8 10

8_9

8_11

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8 10 Further Notes and Views

Knot presentations

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

Minimum Braid Representative: Length is 8, width is 3. Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-8] Hyperbolic Volume 8.65115 A-Polynomial See Data:8 10/A-polynomial

[edit Notes for 8 10's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 10's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-3t^{2}+6t-7+6t^{-1}-3t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+3z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 27, 2 }
Jones polynomial	$-q^{6}+2q^{5}-4q^{4}+5q^{3}-4q^{2}+5q-3+2q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+5z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+9z^{2}a^{-2}-3z^{2}a^{-4}-3z^{2}+6a^{-2}-3a^{-4}-2$
Kauffman polynomial (db, data sources)	$z^{7}a^{-1}+z^{7}a^{-3}+5z^{6}a^{-2}+3z^{6}a^{-4}+2z^{6}+az^{5}+z^{5}a^{-1}+3z^{5}a^{-3}+3z^{5}a^{-5}-13z^{4}a^{-2}-5z^{4}a^{-4}+2z^{4}a^{-6}-6z^{4}-3az^{3}-8z^{3}a^{-1}-9z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}+12z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}+5z^{2}+2az+5za^{-1}+6za^{-3}+2za^{-5}-za^{-7}-6a^{-2}-3a^{-4}-2$
The A2 invariant	$-q^{6}-q^{2}+2q^{-2}+q^{-4}+4q^{-6}+q^{-8}+q^{-10}-q^{-12}-2q^{-14}-q^{-18}$
The G2 invariant	$q^{32}-q^{30}+3q^{28}-4q^{26}+2q^{24}-q^{22}-5q^{20}+9q^{18}-12q^{16}+10q^{14}-6q^{12}-5q^{10}+12q^{8}-16q^{6}+14q^{4}-9q^{2}-2+9q^{-2}-13q^{-4}+9q^{-6}-q^{-8}-4q^{-10}+12q^{-12}-7q^{-14}+3q^{-16}+7q^{-18}-10q^{-20}+19q^{-22}-14q^{-24}+9q^{-26}+7q^{-28}-12q^{-30}+23q^{-32}-19q^{-34}+14q^{-36}-q^{-38}-7q^{-40}+13q^{-42}-17q^{-44}+11q^{-46}-q^{-48}-7q^{-50}+8q^{-52}-8q^{-54}-2q^{-56}+8q^{-58}-14q^{-60}+9q^{-62}-6q^{-64}-3q^{-66}+9q^{-68}-14q^{-70}+14q^{-72}-8q^{-74}+2q^{-76}+2q^{-78}-7q^{-80}+6q^{-82}-6q^{-84}+5q^{-86}-2q^{-88}+q^{-92}-2q^{-94}+2q^{-96}-q^{-98}+q^{-100}$

