10 9

10_8

10_10

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10 9 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X16,8,17,7 X12,3,13,4 X2,15,3,16 X14,5,15,6 X4,13,5,14 X18,10,19,9 X20,12,1,11 X8,18,9,17 X10,20,11,19
Gauss code	1, -4, 3, -6, 5, -1, 2, -9, 7, -10, 8, -3, 6, -5, 4, -2, 9, -7, 10, -8
Dowker-Thistlethwaite code	6 12 14 16 18 20 4 2 8 10
Conway Notation	[5113]

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 1 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 8.2941 A-Polynomial See Data:10 9/A-polynomial

[edit Notes for 10 9's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $4$ Rasmussen s-Invariant -2

[edit Notes for 10 9's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+3t^{3}-5t^{2}+7t-7+7t^{-1}-5t^{-2}+3t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-5z^{6}-7z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 39, 2 }
Jones polynomial	$q^{7}-2q^{6}+3q^{5}-5q^{4}+6q^{3}-6q^{2}+6q-4+3q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-7z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-17z^{4}a^{-2}+5z^{4}a^{-4}+5z^{4}-16z^{2}a^{-2}+7z^{2}a^{-4}+7z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+2z^{8}a^{-4}+2z^{8}+2az^{7}-2z^{7}a^{-1}-2z^{7}a^{-3}+2z^{7}a^{-5}+a^{2}z^{6}-18z^{6}a^{-2}-7z^{6}a^{-4}+2z^{6}a^{-6}-8z^{6}-8az^{5}-2z^{5}a^{-1}-4z^{5}a^{-5}+2z^{5}a^{-7}-4a^{2}z^{4}+31z^{4}a^{-2}+13z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+10z^{4}+7az^{3}+4z^{3}a^{-1}+5z^{3}a^{-3}+4z^{3}a^{-5}-4z^{3}a^{-7}+3a^{2}z^{2}-22z^{2}a^{-2}-8z^{2}a^{-4}+z^{2}a^{-6}-2z^{2}a^{-8}-8z^{2}-az-2za^{-1}-2za^{-3}+za^{-7}+4a^{-2}+2a^{-4}+3$
The A2 invariant	$q^{8}+q^{4}+q^{-2}-q^{-4}+2q^{-6}-q^{-8}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
The G2 invariant	$q^{46}-q^{44}+2q^{42}-3q^{40}+2q^{38}-2q^{36}-q^{34}+6q^{32}-8q^{30}+10q^{28}-9q^{26}+5q^{24}+2q^{22}-10q^{20}+16q^{18}-16q^{16}+14q^{14}-4q^{12}-5q^{10}+15q^{8}-14q^{6}+12q^{4}-4q^{2}-4+10q^{-2}-9q^{-4}+3q^{-6}+6q^{-8}-10q^{-10}+14q^{-12}-9q^{-14}-2q^{-16}+8q^{-18}-16q^{-20}+18q^{-22}-17q^{-24}+7q^{-26}+3q^{-28}-13q^{-30}+18q^{-32}-18q^{-34}+11q^{-36}-3q^{-38}-6q^{-40}+10q^{-42}-12q^{-44}+8q^{-46}-5q^{-50}+6q^{-52}-4q^{-54}-2q^{-56}+7q^{-58}-8q^{-60}+7q^{-62}-4q^{-64}-q^{-66}+6q^{-68}-8q^{-70}+11q^{-72}-8q^{-74}+6q^{-76}-q^{-78}-2q^{-80}+5q^{-82}-8q^{-84}+10q^{-86}-7q^{-88}+4q^{-90}-4q^{-94}+6q^{-96}-6q^{-98}+5q^{-100}-3q^{-102}+q^{-106}-3q^{-108}+2q^{-110}-q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 10_9.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+q^{3}-q+2q^{-1}+q^{-7}-2q^{-9}+q^{-11}-q^{-13}+q^{-15}$
