10 62

10_61

10_63

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10 62 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,19,12,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8 X13,1,14,20 X19,13,20,12 X9,2,10,3
Gauss code	-1, 10, -2, 1, -4, 6, -5, 7, -10, 2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8
Dowker-Thistlethwaite code	4 10 14 16 2 18 20 6 8 12
Conway Notation	[4,3,21]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [0][-12] Hyperbolic Volume 10.1415 A-Polynomial See Data:10 62/A-polynomial

[edit Notes for 10 62's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 62's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-3t^{3}+6t^{2}-8t+9-8t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+5z^{6}+8z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 45, 4 }
Jones polynomial	$-q^{9}+2q^{8}-4q^{7}+6q^{6}-7q^{5}+7q^{4}-6q^{3}+6q^{2}-3q+2-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+7z^{6}a^{-4}-z^{6}a^{-6}-5z^{4}a^{-2}+18z^{4}a^{-4}-5z^{4}a^{-6}-7z^{2}a^{-2}+20z^{2}a^{-4}-8z^{2}a^{-6}-2a^{-2}+7a^{-4}-4a^{-6}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+z^{7}a^{-1}-z^{7}a^{-3}+2z^{7}a^{-5}+4z^{7}a^{-7}-10z^{6}a^{-2}-21z^{6}a^{-4}-7z^{6}a^{-6}+4z^{6}a^{-8}-5z^{5}a^{-1}-9z^{5}a^{-3}-15z^{5}a^{-5}-8z^{5}a^{-7}+3z^{5}a^{-9}+16z^{4}a^{-2}+30z^{4}a^{-4}+6z^{4}a^{-6}-6z^{4}a^{-8}+2z^{4}a^{-10}+7z^{3}a^{-1}+15z^{3}a^{-3}+16z^{3}a^{-5}+5z^{3}a^{-7}-2z^{3}a^{-9}+z^{3}a^{-11}-10z^{2}a^{-2}-23z^{2}a^{-4}-8z^{2}a^{-6}+4z^{2}a^{-8}-z^{2}a^{-10}-2za^{-1}-5za^{-3}-6za^{-5}-za^{-7}+za^{-9}-za^{-11}+2a^{-2}+7a^{-4}+4a^{-6}$
The A2 invariant	$-q^{2}-q^{-2}+q^{-4}+2q^{-6}+q^{-8}+3q^{-10}-q^{-12}+2q^{-14}-2q^{-22}-q^{-26}$
The G2 invariant	$q^{12}-q^{10}+3q^{8}-5q^{6}+4q^{4}-4q^{2}-2+9q^{-2}-16q^{-4}+19q^{-6}-18q^{-8}+5q^{-10}+11q^{-12}-27q^{-14}+30q^{-16}-27q^{-18}+13q^{-20}+8q^{-22}-23q^{-24}+27q^{-26}-20q^{-28}+11q^{-30}+12q^{-32}-20q^{-34}+15q^{-36}-3q^{-38}-3q^{-40}+21q^{-42}-22q^{-44}+21q^{-46}-5q^{-48}-5q^{-50}+25q^{-52}-38q^{-54}+33q^{-56}-19q^{-58}+q^{-60}+18q^{-62}-31q^{-64}+33q^{-66}-22q^{-68}+6q^{-70}+11q^{-72}-23q^{-74}+17q^{-76}-7q^{-78}-7q^{-80}+16q^{-82}-13q^{-84}+6q^{-86}+4q^{-88}-13q^{-90}+16q^{-92}-16q^{-94}+8q^{-96}-q^{-98}-9q^{-100}+12q^{-102}-13q^{-104}+13q^{-106}-10q^{-108}+4q^{-110}-9q^{-114}+10q^{-116}-12q^{-118}+10q^{-120}-5q^{-122}+2q^{-124}+3q^{-126}-6q^{-128}+6q^{-130}-5q^{-132}+4q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

Further Quantum Invariants

Further quantum knot invariants for 10_62.

A1 Invariants.