Further Quantum Invariants

Further quantum knot invariants for 8_10.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+q^{3}-q+2q^{-1}+q^{-3}+q^{-5}+q^{-7}-2q^{-9}+q^{-11}-q^{-13}$
2	$q^{16}-q^{14}-2q^{12}+3q^{10}-5q^{6}+3q^{4}+3q^{2}-5+2q^{-2}+5q^{-4}-2q^{-6}+4q^{-10}+q^{-12}-2q^{-14}-q^{-16}+4q^{-18}-4q^{-20}-4q^{-22}+6q^{-24}-2q^{-26}-4q^{-28}+3q^{-30}-q^{-34}+q^{-36}$
3	$-q^{33}+q^{31}+2q^{29}-4q^{25}-2q^{23}+6q^{21}+5q^{19}-6q^{17}-10q^{15}+3q^{13}+13q^{11}+q^{9}-15q^{7}-6q^{5}+12q^{3}+12q-11q^{-1}-11q^{-3}+8q^{-5}+16q^{-7}-3q^{-9}-13q^{-11}+2q^{-13}+14q^{-15}+q^{-17}-10q^{-19}-3q^{-21}+6q^{-23}+8q^{-25}-4q^{-27}-11q^{-29}-4q^{-31}+14q^{-33}+6q^{-35}-13q^{-37}-13q^{-39}+13q^{-41}+13q^{-43}-10q^{-45}-13q^{-47}+4q^{-49}+11q^{-51}-q^{-53}-6q^{-55}+q^{-57}+3q^{-59}-q^{-63}+q^{-67}-q^{-69}$
4	$q^{56}-q^{54}-2q^{52}+q^{48}+6q^{46}-6q^{42}-6q^{40}-4q^{38}+15q^{36}+12q^{34}-2q^{32}-16q^{30}-25q^{28}+9q^{26}+26q^{24}+24q^{22}-4q^{20}-45q^{18}-22q^{16}+11q^{14}+45q^{12}+32q^{10}-33q^{8}-46q^{6}-26q^{4}+37q^{2}+60+q^{-2}-40q^{-4}-49q^{-6}+13q^{-8}+60q^{-10}+25q^{-12}-26q^{-14}-52q^{-16}-q^{-18}+46q^{-20}+28q^{-22}-16q^{-24}-42q^{-26}-9q^{-28}+27q^{-30}+29q^{-32}-4q^{-34}-30q^{-36}-22q^{-38}+5q^{-40}+32q^{-42}+21q^{-44}-7q^{-46}-40q^{-48}-31q^{-50}+28q^{-52}+46q^{-54}+27q^{-56}-41q^{-58}-66q^{-60}+3q^{-62}+48q^{-64}+55q^{-66}-17q^{-68}-64q^{-70}-19q^{-72}+21q^{-74}+50q^{-76}+10q^{-78}-32q^{-80}-19q^{-82}-4q^{-84}+22q^{-86}+10q^{-88}-7q^{-90}-3q^{-92}-6q^{-94}+4q^{-96}+2q^{-98}-2q^{-100}+q^{-102}-2q^{-104}+q^{-106}-q^{-110}+q^{-112}$
5	$-q^{85}+q^{83}+2q^{81}-q^{77}-3q^{75}-4q^{73}+8q^{69}+8q^{67}+2q^{65}-7q^{63}-16q^{61}-14q^{59}+4q^{57}+26q^{55}+29q^{53}+10q^{51}-23q^{49}-49q^{47}-39q^{45}+7q^{43}+59q^{41}+71q^{39}+31q^{37}-43q^{35}-97q^{33}-82q^{31}+2q^{29}+98q^{27}+127q^{25}+61q^{23}-66q^{21}-153q^{19}-129q^{17}+2q^{15}+149q^{13}+181q^{11}+66q^{9}-107q^{7}-208q^{5}-143q^{3}+53q+209q^{-1}+185q^{-3}+16q^{-5}-176q^{-7}-215q^{-9}-60q^{-11}+146q^{-13}+214q^{-15}+94q^{-17}-100q^{-19}-198q^{-21}-105q^{-23}+78q^{-25}+170q^{-27}+104q^{-29}-55q^{-31}-147q^{-33}-93q^{-35}+40q^{-37}+124q^{-39}+82q^{-41}-32q^{-43}-108q^{-45}-81q^{-47}+14q^{-49}+92q^{-51}+84q^{-53}+12q^{-55}-69q^{-57}-102q^{-59}-55q^{-61}+46q^{-63}+112q^{-65}+105q^{-67}+9q^{-69}-122q^{-71}-164q^{-73}-60q^{-75}+104q^{