2	$q^{22}-q^{20}-q^{18}+3q^{16}-q^{14}-3q^{12}+4q^{10}+q^{8}-5q^{6}+2q^{4}+3q^{2}-4+q^{-2}+4q^{-4}-2q^{-6}-q^{-8}+3q^{-10}+q^{-12}-2q^{-14}-q^{-16}+4q^{-18}-3q^{-20}-3q^{-22}+5q^{-24}-2q^{-26}-2q^{-28}+3q^{-30}-q^{-32}-q^{-34}+2q^{-36}-q^{-38}-q^{-40}+q^{-42}$
3	$q^{45}-q^{43}-q^{41}+q^{39}+2q^{37}-q^{35}-4q^{33}+q^{31}+6q^{29}-7q^{25}-3q^{23}+8q^{21}+7q^{19}-6q^{17}-9q^{15}+2q^{13}+9q^{11}+2q^{9}-9q^{7}-6q^{5}+5q^{3}+10q-q^{-1}-9q^{-3}-q^{-5}+11q^{-7}+4q^{-9}-9q^{-11}-3q^{-13}+8q^{-15}+5q^{-17}-6q^{-19}-5q^{-21}+2q^{-23}+5q^{-25}-6q^{-29}-6q^{-31}+7q^{-33}+6q^{-35}-4q^{-37}-9q^{-39}+2q^{-41}+9q^{-43}+q^{-45}-4q^{-47}-3q^{-49}+2q^{-51}+3q^{-53}+2q^{-55}-3q^{-57}-3q^{-59}+3q^{-63}+q^{-65}-2q^{-67}-2q^{-69}+q^{-71}+2q^{-73}-q^{-77}-q^{-79}+q^{-81}$
4	$q^{76}-q^{74}-q^{72}+q^{70}+2q^{66}-3q^{64}-2q^{62}+3q^{60}+q^{58}+6q^{56}-6q^{54}-9q^{52}+2q^{50}+5q^{48}+16q^{46}-3q^{44}-17q^{42}-10q^{40}-3q^{38}+25q^{36}+14q^{34}-8q^{32}-17q^{30}-21q^{28}+13q^{26}+19q^{24}+13q^{22}+q^{20}-24q^{18}-11q^{16}-5q^{14}+14q^{12}+27q^{10}-16q^{6}-32q^{4}-6q^{2}+36+28q^{-2}-39q^{-6}-27q^{-8}+25q^{-10}+36q^{-12}+11q^{-14}-30q^{-16}-29q^{-18}+11q^{-20}+26q^{-22}+11q^{-24}-17q^{-26}-20q^{-28}+2q^{-30}+18q^{-32}+9q^{-34}-7q^{-36}-15q^{-38}-10q^{-40}+9q^{-42}+16q^{-44}+14q^{-46}-11q^{-48}-32q^{-50}-7q^{-52}+20q^{-54}+41q^{-56}+9q^{-58}-44q^{-60}-34q^{-62}-q^{-64}+50q^{-66}+36q^{-68}-25q^{-70}-39q^{-72}-28q^{-74}+27q^{-76}+43q^{-78}+4q^{-80}-20q^{-82}-35q^{-84}+26q^{-88}+15q^{-90}+2q^{-92}-24q^{-94}-10q^{-96}+9q^{-98}+11q^{-100}+10q^{-102}-11q^{-104}-8q^{-106}-q^{-108}+3q^{-110}+9q^{-112}-3q^{-114}-3q^{-116}-2q^{-118}-q^{-120}+4q^{-122}-q^{-128}-q^{-130}+q^{-132}$
5	$q^{115}-q^{113}-q^{111}+q^{109}-q^{101}-q^{99}+3q^{97}+4q^{95}-q^{93}-4q^{91}-6q^{89}-4q^{87}+4q^{85}+14q^{83}+11q^{81}-4q^{79}-17q^{77}-22q^{75}-8q^{73}+18q^{71}+35q^{69}+24q^{67}-9q^{65}-37q^{63}-42q^{61}-14q^{59}+29q^{57}+54q^{55}+37q^{53}-9q^{51}-46q^{49}-52q^{47}-21q^{45}+24q^{43}+48q^{41}+38q^{39}+10q^{37}-20q^{35}-37q^{33}-36q^{31}-22q^{29}+12q^{27}+47q^{25}+60q^{23}+35q^{21}-28q^{19}-87q^{17}-87q^{15}-9q^{13}+87q^{11}+122q^{9}+61q^{7}-63q^{5}-146q^{3}-107q+29q^{-1}+142q^{-3}+140q^{-5}+18q^{-7}-124q^{-9}-155q^{-11}-47q^{-13}+97q^{-15}+148q^{-17}+71q^{-19}-63q^{-21}-133q^{-23}-75q^{-25}+40q^{-27}+105q^{-29}+69q^{-31}-20q^{-33}-78q^{-35}-59q^{-37}+12q^{-39}+58q^{-41}+40q^{-43}-8q^{-45}-42q^{-47}-35q^{-49}+5q^{-51}+34q^{-53}+38q^{-55}+8q^{-57}-33q^{-59}-52q^{-61}-34q^{-63}+18q^{-65}+74q^{-67}+79q^{-69}+5q^{-71}-91q^{-73}-119q^{-75}-52q^{-77}+86q^{-79}+170q^{-81}+104q^{-83}-65q^{-85}-189q^{-87}-156q^{-89}+16q^{-91}+184q^{-93}+191q^{