10 47

10_46

10_48

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10 47 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X9,17,10,16 X5,15,6,14 X15,7,16,6 X11,19,12,18 X13,1,14,20 X17,11,18,10 X19,13,20,12 X7283
Gauss code	-1, 10, -2, 1, -4, 5, -10, 2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7
Dowker-Thistlethwaite code	4 8 14 2 16 18 20 6 10 12
Conway Notation	[5,21,2]

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [0][-12] Hyperbolic Volume 9.38519 A-Polynomial See Data:10 47/A-polynomial

[edit Notes for 10 47's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $4$ Rasmussen s-Invariant 4

[edit Notes for 10 47's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-3t^{3}+6t^{2}-7t+7-7t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+5z^{6}+8z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, 4 }
Jones polynomial	$-q^{9}+2q^{8}-4q^{7}+5q^{6}-6q^{5}+7q^{4}-5q^{3}+5q^{2}-3q+2-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+7z^{6}a^{-4}-z^{6}a^{-6}-5z^{4}a^{-2}+18z^{4}a^{-4}-5z^{4}a^{-6}-7z^{2}a^{-2}+21z^{2}a^{-4}-8z^{2}a^{-6}-3a^{-2}+9a^{-4}-5a^{-6}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+z^{7}a^{-1}-z^{7}a^{-3}+z^{7}a^{-5}+3z^{7}a^{-7}-10z^{6}a^{-2}-23z^{6}a^{-4}-10z^{6}a^{-6}+3z^{6}a^{-8}-5z^{5}a^{-1}-11z^{5}a^{-3}-14z^{5}a^{-5}-5z^{5}a^{-7}+3z^{5}a^{-9}+15z^{4}a^{-2}+35z^{4}a^{-4}+15z^{4}a^{-6}-3z^{4}a^{-8}+2z^{4}a^{-10}+7z^{3}a^{-1}+20z^{3}a^{-3}+19z^{3}a^{-5}+2z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}-9z^{2}a^{-2}-26z^{2}a^{-4}-15z^{2}a^{-6}+z^{2}a^{-8}-z^{2}a^{-10}-3za^{-1}-8za^{-3}-9za^{-5}-za^{-7}+2za^{-9}-za^{-11}+3a^{-2}+9a^{-4}+5a^{-6}$
The A2 invariant	$-q^{2}-q^{-2}+q^{-6}+q^{-8}+4q^{-10}+q^{-12}+3q^{-14}-q^{-18}-q^{-20}-2q^{-22}-q^{-26}$
The G2 invariant	$q^{12}-q^{10}+3q^{8}-4q^{6}+3q^{4}-3q^{2}-2+7q^{-2}-14q^{-4}+15q^{-6}-14q^{-8}+3q^{-10}+7q^{-12}-19q^{-14}+23q^{-16}-20q^{-18}+9q^{-20}+3q^{-22}-15q^{-24}+18q^{-26}-12q^{-28}+4q^{-30}+8q^{-32}-10q^{-34}+11q^{-36}-6q^{-40}+16q^{-42}-14q^{-44}+15q^{-46}-2q^{-48}-5q^{-50}+18q^{-52}-20q^{-54}+24q^{-56}-12q^{-58}+2q^{-60}+11q^{-62}-17q^{-64}+19q^{-66}-13q^{-68}+5q^{-70}+5q^{-72}-10q^{-74}+8q^{-76}-4q^{-78}-5q^{-80}+8q^{-82}-9q^{-84}+3q^{-88}-9q^{-90}+8q^{-92}-7q^{-94}+2q^{-96}-q^{-98}-4q^{-100}+4q^{-102}-7q^{-104}+6q^{-106}-6q^{-108}+5q^{-110}-3q^{-112}-2q^{-114}+6q^{-116}-10q^{-118}+11q^{-120}-7q^{-122}+3q^{-124}+q^{-126}-5q^{-128}+6q^{-130}-6q^{-132}+5q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

Further Quantum Invariants

Further quantum knot invariants for 10_47.