-77}+204q^{-79}+139q^{-81}-71q^{-83}-236q^{-85}-193q^{-87}+11q^{-89}+218q^{-91}+241q^{-93}+50q^{-95}-176q^{-97}-246q^{-99}-100q^{-101}+113q^{-103}+213q^{-105}+131q^{-107}-46q^{-109}-160q^{-111}-128q^{-113}+q^{-115}+98q^{-117}+99q^{-119}+27q^{-121}-47q^{-123}-64q^{-125}-31q^{-127}+17q^{-129}+32q^{-131}+20q^{-133}-q^{-135}-13q^{-137}-12q^{-139}-4q^{-141}+6q^{-143}+5q^{-145}-q^{-151}-2q^{-153}+q^{-155}+2q^{-157}-q^{-159}+q^{-163}-q^{-165}$
6	$q^{120}-q^{118}-2q^{116}+q^{112}+3q^{110}+q^{108}+4q^{106}-2q^{104}-10q^{102}-7q^{100}-2q^{98}+8q^{96}+10q^{94}+21q^{92}+8q^{90}-16q^{88}-30q^{86}-32q^{84}-12q^{82}+8q^{80}+61q^{78}+66q^{76}+31q^{74}-25q^{72}-81q^{70}-100q^{68}-88q^{66}+27q^{64}+121q^{62}+165q^{60}+121q^{58}+8q^{56}-137q^{54}-260q^{52}-194q^{50}-43q^{48}+171q^{46}+308q^{44}+312q^{42}+121q^{40}-207q^{38}-392q^{36}-416q^{34}-182q^{32}+171q^{30}+505q^{28}+550q^{26}+238q^{24}-187q^{22}-585q^{20}-648q^{18}-349q^{16}+245q^{14}+694q^{12}+721q^{10}+362q^{8}-285q^{6}-773q^{4}-829q^{2}-300+394q^{-2}+838q^{-4}+803q^{-6}+225q^{-8}-492q^{-10}-923q^{-12}-684q^{-14}-47q^{-16}+604q^{-18}+871q^{-20}+539q^{-22}-128q^{-24}-712q^{-26}-727q^{-28}-295q^{-30}+311q^{-32}+677q^{-34}+545q^{-36}+55q^{-38}-455q^{-40}-555q^{-42}-298q^{-44}+159q^{-46}+453q^{-48}+393q^{-50}+60q^{-52}-302q^{-54}-376q^{-56}-203q^{-58}+123q^{-60}+320q^{-62}+280q^{-64}+44q^{-66}-222q^{-68}-309q^{-70}-203q^{-72}+56q^{-74}+255q^{-76}+318q^{-78}+173q^{-80}-83q^{-82}-316q^{-84}-375q^{-86}-189q^{-88}+105q^{-90}+417q^{-92}+486q^{-94}+266q^{-96}-200q^{-98}-580q^{-100}-612q^{-102}-288q^{-104}+321q^{-106}+768q^{-108}+775q^{-110}+203q^{-112}-517q^{-114}-928q^{-116}-808q^{-118}-103q^{-120}+689q^{-122}+1086q^{-124}+708q^{-126}-81q^{-128}-797q^{-130}-1043q^{-132}-579q^{-134}+221q^{-136}+876q^{-138}+865q^{-140}+366q^{-142}-297q^{-144}-761q^{-146}-678q^{-148}-199q^{-150}+362q^{-152}+563q^{-154}+434q^{-156}+85q^{-158}-283q^{-160}-399q^{-162}-258q^{-164}+21q^{-166}+177q^{-168}+223q^{-170}+139q^{-172}-26q^{-174}-120q^{-176}-119q^{-178}-38q^{-180}+10q^{-182}+53q^{-184}+63q^{-186}+17q^{-188}-18q^{-190}-28q^{-192}-12q^{-194}-10q^{-196}+5q^{-198}+17q^{-200}+6q^{-202}-2q^{-204}-5q^{-206}+q^{-208}-4q^{-210}-q^{-212}+5q^{-214}-q^{-218}-2q^{-220}+q^{-222}-q^{-226}+q^{-228}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}-q^{2}+2q^{-2}+q^{-4}+4q^{-6}+q^{-8}+q^{-10}-q^{-12}-2q^{-14}-q^{-18}$