-95}+34q^{-97}-150q^{-99}-194q^{-101}-73q^{-103}+95q^{-105}+175q^{-107}+99q^{-109}-50q^{-111}-134q^{-113}-99q^{-115}+9q^{-117}+91q^{-119}+85q^{-121}+11q^{-123}-55q^{-125}-65q^{-127}-21q^{-129}+32q^{-131}+47q^{-133}+21q^{-135}-14q^{-137}-33q^{-139}-23q^{-141}+8q^{-143}+24q^{-145}+18q^{-147}-16q^{-151}-17q^{-153}-5q^{-155}+11q^{-157}+14q^{-159}+5q^{-161}-4q^{-163}-9q^{-165}-7q^{-167}+q^{-169}+7q^{-171}+4q^{-173}-2q^{-177}-4q^{-179}-q^{-181}+2q^{-183}+2q^{-185}-q^{-191}-q^{-193}+q^{-195}$
6	$q^{162}-q^{160}-q^{158}+q^{156}-2q^{150}+2q^{148}-q^{144}+5q^{142}+2q^{140}-2q^{138}-9q^{136}-2q^{134}-3q^{132}+15q^{128}+14q^{126}+6q^{124}-17q^{122}-15q^{120}-22q^{118}-16q^{116}+21q^{114}+39q^{112}+42q^{110}+4q^{108}-17q^{106}-55q^{104}-69q^{102}-22q^{100}+34q^{98}+85q^{96}+70q^{94}+49q^{92}-31q^{90}-104q^{88}-107q^{86}-58q^{84}+40q^{82}+89q^{80}+134q^{78}+78q^{76}-22q^{74}-99q^{72}-125q^{70}-71q^{68}-23q^{66}+74q^{64}+99q^{62}+79q^{60}+32q^{58}-9q^{56}-25q^{54}-81q^{52}-72q^{50}-76q^{48}-38q^{46}+33q^{44}+131q^{42}+197q^{40}+117q^{38}-7q^{36}-189q^{34}-289q^{32}-238q^{30}-7q^{28}+269q^{26}+379q^{24}+313q^{22}+22q^{20}-311q^{18}-497q^{16}-368q^{14}+q^{12}+358q^{10}+553q^{8}+407q^{6}-5q^{4}-445q^{2}-598-379q^{-2}+52q^{-4}+484q^{-6}+620q^{-8}+353q^{-10}-157q^{-12}-534q^{-14}-558q^{-16}-252q^{-18}+225q^{-20}+554q^{-22}+495q^{-24}+98q^{-26}-314q^{-28}-485q^{-30}-349q^{-32}+11q^{-34}+350q^{-36}+410q^{-38}+176q^{-40}-125q^{-42}-302q^{-44}-267q^{-46}-62q^{-48}+167q^{-50}+236q^{-52}+121q^{-54}-40q^{-56}-142q^{-58}-130q^{-60}-36q^{-62}+77q^{-64}+113q^{-66}+53q^{-68}-34q^{-70}-93q^{-72}-81q^{-74}-20q^{-76}+71q^{-78}+127q^{-80}+106q^{-82}+5q^{-84}-124q^{-86}-191q^{-88}-163q^{-90}+207q^{-94}+327q^{-96}+231q^{-98}-48q^{-100}-325q^{-102}-461q^{-104}-290q^{-106}+115q^{-108}+502q^{-110}+577q^{-112}+264q^{-114}-220q^{-116}-629q^{-118}-639q^{-120}-216q^{-122}+367q^{-124}+697q^{-126}+573q^{-128}+125q^{-130}-429q^{-132}-678q^{-134}-479q^{-136}+8q^{-138}+431q^{-140}+535q^{-142}+349q^{-144}-63q^{-146}-371q^{-148}-399q^{-150}-187q^{-152}+84q^{-154}+234q^{-156}+262q^{-158}+102q^{-160}-70q^{-162}-146q^{-164}-118q^{-166}-42q^{-168}+12q^{-170}+77q^{-172}+48q^{-174}+11q^{-176}-3q^{-178}-2q^{-180}-4q^{-182}-24q^{-184}+2q^{-186}-17q^{-188}-17q^{-190}+q^{-192}+22q^{-194}+24q^{-196}+4q^{-198}+15q^{-200}-16q^{-202}-23q^{-204}-19q^{-206}-q^{-208}+9q^{-210}+7q^{-212}+26q^{-214}+5q^{-216}-4q^{-218}-14q^{-220}-10q^{-222}-8q^{-224}-4q^{-226}+15q^{-228}+7q^{-230}+7q^{-232}-3q^{-234}-3q^{-236}-7q^{-238}-7q^{-240}+5q^{-242}+2q^{-244}+5q^{-246}-3q^{-252}-3q^{-254}+2q^{-256}+2q^{-260}-q^{-266}-q^{-268}+q^{-270}$