Weight	Invariant
1	$-q^{3}+q-q^{-1}+3q^{-3}+q^{-7}-q^{-11}+2q^{-13}-2q^{-15}+q^{-17}-q^{-19}$
2	$q^{12}-q^{10}-2q^{8}+3q^{6}-6q^{2}+3+5q^{-2}-6q^{-4}+2q^{-6}+8q^{-8}-3q^{-10}-2q^{-12}+6q^{-14}-q^{-16}-6q^{-18}+2q^{-20}+4q^{-22}-3q^{-24}-2q^{-26}+5q^{-28}-7q^{-32}+4q^{-34}+2q^{-36}-6q^{-38}+3q^{-40}+q^{-42}-3q^{-44}+2q^{-46}-q^{-50}+q^{-52}$
3	$-q^{27}+q^{25}+2q^{23}-4q^{19}-2q^{17}+6q^{15}+6q^{13}-6q^{11}-12q^{9}+q^{7}+15q^{5}+6q^{3}-17q-14q^{-1}+10q^{-3}+22q^{-5}-2q^{-7}-18q^{-9}-6q^{-11}+20q^{-13}+15q^{-15}-10q^{-17}-17q^{-19}+5q^{-21}+18q^{-23}-20q^{-27}-5q^{-29}+20q^{-31}+7q^{-33}-18q^{-35}-12q^{-37}+19q^{-39}+14q^{-41}-13q^{-43}-19q^{-45}+3q^{-47}+17q^{-49}+7q^{-51}-14q^{-53}-16q^{-55}+7q^{-57}+21q^{-59}-19q^{-63}-7q^{-65}+16q^{-67}+7q^{-69}-8q^{-71}-6q^{-73}+2q^{-75}+3q^{-77}-q^{-85}+q^{-87}+q^{-89}-q^{-91}+q^{-97}-q^{-99}$
4	$q^{48}-q^{46}-2q^{44}+q^{40}+6q^{38}-6q^{34}-6q^{32}-5q^{30}+15q^{28}+14q^{26}-15q^{22}-30q^{20}+3q^{18}+26q^{16}+33q^{14}+10q^{12}-47q^{10}-42q^{8}-11q^{6}+45q^{4}+66q^{2}+1-47q^{-2}-74q^{-4}-16q^{-6}+70q^{-8}+69q^{-10}+28q^{-12}-67q^{-14}-86q^{-16}-7q^{-18}+62q^{-20}+98q^{-22}+13q^{-24}-81q^{-26}-82q^{-28}-13q^{-30}+94q^{-32}+81q^{-34}-23q^{-36}-93q^{-38}-65q^{-40}+51q^{-42}+92q^{-44}+20q^{-46}-70q^{-48}-70q^{-50}+28q^{-52}+83q^{-54}+23q^{-56}-68q^{-58}-66q^{-60}+20q^{-62}+84q^{-64}+36q^{-66}-58q^{-68}-78q^{-70}-20q^{-72}+72q^{-74}+80q^{-76}+4q^{-78}-66q^{-80}-94q^{-82}-10q^{-84}+88q^{-86}+98q^{-88}+17q^{-90}-116q^{-92}-108q^{-94}+19q^{-96}+121q^{-98}+105q^{-100}-55q^{-102}-124q^{-104}-52q^{-106}+57q^{-108}+106q^{-110}+9q^{-112}-61q^{-114}-54q^{-116}-q^{-118}+55q^{-120}+19q^{-122}-12q^{-124}-21q^{-126}-13q^{-128}+16q^{-130}+5q^{-132}+3q^{-134}-2q^{-136}-8q^{-138}+5q^{-140}-q^{-142}+q^{-144}-3q^{-148}+3q^{-150}-q^{-152}-q^{-158}+q^{-160}$
5	$-q^{75}+q^{73}+2q^{71}-q^{67}-3q^{65}-4q^{63}+8q^{59}+8q^{57}+2q^{55}-6q^{53}-16q^{51}-16q^{49}+2q^{47}+25q^{45}+31q^{43}+16q^{41}-16q^{39}-50q^{37}-50q^{35}-9q^{33}+50q^{31}+82q^{29}+63q^{27}-10q^{25}-95q^{23}-123q^{21}-64q^{19}+53q^{17}+151q^{15}+154q^{13}+43q^{11}-115q^{9}-215q^{7}-171q^{5}+6q^{3}+193q+260q^{-1}+157q^{-3}-78q^{-5}-280q^{-7}-292q^{-9}-96q^{-11}+174q^{-13}+356q^{-15}+295q^{-17}+8q^{-19}-300q^{-21}-403q^{-23}-228q^{-25}+135q^{-27}+432q^{-29}+417q^{-31}+81q^{-33}-331q^{-35}-517q^{-37}-306q^{-39}+161q^{-41}+516q^{-43}+467q^{-45}+38q^{-47}-430q^{-49}-554q^{-51}-219q^{-53}+294q^{-55}+557q^{-57}+344q^{-59}-150q^{-61}-498q^{-63}-397q^{-65}+36q^{-67}+410q^{-69}+389q^{-71