1,1	$q^{20}-2q^{18}+6q^{16}-12q^{14}+19q^{12}-28q^{10}+36q^{8}-44q^{6}+39q^{4}-40q^{2}+26-10q^{-2}-8q^{-4}+34q^{-6}-40q^{-8}+70q^{-10}-67q^{-12}+80q^{-14}-70q^{-16}+60q^{-18}-49q^{-20}+24q^{-22}-12q^{-24}-12q^{-26}+23q^{-28}-34q^{-30}+34q^{-32}-34q^{-34}+32q^{-36}-26q^{-38}+18q^{-40}-14q^{-42}+11q^{-44}-6q^{-46}+4q^{-48}-2q^{-50}+q^{-52}$
2,0	$q^{18}-q^{14}+q^{10}-q^{8}-3q^{6}-q^{4}+q^{2}-2-2q^{-2}+3q^{-4}+q^{-6}+q^{-8}+3q^{-10}+6q^{-12}+4q^{-14}+5q^{-16}+6q^{-18}+2q^{-20}-4q^{-22}-3q^{-24}-3q^{-26}-7q^{-28}-4q^{-30}-q^{-36}+2q^{-40}+q^{-42}+q^{-46}$
3,0	$-q^{36}+q^{32}+2q^{30}-3q^{26}-q^{24}+3q^{22}+6q^{20}-7q^{16}-6q^{14}+3q^{12}+8q^{10}-12q^{6}-9q^{4}+3q^{2}+10-q^{-2}-9q^{-4}-3q^{-6}+8q^{-8}+10q^{-10}-q^{-12}-2q^{-14}+3q^{-16}+10q^{-18}+4q^{-20}+4q^{-24}+11q^{-26}+5q^{-28}+q^{-30}+9q^{-34}+4q^{-36}-11q^{-38}-18q^{-40}-9q^{-42}+3q^{-44}-4q^{-46}-13q^{-48}-14q^{-50}+4q^{-52}+10q^{-54}+4q^{-56}-5q^{-58}-4q^{-60}+7q^{-62}+10q^{-64}+4q^{-66}-q^{-68}-2q^{-70}+2q^{-72}+q^{-74}-q^{-76}-q^{-78}-q^{-80}-q^{-84}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-q^{12}+q^{10}+q^{8}-4q^{6}-2q^{2}-7+q^{-2}+3q^{-4}+9q^{-8}+10q^{-10}+6q^{-12}+3q^{-14}+q^{-16}-2q^{-18}-6q^{-20}-6q^{-22}+q^{-24}-4q^{-26}-3q^{-28}+5q^{-30}-2q^{-32}-2q^{-34}+3q^{-36}-q^{-40}+q^{-42}$
1,0,0	$-q^{7}-2q^{3}-q^{-1}+2q^{-3}+2q^{-5}+4q^{-7}+4q^{-9}+2q^{-11}+q^{-13}-2q^{-15}-q^{-17}-3q^{-19}-q^{-23}$
1,0,1	$q^{24}-2q^{22}+5q^{20}-5q^{18}+2q^{16}+6q^{14}-16q^{12}+24q^{10}-21q^{8}+7q^{6}+7q^{4}-39q^{2}+35-47q^{-2}+14q^{-4}+4q^{-6}-21q^{-8}+47q^{-10}-7q^{-12}+34q^{-14}+21q^{-16}+16q^{-18}+19q^{-22}-26q^{-24}-13q^{-26}+7q^{-28}-54q^{-30}+29q^{-32}-29q^{-34}-2q^{-36}+27q^{-38}-32q^{-40}+31q^{-42}-7q^{-44}-10q^{-46}+21q^{-48}-19q^{-50}+5q^{-52}+5q^{-54}-9q^{-56}+9q^{-58}-3q^{-60}-q^{-62}+3q^{-64}-2q^{-66}+q^{-68}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{16}+q^{12}+2q^{10}-2q^{6}-q^{4}-6q^{2}-10-7q^{-2}-4q^{-4}-3q^{-6}+q^{-8}+14q^{-10}+17q^{-12}+15q^{-14}+16q^{-16}+17q^{-18}+2q^{-20}-3q^{-22}-4q^{-24}-11q^{-26}-14q^{-28}-8q^{-30}-4q^{-32}-6q^{-34}-2q^{-36}+3q^{-38}+2q^{-40}-2q^{-42}+2q^{-44}+3q^{-46}+q^{-52}$
1,0,0,0	$-q^{8}-2q^{4}-q^{2}-1-q^{-2}+2q^{-4}+2q^{-6}+5q^{-8}+4q^{-10}+5q^{-12}+2q^{-14}+q^{-16}-2q^{-18}-2q^{-20}-2q^{-22}-3q^{-24}-q^{-28}$