A2 Invariants.

Weight	Invariant
1,0	$q^{8}+q^{4}+q^{-2}-q^{-4}+2q^{-6}-q^{-8}-q^{-12}-q^{-14}+q^{-16}+q^{-20}$
1,1	$q^{28}-2q^{26}+4q^{24}-8q^{22}+15q^{20}-20q^{18}+30q^{16}-38q^{14}+45q^{12}-50q^{10}+50q^{8}-46q^{6}+31q^{4}-14q^{2}-2+30q^{-2}-50q^{-4}+68q^{-6}-76q^{-8}+82q^{-10}-84q^{-12}+74q^{-14}-62q^{-16}+46q^{-18}-32q^{-20}+18q^{-22}-2q^{-24}-4q^{-26}+17q^{-28}-18q^{-30}+18q^{-32}-22q^{-34}+22q^{-36}-26q^{-38}+20q^{-40}-22q^{-42}+25q^{-44}-20q^{-46}+18q^{-48}-14q^{-50}+11q^{-52}-8q^{-54}+4q^{-56}-2q^{-58}+q^{-60}$
2,0	$q^{24}+q^{18}+q^{16}-q^{14}-q^{12}+2q^{10}+q^{8}-2q^{6}+2q^{2}-1-3q^{-2}+q^{-4}-q^{-8}+q^{-10}+3q^{-12}+q^{-14}+q^{-16}+4q^{-18}+q^{-20}-3q^{-22}-q^{-24}+q^{-26}-2q^{-28}-q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-q^{18}+q^{14}-2q^{12}+q^{10}+2q^{8}-q^{6}+4q^{4}+3q^{2}-2+3q^{-2}+2q^{-4}-5q^{-6}+q^{-10}-3q^{-12}-2q^{-14}-2q^{-20}+5q^{-24}-q^{-26}-q^{-28}+6q^{-30}-q^{-32}-3q^{-34}+3q^{-36}-3q^{-40}+q^{-42}-q^{-46}+q^{-48}$
1,0,0	$q^{9}+2q^{5}+2q-q^{-1}+q^{-3}-q^{-5}-q^{-11}-2q^{-15}+q^{-17}-q^{-19}+2q^{-21}+q^{-25}$
1,0,1	$q^{34}-2q^{32}+3q^{30}-3q^{28}+q^{26}+4q^{24}-7q^{22}+13q^{20}-6q^{18}+2q^{16}+9q^{14}-17q^{12}+16q^{10}-14q^{8}+5q^{6}+4q^{4}-11q^{2}+22-17q^{-2}+19q^{-4}-13q^{-6}+3q^{-8}-2q^{-10}-7q^{-12}+8q^{-14}-12q^{-16}+16q^{-18}-18q^{-20}+28q^{-22}-22q^{-24}+16q^{-26}+2q^{-28}-17q^{-30}+26q^{-32}-28q^{-34}+22q^{-36}-6q^{-38}-4q^{-40}+10q^{-42}-6q^{-44}-5q^{-46}+6q^{-48}-7q^{-50}-7q^{-52}+12q^{-54}-12q^{-56}+12q^{-58}-2q^{-60}-4q^{-62}+9q^{-64}-9q^{-66}+6q^{-68}-q^{-70}-3q^{-72}+3q^{-74}-2q^{-76}+q^{-78}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{22}+q^{16}-q^{12}+q^{8}+q^{6}+4q^{4}+3q^{2}+3+4q^{-2}+3q^{-4}-q^{-6}-4q^{-8}-3q^{-12}-6q^{-14}-4q^{-16}+q^{-18}-5q^{-20}-2q^{-22}+3q^{-24}+5q^{-30}+5q^{-32}+3q^{-36}+4q^{-38}-4q^{-42}-3q^{-48}-q^{-50}+q^{-52}+q^{-58}$
1,0,0,0	$q^{10}+2q^{6}+q^{4}+2q^{2}+1+q^{-4}-2q^{-6}-2q^{-10}-2q^{-14}-q^{-18}+q^{-22}+2q^{-26}+q^{-30}$