A1 Invariants.

Weight	Invariant
1	$-q^{3}+q-q^{-1}+2q^{-3}+2q^{-7}+q^{-9}-q^{-11}+q^{-13}-2q^{-15}+q^{-17}-q^{-19}$
2	$q^{12}-q^{10}-2q^{8}+3q^{6}-5q^{2}+3+4q^{-2}-5q^{-4}+5q^{-8}-2q^{-10}-q^{-12}+5q^{-14}+q^{-16}-2q^{-18}+2q^{-20}+3q^{-22}-3q^{-24}-2q^{-26}+3q^{-28}-5q^{-32}+2q^{-34}+q^{-36}-4q^{-38}+2q^{-40}-q^{-42}-2q^{-44}+3q^{-46}-q^{-50}+q^{-52}$
3	$-q^{27}+q^{25}+2q^{23}-4q^{19}-2q^{17}+6q^{15}+5q^{13}-6q^{11}-10q^{9}+2q^{7}+13q^{5}+3q^{3}-13q-8q^{-1}+8q^{-3}+14q^{-5}-3q^{-7}-12q^{-9}-4q^{-11}+10q^{-13}+9q^{-15}-3q^{-17}-11q^{-19}+q^{-21}+11q^{-23}+7q^{-25}-9q^{-27}-6q^{-29}+10q^{-31}+9q^{-33}-10q^{-35}-10q^{-37}+9q^{-39}+9q^{-41}-5q^{-43}-12q^{-45}+q^{-47}+8q^{-49}+6q^{-51}-7q^{-53}-11q^{-55}-2q^{-57}+14q^{-59}+4q^{-61}-11q^{-63}-12q^{-65}+6q^{-67}+11q^{-69}-7q^{-73}-4q^{-75}+6q^{-77}+6q^{-79}-q^{-81}-5q^{-83}-q^{-85}+4q^{-87}+2q^{-89}-2q^{-91}-q^{-93}+q^{-97}-q^{-99}$
4	$q^{48}-q^{46}-2q^{44}+q^{40}+6q^{38}-6q^{34}-6q^{32}-4q^{30}+15q^{28}+12q^{26}-2q^{24}-15q^{22}-25q^{20}+7q^{18}+24q^{16}+23q^{14}+q^{12}-39q^{10}-23q^{8}+2q^{6}+32q^{4}+36q^{2}-10-27q^{-2}-33q^{-4}-3q^{-6}+34q^{-8}+25q^{-10}+14q^{-12}-23q^{-14}-38q^{-16}-12q^{-18}+15q^{-20}+48q^{-22}+25q^{-24}-26q^{-26}-48q^{-28}-29q^{-30}+40q^{-32}+62q^{-34}+11q^{-36}-48q^{-38}-57q^{-40}+12q^{-42}+62q^{-44}+31q^{-46}-30q^{-48}-55q^{-50}-3q^{-52}+49q^{-54}+27q^{-56}-26q^{-58}-44q^{-60}-5q^{-62}+44q^{-64}+29q^{-66}-21q^{-68}-43q^{-70}-23q^{-72}+29q^{-74}+46q^{-76}+14q^{-78}-24q^{-80}-57q^{-82}-27q^{-84}+38q^{-86}+65q^{-88}+32q^{-90}-58q^{-92}-83q^{-94}-13q^{-96}+70q^{-98}+87q^{-100}-12q^{-102}-86q^{-104}-54q^{-106}+29q^{-108}+86q^{-110}+23q^{-112}-46q^{-114}-51q^{-116}-7q^{-118}+52q^{-120}+25q^{-122}-14q^{-124}-28q^{-126}-16q^{-128}+23q^{-130}+14q^{-132}+q^{-134}-11q^{-136}-13q^{-138}+7q^{-140}+4q^{-142}+3q^{-144}-2q^{-146}-5q^{-148}+2q^{-150}+q^{-154}-q^{-158}+q^{-160}$
5	$-q^{75}+q^{73}+2q^{71}-q^{67}-3q^{65}-4q^{63}+8q^{59}+8q^{57}+2q^{55}-7q^{53}-16q^{51}-14q^{49}+4q^{47}+26q^{45}+28q^{43}+10q^{41}-21q^{39}-47q^{37}-38q^{35}+5q^{33}+53q^{31}+64q^{29}+32q^{27}-31q^{25}-81q^{23}-74q^{21}-12q^{19}+64q^{17}+95q^{15}+65q^{13}-14q^{11}-83q^{9}-97q^{7}-48q^{5}+34q^{3}+84q+86q^{-1}+41q^{-3}-30q^{-5}-84q^{-7}-91q^{-9}-49q^{-11}+19q^{-13}+100q^{-15}+127q^{-17}+69q^{-19}-48q^{-21}-152q^{-23}-164q^{-25}-48q^{-27}+135q^{-29}+230q^{-31}+153q^{-33}-56q^{-35}-248q^{-37}-243q^{-39}-39q^{-41}+214q^{-43}+300q^{-45}+136q^{-47}-148q^{-49}-307q^{-51}-207q^{-53}+65q^{-55}+284q^{-57}+256q^{-59}+6q^{-61}-235q^{-63}-261q^{-65}-64q^{-67}+176q^{-69}+248q^{-71}+90q^{-73}-131q^{-75}-211q^{-77}-93q^{-79}+96q^{-81}+171q^{-83}+77q^{-85}-90q^{-87}-150q^{-89}-51q^{-91}+95q^{-93}+138q^{-95}+45q^{-97}-104q^{-99}-157q^{-101}-63q^{-103}+94q^{-105}+172q^{-107}+115q^{-109}-40q^{-111}-179q^{-113}-186q^{-115}-53q^{-117}+134q^{-119}+250q^{-121}+173q^{-123}-47q^{-125}-260q^{-127}-294q^{-129}-81q^{-131}+231q^{-133}+365q^{-135}+203q^{-137}