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{18}-q^{16}+2q^{14}-2q^{12}+3q^{10}-4q^{8}+q^{6}-6q^{4}-6-2q^{-2}-2q^{-4}+q^{-6}+5q^{-8}+3q^{-10}+13q^{-12}+5q^{-14}+14q^{-16}+9q^{-20}-5q^{-22}+3q^{-24}-9q^{-26}-3q^{-28}-7q^{-30}-2q^{-32}-2q^{-34}-3q^{-36}+q^{-38}-3q^{-40}+5q^{-42}-3q^{-44}+2q^{-46}-3q^{-48}+3q^{-50}-q^{-52}+q^{-54}-q^{-56}+q^{-58}$

G2 Invariants.

Weight

Invariant

1,0

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

.

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["8 10"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+3z^{4}+3z^{2}+1$

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]=

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+5z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+9z^{2}a^{-2}-3z^{2}a^{-4}-3z^{2}+6a^{-2}-3a^{-4}-2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{-1}+z^{7}a^{-3}+5z^{6}a^{-2}+3z^{6}a^{-4}+2z^{6}+az^{5}+z^{5}a^{-1}+3z^{5}a^{-3}+3z^{5}a^{-5}-13z^{4}a^{-2}-5z^{4}a^{-4}+2z^{4}a^{-6}-6z^{4}-3az^{3}-8z^{3}a^{-1}-9z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}+12z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}+5z^{2}+2az+5za^{-1}+6za^{-3}+2za^{-5}-za^{-7}-6a^{-2}-3a^{-4}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_143, K11n106, ...}

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

Vassiliev invariants

V2 and V3:

(3, 3)

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 $12$ $24$ $72$ $110$ $10$ $288$ $432$ $64$ $56$ $288$ $288$ $1320$ $120$ ${\frac {19711}{10}}$ ${\frac {618}{5}}$ ${\frac {9062}{15}}$ ${\frac {161}{6}}$ ${\frac {511}{10}}$