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{26}-q^{24}+q^{22}-2q^{20}+3q^{18}-4q^{16}+3q^{14}-4q^{12}+6q^{10}-3q^{8}+6q^{6}-q^{4}+6q^{2}+2+2q^{-2}+2q^{-4}-2q^{-6}+2q^{-8}-8q^{-10}+4q^{-12}-10q^{-14}+5q^{-16}-11q^{-18}+6q^{-20}-8q^{-22}+7q^{-24}-5q^{-26}+5q^{-28}-2q^{-30}+4q^{-32}+2q^{-34}+2q^{-38}-2q^{-40}+5q^{-42}-3q^{-44}+3q^{-46}-4q^{-48}+4q^{-50}-3q^{-52}+2q^{-54}-3q^{-56}+2q^{-58}-2q^{-60}+q^{-62}-q^{-64}+q^{-66}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-q^{44}+2q^{42}-3q^{40}+2q^{38}-2q^{36}-q^{34}+6q^{32}-8q^{30}+10q^{28}-9q^{26}+5q^{24}+2q^{22}-10q^{20}+16q^{18}-16q^{16}+14q^{14}-4q^{12}-5q^{10}+15q^{8}-14q^{6}+12q^{4}-4q^{2}-4+10q^{-2}-9q^{-4}+3q^{-6}+6q^{-8}-10q^{-10}+14q^{-12}-9q^{-14}-2q^{-16}+8q^{-18}-16q^{-20}+18q^{-22}-17q^{-24}+7q^{-26}+3q^{-28}-13q^{-30}+18q^{-32}-18q^{-34}+11q^{-36}-3q^{-38}-6q^{-40}+10q^{-42}-12q^{-44}+8q^{-46}-5q^{-50}+6q^{-52}-4q^{-54}-2q^{-56}+7q^{-58}-8q^{-60}+7q^{-62}-4q^{-64}-q^{-66}+6q^{-68}-8q^{-70}+11q^{-72}-8q^{-74}+6q^{-76}-q^{-78}-2q^{-80}+5q^{-82}-8q^{-84}+10q^{-86}-7q^{-88}+4q^{-90}-4q^{-94}+6q^{-96}-6q^{-98}+5q^{-100}-3q^{-102}+q^{-106}-3q^{-108}+2q^{-110}-q^{-112}+q^{-114}$

.

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 9"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+3t^{3}-5t^{2}+7t-7+7t^{-1}-5t^{-2}+3t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-5z^{6}-7z^{4}-2z^{2}+1$

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, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-7z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-17z^{4}a^{-2}+5z^{4}a^{-4}+5z^{4}-16z^{2}a^{-2}+7z^{2}a^{-4}+7z^{2}-4a^{-2}+2a^{-4}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+4z^{8}a^{-2}+2z^{8}a^{-4}+2z^{8}+2az^{7}-2z^{7}a^{-1}-2z^{7}a^{-3}+2z^{7}a^{-5}+a^{2}z^{6}-18z^{6}a^{-2}-7z^{6}a^{-4}+2z^{6}a^{-6}-8z^{6}-8az^{5}-2z^{5}a^{-1}-4z^{5}a^{-5}+2z^{5}a^{-7}-4a^{2}z^{4}+31z^{4}a^{-2}+13z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+10z^{4}+7az^{3}+4z^{3}a^{-1}+5z^{3}a^{-3}+4z^{3}a^{-5}-4z^{3}a^{-7}+3a^{2}z^{2}-22z^{2}a^{-2}-8z^{2}a^{-4}+z^{2}a^{-6}-2z^{2}a^{-8}-8z^{2}-az-2za^{-1}-2za^{-3}+za^{-7}+4a^{-2}+2a^{-4}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(-2, -2)

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 $-8$ $-16$ $32$ ${\frac {260}{3}}$ ${\frac {220}{3}}$ $128$ ${\frac {1088}{3}}$ ${\frac {320}{3}}$ $144$ $-{\frac {256}{3}}$ $128$ $-{\frac {2080}{3}}$ $-{\frac {1760}{3}}$ $-{\frac {1231}{15}}$ ${\frac {3588}{5}}$ $-{\frac {54844}{45}}$ ${\frac {2911}{9}}$ $-{\frac {6271}{15}}$