-131q^{-139}-380q^{-141}-303q^{-143}+38q^{-145}+333q^{-147}+334q^{-149}+52q^{-151}-250q^{-153}-315q^{-155}-99q^{-157}+175q^{-159}+257q^{-161}+105q^{-163}-115q^{-165}-188q^{-167}-89q^{-169}+74q^{-171}+139q^{-173}+62q^{-175}-54q^{-177}-97q^{-179}-43q^{-181}+38q^{-183}+67q^{-185}+32q^{-187}-27q^{-189}-49q^{-191}-26q^{-193}+17q^{-195}+31q^{-197}+18q^{-199}-5q^{-201}-19q^{-203}-14q^{-205}+q^{-207}+12q^{-209}+6q^{-211}+q^{-213}-3q^{-215}-5q^{-217}+3q^{-221}+q^{-223}-q^{-229}+q^{-233}-q^{-235}$
6	$q^{108}-q^{106}-2q^{104}+q^{100}+3q^{98}+q^{96}+4q^{94}-2q^{92}-10q^{90}-7q^{88}-2q^{86}+8q^{84}+10q^{82}+21q^{80}+8q^{78}-16q^{76}-30q^{74}-32q^{72}-11q^{70}+8q^{68}+59q^{66}+64q^{64}+30q^{62}-23q^{60}-78q^{58}-93q^{56}-82q^{54}+23q^{52}+108q^{50}+147q^{48}+110q^{46}+11q^{44}-106q^{42}-213q^{40}-164q^{38}-53q^{36}+109q^{34}+216q^{32}+229q^{30}+127q^{28}-85q^{26}-209q^{24}-263q^{22}-171q^{20}-13q^{18}+167q^{16}+265q^{14}+206q^{12}+103q^{10}-67q^{8}-176q^{6}-243q^{4}-193q^{2}-75+42q^{-2}+200q^{-4}+278q^{-6}+284q^{-8}+122q^{-10}-113q^{-12}-346q^{-14}-484q^{-16}-361q^{-18}-17q^{-20}+434q^{-22}+678q^{-24}+597q^{-26}+161q^{-28}-465q^{-30}-888q^{-32}-837q^{-34}-251q^{-36}+504q^{-38}+1045q^{-40}+1027q^{-42}+359q^{-44}-566q^{-46}-1199q^{-48}-1118q^{-50}-396q^{-52}+602q^{-54}+1296q^{-56}+1197q^{-58}+367q^{-60}-691q^{-62}-1316q^{-64}-1180q^{-66}-317q^{-68}+776q^{-70}+1356q^{-72}+1090q^{-74}+161q^{-76}-834q^{-78}-1323q^{-80}-971q^{-82}+26q^{-84}+949q^{-86}+1228q^{-88}+723q^{-90}-212q^{-92}-997q^{-94}-1099q^{-96}-439q^{-98}+443q^{-100}+965q^{-102}+820q^{-104}+153q^{-106}-578q^{-108}-861q^{-110}-491q^{-112}+154q^{-114}+598q^{-116}+578q^{-118}+153q^{-120}-336q^{-122}-521q^{-124}-253q^{-126}+185q^{-128}+419q^{-130}+298q^{-132}-74q^{-134}-403q^{-136}-425q^{-138}-90q^{-140}+350q^{-142}+558q^{-144}+391q^{-146}-60q^{-148}-510q^{-150}-679q^{-152}-444q^{-154}+100q^{-156}+627q^{-158}+836q^{-160}+576q^{-162}-44q^{-164}-716q^{-166}-1066q^{-168}-791q^{-170}-10q^{-172}+862q^{-174}+1295q^{-176}+997q^{-178}+52q^{-180}-1039q^{-182}-1516q^{-184}-1090q^{-186}+34q^{-188}+1155q^{-190}+1607q^{-192}+1067q^{-194}-184q^{-196}-1246q^{-198}-1514q^{-200}-831q^{-202}+302q^{-204}+1199q^{-206}+1278q^{-208}+515q^{-210}-424q^{-212}-997q^{-214}-864q^{-216}-262q^{-218}+432q^{-220}+722q^{-222}+468q^{-224}+42q^{-226}-324q^{-228}-374q^{-230}-217q^{-232}+52q^{-234}+186q^{-236}+114q^{-238}+37q^{-240}-39q^{-242}-21q^{-244}-9q^{-246}+32q^{-248}+6q^{-250}-67q^{-252}-54q^{-254}-25q^{-256}+49q^{-258}+61q^{-260}+66q^{-262}+11q^{-264}-61q^{-266}-58q^{-268}-40q^{-270}+15q^{-272}+30q^{-274}+47q^{-276}+22q^{-278}-16q^{-280}-20q^{-282}-24q^{-284}-3q^{-286}+2q^{-288}+17q^{-290}+10q^{-292}-2q^{-294}-2q^{-296}-8q^{-298}-q^{-300}-2q^{-302}+4q^{-304}+2q^{-306}-q^{-308}+q^{-310}-3q^{-312}+q^{-318}-q^{-322}+q^{-324}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{2}-q^{-2}+q^{-6}+q^{-8}+4q^{-10}+q^{-12}+3q^{-14}-q^{-18}-q^{-20}-2q^{-22}-q^{-26}$