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 8 10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{17}-2q^{16}+q^{15}+4q^{14}-9q^{13}+3q^{12}+12q^{11}-19q^{10}+3q^{9}+20q^{8}-24q^{7}+2q^{6}+23q^{5}-21q^{4}-2q^{3}+21q^{2}-14q-5+14q^{-1}-6q^{-2}-5q^{-3}+6q^{-4}-q^{-5}-2q^{-6}+q^{-7}$
3	$-q^{33}+2q^{32}-q^{31}-q^{30}+5q^{28}-3q^{27}-8q^{26}+5q^{25}+17q^{24}-10q^{23}-25q^{22}+8q^{21}+40q^{20}-10q^{19}-51q^{18}+8q^{17}+59q^{16}-2q^{15}-69q^{14}+q^{13}+66q^{12}+10q^{11}-71q^{10}-8q^{9}+59q^{8}+21q^{7}-58q^{6}-20q^{5}+44q^{4}+31q^{3}-39q^{2}-28q+25+31q^{-1}-16q^{-2}-28q^{-3}+7q^{-4}+22q^{-5}-16q^{-7}-3q^{-8}+9q^{-9}+4q^{-10}-5q^{-11}-2q^{-12}+q^{-13}+2q^{-14}-q^{-15}$
4	$q^{54}-2q^{53}+q^{52}+q^{51}-3q^{50}+4q^{49}-5q^{48}+5q^{47}+3q^{46}-13q^{45}+7q^{44}-9q^{43}+22q^{42}+15q^{41}-39q^{40}-8q^{39}-22q^{38}+64q^{37}+55q^{36}-68q^{35}-48q^{34}-67q^{33}+111q^{32}+127q^{31}-75q^{30}-93q^{29}-136q^{28}+136q^{27}+195q^{26}-56q^{25}-111q^{24}-195q^{23}+127q^{22}+228q^{21}-28q^{20}-100q^{19}-222q^{18}+100q^{17}+220q^{16}-2q^{15}-67q^{14}-224q^{13}+64q^{12}+187q^{11}+24q^{10}-23q^{9}-206q^{8}+17q^{7}+136q^{6}+50q^{5}+28q^{4}-171q^{3}-30q^{2}+74q+59+69q^{-1}-112q^{-2}-53q^{-3}+11q^{-4}+39q^{-5}+82q^{-6}-47q^{-7}-40q^{-8}-23q^{-9}+6q^{-10}+59q^{-11}-6q^{-12}-12q^{-13}-21q^{-14}-11q^{-15}+25q^{-16}+3q^{-17}+2q^{-18}-7q^{-19}-8q^{-20}+6q^{-21}+q^{-22}+2q^{-23}-q^{-24}-2q^{-25}+q^{-26}$
5	$-q^{80}+2q^{79}-q^{78}-q^{77}+3q^{76}-q^{75}-4q^{74}+3q^{73}-q^{71}+8q^{70}-14q^{68}-5q^{67}-q^{66}+11q^{65}+29q^{64}+12q^{63}-29q^{62}-53q^{61}-34q^{60}+28q^{59}+103q^{58}+84q^{57}-30q^{56}-150q^{55}-163q^{54}-4q^{53}+217q^{52}+261q^{51}+52q^{50}-250q^{49}-376q^{48}-150q^{47}+287q^{46}+487q^{45}+243q^{44}-273q^{43}-583q^{42}-354q^{41}+244q^{40}+652q^{39}+453q^{38}-208q^{37}-683q^{36}-518q^{35}+140q^{34}+694q^{33}+584q^{32}-112q^{31}-676q^{30}-584q^{29}+39q^{28}+647q^{27}+617q^{26}-31q^{25}-604q^{24}-576q^{23}-39q^{22}+552q^{21}+590q^{20}+45q^{19}-490q^{18}-534q^{17}-123q^{16}+419q^{15}+536q^{14}+137q^{13}-331q^{12}-468q^{11}-215q^{10}+236q^{9}+443q^{8}+235q^{7}-137q^{6}-348q^{5}-283q^{4}+30q^{3}+288q^{2}+274q+55-179q^{-1}-259q^{-2}-126q^{-3}+92q^{-4}+209q^{-5}+156q^{-6}-6q^{-7}-144q^{-8}-158q^{-9}-55q^{-10}+78q^{-11}+132q^{-12}+81q^{-13}-17q^{-14}-92q^{-15}-84q^{-16}-18q^{-17}+48q^{-18}+66q^{-19}+37q^{-20}-18q^{-21}-44q^{-22}-30q^{-23}-4q^{-24}+20q^{-25}+27q^{-26}+8q^{-27}-11q^{-28}-11q^{-29}-7q^{-30}-2q^{-31}+9q^{-32}+6q^{-33}-2q^{-34}-2q^{-35}-q^{-36}-2q^{-37}+q^{-38}+2q^{-39}-q^{-40}$
6	$q^{111}-2q^{110}+q^{109}+q^{108}-3q^{107}+q^{106}+q^{105}+6q^{104}-8q^{103}-2q^{102}+6q^{101}-9q^{100}+4q^{99}+9q^{98}+17q^{97}-20q^{96}-17q^{95}+4q^{94}-25q^{93}+14q^{92}+44q^{91}+63q^{90}-30q^{89}-60q^{88}-44q^{87}-106q^{86}+13q^{85}+138q^{84}+228q^{83}+54q^{82}-106q^{81}-200q^{80}-385q^{79}-128q^{78}+254q^{77}+596q^{76}+403q^{75}+23q^{74}-401q^{73}-946q^{72}-607q^{71}+171q^{70}+1060q^{69}+1066q^{68}+522q^{67}-390q^{66}-1601q^{65}-1407q^{64}-293q^{63}+1306q^{62}+1782q^{61}+1311q^{60}-12q^{59}-1998q^{58}-2199q^{57}-998q^{56}+1186q^{55}+2193q^{54}+2031q^{53}+560q^{52}-2005q^{51}-2646q^{50}-1607q^{49}+862q^{48}+2225q^{47}+2411q^{46}+1026q^{45}-1785q^{44}-2715q^{43}-1919q^{42}+568q^{41}+2039q^{40}+2470q^{39}+1259q^{38}-1529q^{37}-2570