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

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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=$2 is the signature of 10 9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{20}-2q^{19}+4q^{17}-5q^{16}+8q^{14}-10q^{13}+15q^{11}-18q^{10}+22q^{8}-23q^{7}-q^{6}+25q^{5}-21q^{4}-5q^{3}+24q^{2}-15q-8+19q^{-1}-8q^{-2}-9q^{-3}+12q^{-4}-2q^{-5}-6q^{-6}+5q^{-7}-2q^{-9}+q^{-10}$
3	$q^{39}-2q^{38}+q^{36}+3q^{35}-3q^{34}-3q^{33}+q^{32}+6q^{31}-q^{30}-6q^{29}-2q^{28}+6q^{27}+4q^{26}-5q^{25}-3q^{24}+q^{23}+3q^{22}+5q^{20}-6q^{19}-8q^{18}+5q^{17}+15q^{16}-5q^{15}-21q^{14}+5q^{13}+21q^{12}-24q^{10}-2q^{9}+20q^{8}+11q^{7}-21q^{6}-13q^{5}+14q^{4}+24q^{3}-14q^{2}-25q+6+32q^{-1}-3q^{-2}-30q^{-3}-5q^{-4}+29q^{-5}+8q^{-6}-23q^{-7}-12q^{-8}+18q^{-9}+11q^{-10}-10q^{-11}-11q^{-12}+7q^{-13}+7q^{-14}-3q^{-15}-5q^{-16}+2q^{-17}+2q^{-18}-2q^{-20}+q^{-21}$
4	$q^{64}-2q^{63}+q^{61}+5q^{59}-7q^{58}-q^{57}+17q^{54}-13q^{53}-5q^{52}-7q^{51}-3q^{50}+38q^{49}-12q^{48}-7q^{47}-26q^{46}-17q^{45}+64q^{44}+q^{43}+4q^{42}-52q^{41}-52q^{40}+79q^{39}+25q^{38}+43q^{37}-68q^{36}-107q^{35}+68q^{34}+39q^{33}+104q^{32}-54q^{31}-158q^{30}+35q^{29}+29q^{28}+157q^{27}-22q^{26}-179q^{25}+8q^{24}+4q^{23}+178q^{22}+3q^{21}-177q^{20}+q^{19}-15q^{18}+173q^{17}+11q^{16}-161q^{15}+10q^{14}-31q^{13}+151q^{12}+14q^{11}-133q^{10}+25q^{9}-46q^{8}+111q^{7}+13q^{6}-92q^{5}+50q^{4}-57q^{3}+59q^{2}+q-53+78q^{-1}-49q^{-2}+17q^{-3}-25q^{-4}-37q^{-5}+94q^{-6}-22q^{-7}+4q^{-8}-44q^{-9}-43q^{-10}+81q^{-11}+3q^{-12}+16q^{-13}-38q^{-14}-49q^{-15}+47q^{-16}+7q^{-17}+25q^{-18}-16q^{-19}-38q^{-20}+19q^{-21}+18q^{-23}-2q^{-24}-19q^{-25}+8q^{-26}-3q^{-27}+7q^{-28}+q^{-29}-7q^{-30}+3q^{-31}-q^{-32}+2q^{-33}-2q^{-35}+q^{-36}$
5	$q^{95}-2q^{94}+q^{92}+2q^{90}+q^{89}-5q^{88}-3q^{87}+3q^{86}+2q^{85}+6q^{84}+4q^{83}-11q^{82}-11q^{81}+q^{80}+7q^{79}+15q^{78}+13q^{77}-14q^{76}-27q^{75}-11q^{74}+8q^{73}+31q^{72}+31q^{71}-8q^{70}-43q^{69}-42q^{68}-2q^{67}+50q^{66}+66q^{65}+18q^{64}-58q^{63}-95q^{62}-46q^{61}+60q^{60}+132q^{59}+92q^{58}-52q^{57}-177q^{56}-154q^{55}+25q^{54}+216q^{53}+241q^{52}+24q^{51}-257q^{50}-322q^{49}-96q^{48}+260q^{47}+425q^{46}+181q^{45}-264q^{44}-490q^{43}-268q^{42}+227q^{41}+549q^{40}+350q^{39}-198q^{38}-574q^{37}-406q^{36}+160q^{35}+577q^{34}+446q^{33}-124q^{32}-579q^{31}-462q^{30}+108q^{29}+559q^{28}+465q^{27}-83q^{26}-549q^{25}-466q^{24}+79q^{23}+519q^{22}+458q^{21}-49q^{20}-501q^{19}-448q^{18}+33q^{17}+448q^{16}+439q^{15}+9q^{14}-412q^{13}-412q^{12}-32q^{11}+333q^{10}+381q^{9}+79q^{8}-278q^{7}-335q^{6}-83q^{5}+189q^{4}+273q^{3}+110q^{2}-136q-213-81q^{-1}+76q^{-2}+137q^{-3}+71q^{-4}-53q^{-5}-89q^{-6}-20q^{-7}+41