30}+3q^{-38}+3q^{-40}+q^{-42}-2q^{-44}+2q^{-46}+7q^{-48}+6q^{-50}-4q^{-52}-5q^{-54}+q^{-56}+8q^{-58}+q^{-60}-10q^{-62}-8q^{-64}+2q^{-66}+5q^{-68}-3q^{-70}-7q^{-72}-2q^{-74}+5q^{-76}+2q^{-78}-3q^{-80}-2q^{-82}+5q^{-84}+4q^{-86}-3q^{-88}-6q^{-90}+2q^{-92}+6q^{-94}-q^{-96}-7q^{-98}-2q^{-100}+7q^{-102}+4q^{-104}-5q^{-106}-7q^{-108}+2q^{-110}+8q^{-112}+2q^{-114}-5q^{-116}-5q^{-118}+2q^{-120}+5q^{-122}+q^{-124}-2q^{-126}-2q^{-128}+q^{-132}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-30}+3q^{-34}+q^{-36}+7q^{-38}+q^{-40}+10q^{-42}+10q^{-46}-4q^{-48}+4q^{-50}-8q^{-52}-8q^{-56}-6q^{-58}-2q^{-60}-6q^{-62}+5q^{-64}-8q^{-66}+11q^{-68}-8q^{-70}+12q^{-72}-11q^{-74}+10q^{-76}-9q^{-78}+8q^{-80}-7q^{-82}+2q^{-84}-2q^{-86}+3q^{-90}-5q^{-92}+4q^{-94}-5q^{-96}+8q^{-98}-5q^{-100}+4q^{-102}-5q^{-104}+5q^{-106}-2q^{-108}+q^{-110}-2q^{-112}+q^{-114}$

G2 Invariants.

Weight

Invariant

1,0

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

.

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, $q\leftrightarrow q^{-1}$): {...}

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 $24$ $112$ $288$ $668$ $84$ $2688$ ${\frac {13408}{3}}$ ${\frac {2176}{3}}$ $496$ $2304$ $6272$ $16032$ $2016$ ${\frac {154271}{5}}$ ${\frac {23548}{15}}$ ${\frac {153164}{15}}$ ${\frac {769}{3}}$ ${\frac {6111}{5}}$

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

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=5$ $i=7$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 16]]
Out[2]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[16, 6, 17, 5], X[18, 8, 1, 7], X[6, 18, 7, 17], X[10, 16, 11, 15], X[14, 10, 15, 9], X[8, 14, 9, 13], X[2, 12, 3, 11]]
In[3]:=	GaussCode[Knot[9, 16]]
Out[3]=	GaussCode[1, -9, 2, -1, 3, -5, 4, -8, 7, -6, 9, -2, 8, -7, 6, -3, 5, -4]
In[4]:=	DTCode[Knot[9, 16]]
Out[4]=	DTCode[4, 12, 16, 18, 14, 2, 8, 10, 6]
In[5]:=	br = BR[Knot[9, 16]]
Out[5]=	BR[3, {1, 1, 1, 1, 2, 2, -1, 2, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[9, 16]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[9, 16]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 16]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 3, {4, 7}, 1}
In[10]:=	alex = Alexander[Knot[9, 16]][t]
Out[10]=	2 5 8 2 3 -9 + -- - -- + - + 8 t - 5 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[9, 16]][z]
Out[11]=	2 4 6 1 + 6 z + 7 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 16]}
In[13]:=	{KnotDet[Knot[9, 16]], KnotSignature[Knot[9, 16]]}
Out[13]=	{39, 6}
In[14]:=	Jones[Knot[9, 16]][q]
Out[14]=	3 4 5 6 7 8 9 10 11 12 q - q + 4 q - 5 q + 6 q - 7 q + 6 q - 5 q + 3 q - q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 16]}
In[16]:=	A2Invariant[Knot[9, 16]][q]
Out[16]=	10 14 16 18 20 22 26 34 36 q + 3 q + q + 2 q + q - 2 q - 3 q + q - q
In[17]:=	HOMFLYPT[Knot[9, 16]][a, z]
Out[17]=	2 2 4 4 4 6 6 -3 4 2 z 8 z z 3 z 5 z z z -- + -- - ---- + ---- - --- + ---- + ---- + -- + -- 8 6 10 6 10 8 6 8 6 a a a a a a a a a
In[18]:=	Kauffman[Knot[9, 16]][a, z]
Out[18]=	2 2 2 2 2 -3 4 2 z 2 z 4 z 4 z z 2 z z 6 z 8 z -- - -- + --- + --- + --- + --- - --- + ---- + --- + ---- + ---- + 8 6 13 11 9 7 14 12 10 8 6 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 z 5 z 5 z z 2 z 3 z 6 z 8 z 4 z 5 z --- - ---- - ---- - -- - ---- + ---- - ---- - ---- - ---- - ---- + 15 13 11 9 7 14 12 10 8 6 a a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 7 5 z z 8 z 2 z 5 z 3 z z z 3 z 4 z z ---- - --- - ---- - ---- + ---- + ---- - -- + -- + ---- + ---- + -- + 13 11 9 7 12 10 8 6 11 9 7 a a a a a a a a a a a 8 8 z z --- + -- 10 8 a a
In[19]:=	{Vassiliev[2][Knot[9, 16]], Vassiliev[3][Knot[9, 16]]}
Out[19]=	{6, 14}
In[20]:=	Kh[Knot[9, 16]][q, t]
Out[20]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + q t + 3 q t + q t + 2 q t + 3 q t + 4 q t + 15 4 15 5 17 5 17 6 19 6 19 7 2 q t + 3 q t + 4 q t + 3 q t + 3 q t + 2 q t + 21 7 21 8 23 8 25 9 3 q t + q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[9, 16], 2][q]
Out[21]=	6 7 9 10 11 12 13 14 15 q - q + 5 q - 4 q - 5 q + 15 q - 7 q - 17 q + 27 q - 16 17 18 19 20 21 22 23 4 q - 32 q + 35 q + 3 q - 42 q + 35 q + 10 q - 42 q + 24 25 26 27 28 29 30 31 28 q + 12 q - 31 q + 16 q + 7 q - 14 q + 6 q + 2 q - 32 33 3 q + q

See/edit the Rolfsen_Splice_Template.