1,1	$q^{12}-2q^{10}+6q^{8}-12q^{6}+21q^{4}-34q^{2}+48-62q^{-2}+65q^{-4}-72q^{-6}+58q^{-8}-42q^{-10}+14q^{-12}+16q^{-14}-38q^{-16}+78q^{-18}-80q^{-20}+108q^{-22}-94q^{-24}+102q^{-26}-84q^{-28}+60q^{-30}-48q^{-32}+16q^{-34}-9q^{-36}-14q^{-38}+14q^{-40}-20q^{-42}+17q^{-44}-12q^{-46}+12q^{-48}-14q^{-50}+19q^{-52}-20q^{-54}+22q^{-56}-28q^{-58}+26q^{-60}-22q^{-62}+18q^{-64}-14q^{-66}+11q^{-68}-6q^{-70}+4q^{-72}-2q^{-74}+q^{-76}$
2,0	$q^{10}-q^{6}+q^{2}-1-3q^{-2}+2q^{-6}-2q^{-8}-3q^{-10}+q^{-12}+q^{-14}-q^{-16}+q^{-18}+5q^{-20}+3q^{-22}+5q^{-24}+5q^{-26}+5q^{-28}+q^{-30}+4q^{-32}+4q^{-34}-2q^{-36}-3q^{-38}-2q^{-40}-4q^{-42}-8q^{-44}-6q^{-46}-3q^{-48}-q^{-50}-q^{-52}+2q^{-56}+2q^{-58}+2q^{-60}+q^{-62}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{6}-q^{4}+q^{2}+1-3q^{-2}-3q^{-6}-6q^{-8}-q^{-10}-q^{-12}+9q^{-16}+8q^{-18}+7q^{-20}+11q^{-22}+6q^{-24}-2q^{-26}-q^{-28}-4q^{-30}-5q^{-32}-7q^{-34}-2q^{-36}-5q^{-40}+3q^{-44}-3q^{-46}-q^{-48}+5q^{-50}-2q^{-52}-2q^{-54}+3q^{-56}-q^{-60}+q^{-62}$
1,0,0	$-q-2q^{-3}-2q^{-7}+2q^{-9}+q^{-11}+4q^{-13}+4q^{-15}+4q^{-17}+3q^{-19}-4q^{-25}-q^{-27}-3q^{-29}-q^{-33}$
1,0,1	$q^{12}-2q^{10}+5q^{8}-5q^{6}+4q^{4}+2q^{2}-10+20q^{-2}-22q^{-4}+18q^{-6}-12q^{-8}-11q^{-10}+7q^{-12}-35q^{-14}+17q^{-16}-39q^{-18}+23q^{-20}-18q^{-22}+31q^{-24}+22q^{-26}+22q^{-28}+61q^{-30}-9q^{-32}+66q^{-34}-34q^{-36}+24q^{-38}-38q^{-40}-20q^{-42}-19q^{-44}-44q^{-46}+18q^{-48}-51q^{-50}+34q^{-52}-20q^{-54}+7q^{-56}+14q^{-58}-6q^{-60}+14q^{-62}-5q^{-64}+8q^{-66}-6q^{-68}+10q^{-70}-12q^{-72}+8q^{-74}-3q^{-76}-10q^{-78}+17q^{-80}-15q^{-82}+5q^{-84}+5q^{-86}-9q^{-88}+9q^{-90}-3q^{-92}-q^{-94}+3q^{-96}-2q^{-98}+q^{-100}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{4}+1+2q^{-2}+q^{-4}-q^{-6}-q^{-8}-5q^{-10}-8q^{-12}-10q^{-14}-11q^{-16}-7q^{-18}-q^{-20}+6q^{-22}+11q^{-24}+21q^{-26}+24q^{-28}+22q^{-30}+13q^{-32}+14q^{-34}+4q^{-36}-8q^{-38}-10q^{-40}-10q^{-42}-16q^{-44}-14q^{-46}-6q^{-48}-8q^{-50}-6q^{-52}+3q^{-56}-2q^{-58}+5q^{-62}+2q^{-64}-2q^{-66}+2q^{-68}+3q^{-70}+q^{-76}$
1,0,0,0	$-1-2q^{-4}-q^{-6}-2q^{-8}-q^{-10}+q^{-12}+q^{-14}+5q^{-16}+4q^{-18}+7q^{-20}+4q^{-22}+4q^{-24}-q^{-28}-3q^{-30}-4q^{-32}-2q^{-34}-3q^{-36}-q^{-40}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{6}+q^{4}-3q^{2}+3-5q^{-2}+6q^{-4}-7q^{-6}+6q^{-8}-5q^{-10}+5q^{-12}+q^{-16}+6q^{-18}-5q^{-20}+11q^{-22}-10q^{-24}+12q^{-26}-11q^{-28}+10q^{-30}-9q^{-32}+5q^{-34}-4q^{-36}+q^{-40}-4q^{-42}+5q^{-44}-5q^{-46}+5q^{-48}-5q^{-50}+4q^{-52}-4q^{-54}+3q^{-56}-2q^{-58}+q^{-60}-q^{-62}$
1,0	$q^{12}-q^{8}-q^{6}+2q^{4}+2q^{2}-2-4q^{-2}+4q^{-6}-7q^{-10}-5q^{-12}+3q^{-14}+5q^{-16}-q^{-18}-7q^{-20}+7q^{-24}+8q^{-26}-q^{-30}+3q^{-32}+8q^{-34}+4q^{-36}+4q^{-42}+q^{-44}-4q^{-46}-4q^{-48}+q^{-50}+2q^{-52}-4q^{-54}-7q^{-56}-2q^{-58}+4q^{-60}-5q^{-64}-4q^{-66}+2q^{-68}+4q^{-70}-4q^{-74}-3q^{-76}+2q^{-78}+5q^{-80}+q^{-82}-3q^{-84}-3q^{-86}+q^{-88}+3q^{-90}+q^{-92}-q^{-94}-q^{-96}+q^{-100}$