}+27q^{-73}-324q^{-75}-346q^{-77}-39q^{-79}+267q^{-81}+279q^{-83}+17q^{-85}-254q^{-87}-251q^{-89}+24q^{-91}+277q^{-93}+250q^{-95}-27q^{-97}-297q^{-99}-307q^{-101}-26q^{-103}+292q^{-105}+372q^{-107}+147q^{-109}-205q^{-111}-414q^{-113}-320q^{-115}+31q^{-117}+390q^{-119}+486q^{-121}+209q^{-123}-257q^{-125}-574q^{-127}-472q^{-129}+37q^{-131}+566q^{-133}+657q^{-135}+215q^{-137}-425q^{-139}-736q^{-141}-437q^{-143}+230q^{-145}+686q^{-147}+550q^{-149}-25q^{-151}-538q^{-153}-566q^{-155}-121q^{-157}+366q^{-159}+482q^{-161}+188q^{-163}-209q^{-165}-358q^{-167}-189q^{-169}+98q^{-171}+239q^{-173}+151q^{-175}-37q^{-177}-141q^{-179}-102q^{-181}+3q^{-183}+75q^{-185}+67q^{-187}+9q^{-189}-37q^{-191}-38q^{-193}-9q^{-195}+12q^{-197}+19q^{-199}+10q^{-201}-4q^{-203}-11q^{-205}-5q^{-207}+3q^{-209}+q^{-211}+2q^{-213}+2q^{-215}-3q^{-217}+2q^{-221}-q^{-223}-q^{-225}+q^{-227}+q^{-233}-q^{-235}$
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}+9q^{82}+21q^{80}+10q^{78}-14q^{76}-29q^{74}-34q^{72}-15q^{70}+q^{68}+55q^{66}+70q^{64}+44q^{62}-8q^{60}-70q^{58}-102q^{56}-116q^{54}-17q^{52}+89q^{50}+171q^{48}+173q^{46}+91q^{44}-55q^{42}-247q^{40}-282q^{38}-203q^{36}+16q^{34}+247q^{32}+411q^{30}+385q^{28}+96q^{26}-229q^{24}-519q^{22}-551q^{20}-331q^{18}+135q^{16}+586q^{14}+740q^{12}+578q^{10}+57q^{8}-520q^{6}-942q^{4}-861q^{2}-335+378q^{-2}+1012q^{-4}+1153q^{-6}+736q^{-8}-168q^{-10}-1003q^{-12}-1400q^{-14}-1133q^{-16}-205q^{-18}+911q^{-20}+1665q^{-22}+1519q^{-24}+606q^{-26}-723q^{-28}-1808q^{-30}-1964q^{-32}-1021q^{-34}+590q^{-36}+1918q^{-38}+2312q^{-40}+1416q^{-42}-395q^{-44}-2090q^{-46}-2632q^{-48}-1644q^{-50}+292q^{-52}+2186q^{-54}+2872q^{-56}+1831q^{-58}-365q^{-60}-2361q^{-62}-2948q^{-64}-1810q^{-66}+481q^{-68}+2522q^{-70}+2988q^{-72}+1571q^{-74}-774q^{-76}-2595q^{-78}-2809q^{-80}-1231q^{-82}+1113q^{-84}+2660q^{-86}+2420q^{-88}+688q^{-90}-1400q^{-92}-2508q^{-94}-1920q^{-96}-101q^{-98}+1635q^{-100}+2122q^{-102}+1210q^{-104}-423q^{-106}-1612q^{-108}-1575q^{-110}-440q^{-112}+845q^{-114}+1317q^{-116}+802q^{-118}-256q^{-120}-975q^{-122}-842q^{-124}+14q^{-126}+829q^{-128}+885q^{-130}+198q^{-132}-740q^{-134}-1144q^{-136}-699q^{-138}+375q^{-140}+1303q^{-142}+1363q^{-144}+508q^{-146}-775q^{-148}-1646q^{-150}-1587q^{-152}-522q^{-154}+961q^{-156}+1974q^{-158}+1906q^{-160}+711q^{-162}-988q^{-164}-2316q^{-166}-2405q^{-168}-1044q^{-170}+1076q^{-172}+2735q^{-174}+2889q^{-176}+1336q^{-178}-1200q^{-180}-3200q^{-182}-3303q^{-184}-1366q^{-186}+1472q^{-188}+3479q^{-190}+3417q^{-192}+1200q^{-194