q^{-8}+41q^{-9}-7q^{-10}-53q^{-11}-30q^{-12}+43q^{-13}+66q^{-14}+28q^{-15}-42q^{-16}-87q^{-17}-44q^{-18}+42q^{-19}+83q^{-20}+58q^{-21}-14q^{-22}-77q^{-23}-68q^{-24}-3q^{-25}+52q^{-26}+64q^{-27}+23q^{-28}-31q^{-29}-51q^{-30}-28q^{-31}+9q^{-32}+36q^{-33}+28q^{-34}-3q^{-35}-18q^{-36}-17q^{-37}-8q^{-38}+10q^{-39}+14q^{-40}+2q^{-41}-5q^{-42}-2q^{-43}-5q^{-44}+6q^{-46}-3q^{-48}+q^{-49}-q^{-51}+2q^{-52}-2q^{-54}+q^{-55}$
6	$q^{132}-2q^{131}+q^{129}+2q^{127}-2q^{126}+3q^{125}-7q^{124}+4q^{122}+7q^{120}-5q^{119}+6q^{118}-19q^{117}+8q^{115}+17q^{113}-5q^{112}+14q^{111}-38q^{110}-4q^{109}+6q^{108}-4q^{107}+27q^{106}+4q^{105}+35q^{104}-57q^{103}-2q^{102}-4q^{101}-22q^{100}+23q^{99}+11q^{98}+66q^{97}-68q^{96}+18q^{95}-6q^{94}-43q^{93}+5q^{92}+11q^{91}+85q^{90}-94q^{89}+38q^{88}-4q^{87}-44q^{86}+19q^{85}+48q^{84}+114q^{83}-159q^{82}-16q^{81}-80q^{80}-72q^{79}+95q^{78}+220q^{77}+274q^{76}-187q^{75}-166q^{74}-351q^{73}-284q^{72}+123q^{71}+528q^{70}+686q^{69}-q^{68}-270q^{67}-774q^{66}-771q^{65}-76q^{64}+777q^{63}+1240q^{62}+447q^{61}-146q^{60}-1104q^{59}-1354q^{58}-499q^{57}+787q^{56}+1649q^{55}+931q^{54}+167q^{53}-1179q^{52}-1741q^{51}-904q^{50}+616q^{49}+1785q^{48}+1208q^{47}+446q^{46}-1083q^{45}-1861q^{44}-1111q^{43}+453q^{42}+1757q^{41}+1275q^{40}+575q^{39}-982q^{38}-1840q^{37}-1167q^{36}+362q^{35}+1696q^{34}+1263q^{33}+634q^{32}-895q^{31}-1780q^{30}-1203q^{29}+249q^{28}+1602q^{27}+1251q^{26}+736q^{25}-734q^{24}-1665q^{23}-1272q^{22}+20q^{21}+1397q^{20}+1216q^{19}+913q^{18}-433q^{17}-1431q^{16}-1332q^{15}-319q^{14}+1037q^{13}+1080q^{12}+1084q^{11}-21q^{10}-1034q^{9}-1273q^{8}-648q^{7}+560q^{6}+774q^{5}+1108q^{4}+356q^{3}-524q^{2}-1006q-784+128q^{-1}+343q^{-2}+889q^{-3}+509q^{-4}-84q^{-5}-594q^{-6}-638q^{-7}-67q^{-8}-14q^{-9}+520q^{-10}+380q^{-11}+102q^{-12}-261q^{-13}-347q^{-14}-q^{-15}-124q^{-16}+244q^{-17}+149q^{-18}+51q^{-19}-161q^{-20}-165q^{-21}+123q^{-22}-44q^{-23}+178q^{-24}+51q^{-25}-20q^{-26}-199q^{-27}-161q^{-28}+114q^{-29}+12q^{-30}+194q^{-31}+92q^{-32}+27q^{-33}-179q^{-34}-186q^{-35}+17q^{-36}-36q^{-37}+140q^{-38}+118q^{-39}+104q^{-40}-79q^{-41}-130q^{-42}-28q^{-43}-85q^{-44}+42q^{-45}+69q^{-46}+107q^{-47}-6q^{-48}-50q^{-49}-7q^{-50}-70q^{-51}-9q^{-52}+13q^{-53}+60q^{-54}+8q^{-55}-12q^{-56}+14q^{-57}-32q^{-58}-12q^{-59}-5q^{-60}+23q^{-61}+2q^{-62}-5q^{-63}+12q^{-64}-9q^{-65}-4q^{-66}-4q^{-67}+8q^{-68}-q^{-69}-4q^{-70}+5q^{-71}-2q^{-72}-q^{-74}+2q^{-75}-2q^{-77}+q^{-78}$