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{12}-q^{10}+3q^{8}-4q^{6}+3q^{4}-3q^{2}-2+7q^{-2}-14q^{-4}+15q^{-6}-14q^{-8}+3q^{-10}+7q^{-12}-19q^{-14}+23q^{-16}-20q^{-18}+9q^{-20}+3q^{-22}-15q^{-24}+18q^{-26}-12q^{-28}+4q^{-30}+8q^{-32}-10q^{-34}+11q^{-36}-6q^{-40}+16q^{-42}-14q^{-44}+15q^{-46}-2q^{-48}-5q^{-50}+18q^{-52}-20q^{-54}+24q^{-56}-12q^{-58}+2q^{-60}+11q^{-62}-17q^{-64}+19q^{-66}-13q^{-68}+5q^{-70}+5q^{-72}-10q^{-74}+8q^{-76}-4q^{-78}-5q^{-80}+8q^{-82}-9q^{-84}+3q^{-88}-9q^{-90}+8q^{-92}-7q^{-94}+2q^{-96}-q^{-98}-4q^{-100}+4q^{-102}-7q^{-104}+6q^{-106}-6q^{-108}+5q^{-110}-3q^{-112}-2q^{-114}+6q^{-116}-10q^{-118}+11q^{-120}-7q^{-122}+3q^{-124}+q^{-126}-5q^{-128}+6q^{-130}-6q^{-132}+5q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