}-1800q^{-196}-3590q^{-198}-3103q^{-200}-736q^{-202}+1968q^{-204}+3351q^{-206}+2525q^{-208}+154q^{-210}-2040q^{-212}-2737q^{-214}-1677q^{-216}+270q^{-218}+1841q^{-220}+2002q^{-222}+853q^{-224}-580q^{-226}-1393q^{-228}-1189q^{-230}-307q^{-232}+632q^{-234}+935q^{-236}+546q^{-238}-52q^{-240}-471q^{-242}-487q^{-244}-209q^{-246}+173q^{-248}+314q^{-250}+190q^{-252}+7q^{-254}-133q^{-256}-149q^{-258}-82q^{-260}+56q^{-262}+99q^{-264}+51q^{-266}+8q^{-268}-35q^{-270}-45q^{-272}-38q^{-274}+16q^{-276}+32q^{-278}+13q^{-280}+9q^{-282}-6q^{-284}-12q^{-286}-17q^{-288}+5q^{-290}+9q^{-292}+q^{-294}+5q^{-296}-q^{-298}-q^{-300}-6q^{-302}+2q^{-304}+2q^{-306}-2q^{-308}+2q^{-310}-q^{-312}+q^{-314}-q^{-316}-q^{-322}+q^{-324}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{2}-q^{-2}+q^{-4}+2q^{-6}+q^{-8}+3q^{-10}-q^{-12}+2q^{-14}-2q^{-22}-q^{-26}$
1,1	$q^{12}-2q^{10}+6q^{8}-14q^{6}+25q^{4}-40q^{2}+58-80q^{-2}+85q^{-4}-92q^{-6}+82q^{-8}-62q^{-10}+33q^{-12}+14q^{-14}-38q^{-16}+90q^{-18}-104q^{-20}+138q^{-22}-144q^{-24}+140q^{-26}-135q^{-28}+98q^{-30}-80q^{-32}+42q^{-34}-10q^{-36}-14q^{-38}+32q^{-40}-38q^{-42}+39q^{-44}-42q^{-46}+36q^{-48}-34q^{-50}+34q^{-52}-32q^{-54}+30q^{-56}-28q^{-58}+24q^{-60}-20q^{-62}+16q^{-64}-12q^{-66}+9q^{-68}-6q^{-70}+4q^{-72}-2q^{-74}+q^{-76}$
2,0	$q^{10}-q^{6}+q^{2}-2-4q^{-2}-q^{-4}+q^{-6}-q^{-8}-q^{-10}+6q^{-12}+5q^{-14}+3q^{-16}+4q^{-18}+4q^{-20}-2q^{-22}-2q^{-24}-q^{-30}+3q^{-32}+5q^{-34}-2q^{-36}-2q^{-38}+q^{-40}-2q^{-42}-5q^{-44}-3q^{-46}-q^{-48}-q^{-50}-q^{-52}+q^{-56}+q^{-60}+q^{-62}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{6}-q^{4}+q^{2}-4q^{-2}+q^{-4}-2q^{-6}-4q^{-8}+3q^{-10}+q^{-12}-q^{-14}+9q^{-16}+4q^{-18}+2q^{-20}+9q^{-22}+4q^{-24}-2q^{-26}-q^{-28}-2q^{-30}-4q^{-32}-7q^{-34}-q^{-36}+2q^{-38}-5q^{-40}+q^{-42}+5q^{-44}-5q^{-46}-q^{-48}+5q^{-50}-3q^{-52}-2q^{-54}+3q^{-56}-q^{-60}+q^{-62}$
1,0,0	$-q-2q^{-3}+q^{-5}-q^{-7}+3q^{-9}+q^{-11}+3q^{-13}+2q^{-15}+2q^{-17}+2q^{-19}+q^{-23}-3q^{-25}-3q^{-29}-q^{-33}$
1,0,1	$q^{12}-2q^{10}+5q^{8}-7q^{6}+6q^{4}-9+23q^{-2}-32q^{-4}+27q^{-6}-25q^{-8}-5q^{-10}+15q^{-12}-47q^{-14}+48q^{-16}-49q^{-18}+42q^{-20}-7q^{-22}+22q^{-24}+31q^{-26}+2q^{-28}+55q^{-30}-31q^{-32}+50q^{-34}-51q^{-36}+17q^{-38}-43q^{-40}-17q^{-42}+6q^{-44}-43q^{-46}+46q^{-48}-38q^{-50}+30q^{-52}-6q^{-54}-10q^{-56}+16q^{-58}-18q^{-60}+8q^{-62}+4q^{-64}-5q^{-66}+7q^{-68}+3q^{-70}-10q^{-72}+10q^{-74}-6q^{-76}-3q^{-78}+10q^{-80}-10q^{-82}+4q^{-84}+3q^{-86}-7q^{-88}+7q^{-90}-3q^{-92}-q^{-94}+3q^{-96}-2q^{-98}+q^{-100}$