7	$q^{175}-2q^{174}+q^{172}+2q^{170}-2q^{169}+q^{167}-4q^{166}+q^{165}+2q^{164}+7q^{162}-4q^{161}-5q^{160}+2q^{159}-10q^{158}+4q^{157}+5q^{156}+17q^{154}-3q^{153}-7q^{152}-q^{151}-26q^{150}+7q^{148}-4q^{147}+34q^{146}+10q^{145}+7q^{144}+11q^{143}-49q^{142}-21q^{141}-16q^{140}-28q^{139}+36q^{138}+28q^{137}+52q^{136}+69q^{135}-31q^{134}-37q^{133}-65q^{132}-110q^{131}-17q^{130}+4q^{129}+88q^{128}+178q^{127}+80q^{126}+38q^{125}-79q^{124}-218q^{123}-157q^{122}-140q^{121}+14q^{120}+245q^{119}+251q^{118}+259q^{117}+89q^{116}-200q^{115}-283q^{114}-398q^{113}-272q^{112}+68q^{111}+253q^{110}+500q^{109}+477q^{108}+159q^{107}-89q^{106}-492q^{105}-657q^{104}-470q^{103}-246q^{102}+322q^{101}+741q^{100}+799q^{99}+709q^{98}+73q^{97}-617q^{96}-1044q^{95}-1278q^{94}-698q^{93}+240q^{92}+1129q^{91}+1837q^{90}+1462q^{89}+390q^{88}-939q^{87}-2250q^{86}-2297q^{85}-1256q^{84}+516q^{83}+2479q^{82}+3033q^{81}+2154q^{80}+137q^{79}-2416q^{78}-3588q^{77}-3041q^{76}-887q^{75}+2183q^{74}+3905q^{73}+3729q^{72}+1594q^{71}-1792q^{70}-3979q^{69}-4205q^{68}-2202q^{67}+1379q^{66}+3918q^{65}+4468q^{64}+2605q^{63}-1036q^{62}-3750q^{61}-4540q^{60}-2865q^{59}+768q^{58}+3591q^{57}+4546q^{56}+2971q^{55}-619q^{54}-3459q^{53}-4482q^{52}-3010q^{51}+525q^{50}+3352q^{49}+4430q^{48}+3034q^{47}-463q^{46}-3278q^{45}-4374q^{44}-3049q^{43}+379q^{42}+3163q^{41}+4322q^{40}+3109q^{39}-246q^{38}-3007q^{37}-4240q^{36}-3197q^{35}+22q^{34}+2773q^{33}+4124q^{32}+3300q^{31}+274q^{30}-2402q^{29}-3922q^{28}-3444q^{27}-685q^{26}+1949q^{25}+3645q^{24}+3528q^{23}+1132q^{22}-1318q^{21}-3221q^{20}-3588q^{19}-1652q^{18}+631q^{17}+2687q^{16}+3507q^{15}+2095q^{14}+149q^{13}-1976q^{12}-3279q^{11}-2476q^{10}-923q^{9}+1198q^{8}+2888q^{7}+2633q^{6}+1580q^{5}-344q^{4}-2271q^{3}-2604q^{2}-2100q-444+1600q^{-1}+2314q^{-2}+2307q^{-3}+1094q^{-4}-829q^{-5}-1825q^{-6}-2306q^{-7}-1525q^{-8}+196q^{-9}+1250q^{-10}+2003q^{-11}+1657q^{-12}+318q^{-13}-652q^{-14}-1582q^{-15}-1579q^{-16}-563q^{-17}+197q^{-18}+1087q^{-19}+1299q^{-20}+615q^{-21}+102q^{-22}-688q^{-23}-965q^{-24}-488q^{-25}-205q^{-26}+396q^{-27}+675q^{-28}+323q^{-29}+168q^{-30}-280q^{-31}-478q^{-32}-153q^{-33}-92q^{-34}+242q^{-35}+402q^{-36}+94q^{-37}+26q^{-38}-269q^{-39}-395q^{-40}-94q^{-41}-18q^{-42}+256q^{-43}+399q^{-44}+147q^{-45}+78q^{-46}-196q^{-47}-389q^{-48}-205q^{-49}-144q^{-50}+117q^{-51}+316q^{-52}+203q^{-53}+206q^{-54}+q^{-55}-224q^{-56}-189q^{-57}-225q^{-58}-59q^{-59}+130q^{-60}+116q^{-61}+193q^{-62}+107q^{-63}-41q^{-64}-62q^{-65}-159q^{-66}-106q^{-67}+12q^{-68}+15q^{-69}+91q^{-70}+81q^{-71}+21q^{-72}+22q^{-73}-63q^{-74}-66q^{-75}-12q^{-76}-17q^{-77}+27q^{-78}+28q^{-79}+11q^{-80}+31q^{-81}-12q^{-82}-28q^{-83}-6q^{-84}-11q^{-85}+8q^{-86}+5q^{-87}-3q^{-88}+16q^{-89}+q^{-90}-8q^{-91}-2q^{-92}-4q^{-93}+4q^{-94}+q^{-95}-5q^{-96}+4q^{-97}+2q^{-98}-2q^{-99}-q^{-101}+2q^{-102}-2q^{-104}+q^{-105}$

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

See/edit the Rolfsen_Splice_Template.