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

In[1]:=

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

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 47"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-3t^{3}+6t^{2}-7t+7-7t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+5z^{6}+8z^{4}+6z^{2}+1$

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]=

{ 41, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-z^{6}a^{-2}+7z^{6}a^{-4}-z^{6}a^{-6}-5z^{4}a^{-2}+18z^{4}a^{-4}-5z^{4}a^{-6}-7z^{2}a^{-2}+21z^{2}a^{-4}-8z^{2}a^{-6}-3a^{-2}+9a^{-4}-5a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+z^{7}a^{-1}-z^{7}a^{-3}+z^{7}a^{-5}+3z^{7}a^{-7}-10z^{6}a^{-2}-23z^{6}a^{-4}-10z^{6}a^{-6}+3z^{6}a^{-8}-5z^{5}a^{-1}-11z^{5}a^{-3}-14z^{5}a^{-5}-5z^{5}a^{-7}+3z^{5}a^{-9}+15z^{4}a^{-2}+35z^{4}a^{-4}+15z^{4}a^{-6}-3z^{4}a^{-8}+2z^{4}a^{-10}+7z^{3}a^{-1}+20z^{3}a^{-3}+19z^{3}a^{-5}+2z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}-9z^{2}a^{-2}-26z^{2}a^{-4}-15z^{2}a^{-6}+z^{2}a^{-8}-z^{2}a^{-10}-3za^{-1}-8za^{-3}-9za^{-5}-za^{-7}+2za^{-9}-za^{-11}+3a^{-2}+9a^{-4}+5a^{-6}$