q^{36}-1982q^{35}+369q^{34}+1780q^{33}+2364q^{32}+1346q^{31}-1263q^{30}-2334q^{29}-1942q^{28}+172q^{27}+1454q^{26}+2191q^{25}+1420q^{24}-901q^{23}-2001q^{22}-1882q^{21}-122q^{20}+997q^{19}+1934q^{18}+1520q^{17}-391q^{16}-1511q^{15}-1750q^{14}-488q^{13}+391q^{12}+1502q^{11}+1535q^{10}+193q^{9}-844q^{8}-1418q^{7}-755q^{6}-260q^{5}+865q^{4}+1296q^{3}+624q^{2}-127q-840-720q^{-1}-704q^{-2}+171q^{-3}+767q^{-4}+680q^{-5}+361q^{-6}-193q^{-7}-361q^{-8}-731q^{-9}-278q^{-10}+173q^{-11}+381q^{-12}+424q^{-13}+205q^{-14}+64q^{-15}-419q^{-16}-323q^{-17}-161q^{-18}+28q^{-19}+190q^{-20}+229q^{-21}+249q^{-22}-91q^{-23}-132q^{-24}-165q^{-25}-109q^{-26}-24q^{-27}+78q^{-28}+183q^{-29}+32q^{-30}+13q^{-31}-52q^{-32}-65q^{-33}-68q^{-34}-16q^{-35}+68q^{-36}+20q^{-37}+32q^{-38}+4q^{-39}-9q^{-40}-33q^{-41}-21q^{-42}+15q^{-43}+12q^{-45}+6q^{-46}+5q^{-47}-9q^{-48}-8q^{-49}+4q^{-50}-2q^{-51}+2q^{-52}+q^{-53}+2q^{-54}-q^{-55}-2q^{-56}+q^{-57}$
7	$-q^{147}+2q^{146}-q^{145}-q^{144}+3q^{143}-q^{142}-q^{141}-3q^{140}-q^{139}+10q^{138}-3q^{137}-5q^{136}+5q^{135}-5q^{134}-2q^{133}-8q^{132}+33q^{130}+3q^{129}-13q^{128}-4q^{127}-29q^{126}-13q^{125}-21q^{124}+13q^{123}+96q^{122}+52q^{121}+8q^{120}-34q^{119}-131q^{118}-116q^{117}-95q^{116}+25q^{115}+261q^{114}+276q^{113}+212q^{112}+2q^{111}-378q^{110}-509q^{109}-510q^{108}-162q^{107}+512q^{106}+890q^{105}+980q^{104}+500q^{103}-550q^{102}-1333q^{101}-1700q^{100}-1131q^{99}+415q^{98}+1774q^{97}+2613q^{96}+2089q^{95}+51q^{94}-2106q^{93}-3671q^{92}-3343q^{91}-825q^{90}+2168q^{89}+4631q^{88}+4818q^{87}+2013q^{86}-1899q^{85}-5485q^{84}-6312q^{83}-3356q^{82}+1267q^{81}+5938q^{80}+7656q^{79}+4877q^{78}-363q^{77}-6095q^{76}-8724q^{75}-6237q^{74}-678q^{73}+5881q^{72}+9406q^{71}+7380q^{70}+1744q^{69}-5432q^{68}-9753q^{67}-8236q^{66}-2654q^{65}+4922q^{64}+9773q^{63}+8701q^{62}+3378q^{61}-4308q^{60}-9623q^{59}-8990q^{58}-3863q^{57}+3885q^{56}+9349q^{55}+8943q^{54}+4156q^{53}-3406q^{52}-9030q^{51}-8916q^{50}-4321q^{49}+3157q^{48}+8713q^{47}+8661q^{46}+4390q^{45}-2770q^{44}-8353q^{43}-8548q^{42}-4474q^{41}+2528q^{40}+7984q^{39}+8242q^{38}+4565q^{37}-2034q^{36}-7511q^{35}-8097q^{34}-4735q^{33}+1607q^{32}+6945q^{31}+7724q^{30}+4942q^{29}-870q^{28}-6211q^{27}-7459q^{26}-5180q^{25}+177q^{24}+5333q^{23}+6882q^{22}+5385q^{21}+766q^{20}-4265q^{19}-6306q^{18}-5507q^{17}-1574q^{16}+3085q^{15}+5380q^{14}+5419q^{13}+2444q^{12}-1778q^{11}-4386q^{10}-5126q^{9}-3029q^{8}+566q^{7}+3117q^{6}+4495q^{5}+3415q^{4}+586q^{3}-1862q^{2}-3669q-3388-1398q^{-1}+620q^{-2}+2585q^{-3}+3042q^{-4}+1904q^{-5}+400q^{-6}-1494q^{-7}-2398q^{-8}-1972q^{-9}-1098q^{-10}+483q^{-11}+1570q^{-12}+1706q^{-13}+1431q^{-14}+283q^{-15}-763q^{-16}-1208q^{-17}-1380q^{-18}-724q^{-19}+88q^{-20}+618q^{-21}+1094q^{-22}+845q^{-23}+334q^{-24}-108q^{-25}-678q^{-26}-725q^{-27}-504q^{-28}-223q^{-29}+290q^{-30}+467q^{-31}+459q^{-32}+383q^{-33}-4q^{-34}-227q^{-35}-323q^{-36}-357q^{-37}-118q^{-38}+22q^{-39}+144q^{-40}+274q^{-41}+166q^{-42}+61q^{-43}-44q^{-44}-157q^{-45}-106q^{-46}-85q^{-47}-43q^{-48}+71q^{-49}+76q^{-50}+74q^{-51}+38q^{-52}-30q^{-53}-20q^{-54}-34q^{-55}-45q^{-56}-4q^{-57}+10q^{-58}+26q^{-59}+24q^{-60}-5q^{-61}+3q^{-62}-2q^{-63}-14q^{-64}-6q^{-65}-4q^{-66}+6q^{-67}+8q^{-68}-2q^{-69}+2q^{-71}-2q^{-72}-q^{-73}-2q^{-74}+q^{-75}+2q^{-76}-q^{-77}$