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{6}-q^{4}+2q^{2}-3+4q^{-2}-6q^{-4}+4q^{-6}-8q^{-8}+5q^{-10}-9q^{-12}+3q^{-14}-6q^{-16}+5q^{-18}+3q^{-22}+8q^{-24}+4q^{-26}+15q^{-28}-q^{-30}+16q^{-32}-8q^{-34}+13q^{-36}-14q^{-38}+7q^{-40}-15q^{-42}+3q^{-44}-12q^{-46}+2q^{-48}-5q^{-50}+q^{-54}-3q^{-56}+4q^{-58}-4q^{-60}+6q^{-62}-6q^{-64}+4q^{-66}-5q^{-68}+6q^{-70}-4q^{-72}+2q^{-74}-3q^{-76}+3q^{-78}-q^{-80}+q^{-82}-q^{-84}+q^{-86}$

G2 Invariants.

Weight

Invariant

1,0

$q^{12}-q^{10}+3q^{8}-5q^{6}+4q^{4}-4q^{2}-2+9q^{-2}-16q^{-4}+19q^{-6}-18q^{-8}+5q^{-10}+11q^{-12}-27q^{-14}+30q^{-16}-27q^{-18}+13q^{-20}+8q^{-22}-23q^{-24}+27q^{-26}-20q^{-28}+11q^{-30}+12q^{-32}-20q^{-34}+15q^{-36}-3q^{-38}-3q^{-40}+21q^{-42}-22q^{-44}+21q^{-46}-5q^{-48}-5q^{-50}+25q^{-52}-38q^{-54}+33q^{-56}-19q^{-58}+q^{-60}+18q^{-62}-31q^{-64}+33q^{-66}-22q^{-68}+6q^{-70}+11q^{-72}-23q^{-74}+17q^{-76}-7q^{-78}-7q^{-80}+16q^{-82}-13q^{-84}+6q^{-86}+4q^{-88}-13q^{-90}+16q^{-92}-16q^{-94}+8q^{-96}-q^{-98}-9q^{-100}+12q^{-102}-13q^{-104}+13q^{-106}-10q^{-108}+4q^{-110}-9q^{-114}+10q^{-116}-12q^{-118}+10q^{-120}-5q^{-122}+2q^{-124}+3q^{-126}-6q^{-128}+6q^{-130}-5q^{-132}+4q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

.