Vassiliev invariants

V2 and V3:

(6, 11)

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 $24$ $88$ $288$ $540$ $76$ $2112$ ${\frac {10096}{3}}$ ${\frac {1696}{3}}$ $440$ $2304$ $3872$ $12960$ $1824$ ${\frac {110351}{5}}$ ${\frac {11108}{15}}$ ${\frac {40988}{5}}$ $155$ ${\frac {5151}{5}}$

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=$4 is the signature of 10 47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 47]]
Out[2]=	10
In[3]:=	PD[Knot[10, 47]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[9, 17, 10, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[11, 19, 12, 18], X[13, 1, 14, 20], X[17, 11, 18, 10], X[19, 13, 20, 12], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[10, 47]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7]
In[5]:=	BR[Knot[10, 47]]
Out[5]=	BR[3, {1, 1, 1, 1, 1, -2, 1, 1, -2, -2}]
In[6]:=	alex = Alexander[Knot[10, 47]][t]
Out[6]=	-4 3 6 7 2 3 4 7 + t - -- + -- - - - 7 t + 6 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 47]][z]
Out[7]=	2 4 6 8 1 + 6 z + 8 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 47]}
In[9]:=	{KnotDet[Knot[10, 47]], KnotSignature[Knot[10, 47]]}
Out[9]=	{41, 4}
In[10]:=	J=Jones[Knot[10, 47]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 2 - - - 3 q + 5 q - 5 q + 7 q - 6 q + 5 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 47]}
In[12]:=	A2Invariant[Knot[10, 47]][q]
Out[12]=	-2 2 6 8 10 12 14 18 20 22 26 -q - q + q + q + 4 q + q + 3 q - q - q - 2 q - q
In[13]:=	Kauffman[Knot[10, 47]][a, z]
Out[13]=	2 2 2 5 9 3 z 2 z z 9 z 8 z 3 z z z 15 z -- + -- + -- - --- + --- - -- - --- - --- - --- - --- + -- - ----- - 6 4 2 11 9 7 5 3 a 10 8 6 a a a a a a a a a a a 2 2 3 3 3 3 3 3 4 26 z 9 z z 3 z 2 z 19 z 20 z 7 z 2 z ----- - ---- + --- - ---- + ---- + ----- + ----- + ---- + ---- - 4 2 11 9 7 5 3 a 10 a a a a a a a a 4 4 4 4 5 5 5 5 5 3 z 15 z 35 z 15 z 3 z 5 z 14 z 11 z 5 z ---- + ----- + ----- + ----- + ---- - ---- - ----- - ----- - ---- + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 7 8 8 3 z 10 z 23 z 10 z 3 z z z z 3 z 5 z ---- - ----- - ----- - ----- + ---- + -- - -- + -- + ---- + ---- + 8 6 4 2 7 5 3 a 6 4 a a a a a a a a a 8 9 9 2 z z z ---- + -- + -- 2 5 3 a a a
In[14]:=	{Vassiliev[2][Knot[10, 47]], Vassiliev[3][Knot[10, 47]]}
Out[14]=	{0, 11}
In[15]:=	Kh[Knot[10, 47]][q, t]
Out[15]=	3 3 5 1 1 q 2 q q 5 7 3 q + 3 q + ----- + ---- + -- + --- + -- + 3 q t + 2 q t + 3 3 2 2 t t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 4 q t + 3 q t + 2 q t + 4 q t + 3 q t + 2 q t + 13 5 15 5 15 6 17 6 19 7 q t + 3 q t + q t + q t + q t