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 62"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-3t^{3}+6t^{2}-8t+9-8t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 45, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-z^{6}a^{-2}+7z^{6}a^{-4}-z^{6}a^{-6}-5z^{4}a^{-2}+18z^{4}a^{-4}-5z^{4}a^{-6}-7z^{2}a^{-2}+20z^{2}a^{-4}-8z^{2}a^{-6}-2a^{-2}+7a^{-4}-4a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+z^{7}a^{-1}-z^{7}a^{-3}+2z^{7}a^{-5}+4z^{7}a^{-7}-10z^{6}a^{-2}-21z^{6}a^{-4}-7z^{6}a^{-6}+4z^{6}a^{-8}-5z^{5}a^{-1}-9z^{5}a^{-3}-15z^{5}a^{-5}-8z^{5}a^{-7}+3z^{5}a^{-9}+16z^{4}a^{-2}+30z^{4}a^{-4}+6z^{4}a^{-6}-6z^{4}a^{-8}+2z^{4}a^{-10}+7z^{3}a^{-1}+15z^{3}a^{-3}+16z^{3}a^{-5}+5z^{3}a^{-7}-2z^{3}a^{-9}+z^{3}a^{-11}-10z^{2}a^{-2}-23z^{2}a^{-4}-8z^{2}a^{-6}+4z^{2}a^{-8}-z^{2}a^{-10}-2za^{-1}-5za^{-3}-6za^{-5}-za^{-7}+za^{-9}-za^{-11}+2a^{-2}+7a^{-4}+4a^{-6}$

Vassiliev invariants

V2 and V3:

(5, 9)

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 $20$ $72$ $200$ ${\frac {1126}{3}}$ ${\frac {98}{3}}$ 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 1440} $2192$ $352$ $200$ ${\frac {4000}{3}}$ $2592$ ${\frac {22520}{3}}$ ${\frac {1960}{3}}$ ${\frac {79231}{6}}$ ${\frac {3514}{3}}$ ${\frac {32246}{9}}$ ${\frac {2149}{18}}$ ${\frac {1855}{6}}$

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 62. 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               3 1     2 11             4 3       -1 9           3 3         0 7         3 4           1 5       3 3             0 3     1 4               3 1   1 2                 -1 -1   1                   1 -3 1                     -1

Integral Khovanov Homology (db, data source)

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

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, 62]]
Out[2]=	10
In[3]:=	PD[Knot[10, 62]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 19, 12, 18], X[5, 15, 6, 14], X[7, 17, 8, 16], X[15, 7, 16, 6], X[17, 9, 18, 8], X[13, 1, 14, 20], X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[10, 62]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 6, -5, 7, -10, 2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8]
In[5]:=	BR[Knot[10, 62]]
Out[5]=	BR[3, {1, 1, 1, 1, -2, 1, 1, 1, -2, -2}]
In[6]:=	alex = Alexander[Knot[10, 62]][t]
Out[6]=	-4 3 6 8 2 3 4 9 + t - -- + -- - - - 8 t + 6 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 62]][z]
Out[7]=	2 4 6 8 1 + 5 z + 8 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 62], Knot[11, NonAlternating, 76], Knot[11, NonAlternating, 78]}
In[9]:=	{KnotDet[Knot[10, 62]], KnotSignature[Knot[10, 62]]}
Out[9]=	{45, 4}
In[10]:=	J=Jones[Knot[10, 62]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 2 - - - 3 q + 6 q - 6 q + 7 q - 7 q + 6 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 62]}
In[12]:=	A2Invariant[Knot[10, 62]][q]
Out[12]=	-2 2 4 6 8 10 12 14 22 26 -q - q + q + 2 q + q + 3 q - q + 2 q - 2 q - q
In[13]:=	Kauffman[Knot[10, 62]][a, z]
Out[13]=	2 2 2 4 7 2 z z z 6 z 5 z 2 z z 4 z 8 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 23 z 10 z z 2 z 5 z 16 z 15 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 6 z 6 z 30 z 16 z 3 z 8 z 15 z 9 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 4 z 7 z 21 z 10 z 4 z 2 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, 62]], Vassiliev[3][Knot[10, 62]]}
Out[14]=	{0, 9}
In[15]:=	Kh[Knot[10, 62]][q, t]
Out[15]=	3 3 5 1 1 q 2 q q 5 7 4 q + 3 q + ----- + ---- + -- + --- + -- + 3 q t + 3 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 + 3 q t + 4 q t + 3 q t + 3 q t + 13 5 15 5 15 6 17 6 19 7 q t + 3 q t + q t + q t + q t