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[8, 10]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[9, 15, 10, 14], X[5, 13, 6, 12], X[13, 7, 14, 6], X[11, 1, 12, 16], X[15, 11, 16, 10], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[8, 10]]
Out[3]=	GaussCode[-1, 8, -2, 1, -4, 5, -8, 2, -3, 7, -6, 4, -5, 3, -7, 6]
In[4]:=	DTCode[Knot[8, 10]]
Out[4]=	DTCode[4, 8, 12, 2, 14, 16, 6, 10]
In[5]:=	br = BR[Knot[8, 10]]
Out[5]=	BR[3, {1, 1, 1, -2, 1, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 8}
In[7]:=	BraidIndex[Knot[8, 10]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[8, 10]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[8, 10]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[8, 10]][t]
Out[10]=	-3 3 6 2 3 -7 + t - -- + - + 6 t - 3 t + t 2 t t
In[11]:=	Conway[Knot[8, 10]][z]
Out[11]=	2 4 6 1 + 3 z + 3 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 10], Knot[10, 143], Knot[11, NonAlternating, 106]}
In[13]:=	{KnotDet[Knot[8, 10]], KnotSignature[Knot[8, 10]]}
Out[13]=	{27, 2}
In[14]:=	Jones[Knot[8, 10]][q]
Out[14]=	-2 2 2 3 4 5 6 -3 - q + - + 5 q - 4 q + 5 q - 4 q + 2 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 10]}
In[16]:=	A2Invariant[Knot[8, 10]][q]
Out[16]=	-6 -2 2 4 6 8 10 12 14 18 -q - q + 2 q + q + 4 q + q + q - q - 2 q - q
In[17]:=	HOMFLYPT[Knot[8, 10]][a, z]
Out[17]=	2 2 4 4 6 3 6 2 3 z 9 z 4 z 5 z z -2 - -- + -- - 3 z - ---- + ---- - z - -- + ---- + -- 4 2 4 2 4 2 2 a a a a a a a
In[18]:=	Kauffman[Knot[8, 10]][a, z]
Out[18]=	2 2 3 6 z 2 z 6 z 5 z 2 z 6 z -2 - -- - -- - -- + --- + --- + --- + 2 a z + 5 z - -- + ---- + 4 2 7 5 3 a 6 4 a a a a a a a 2 3 3 3 3 4 4 12 z z 3 z 9 z 8 z 3 4 2 z 5 z ----- + -- - ---- - ---- - ---- - 3 a z - 6 z + ---- - ---- - 2 7 5 3 a 6 4 a a a a a a 4 5 5 5 6 6 7 7 13 z 3 z 3 z z 5 6 3 z 5 z z z ----- + ---- + ---- + -- + a z + 2 z + ---- + ---- + -- + -- 2 5 3 a 4 2 3 a a a a a a a
In[19]:=	{Vassiliev[2][Knot[8, 10]], Vassiliev[3][Knot[8, 10]]}
Out[19]=	{3, 3}
In[20]:=	Kh[Knot[8, 10]][q, t]
Out[20]=	3 1 1 1 2 q 3 5 3 q + 3 q + ----- + ----- + ---- + --- + - + 2 q t + 2 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 13 5 3 q t + 2 q t + q t + 3 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[8, 10], 2][q]
Out[21]=	-7 2 -5 6 5 6 14 2 3 4 -5 + q - -- - q + -- - -- - -- + -- - 14 q + 21 q - 2 q - 21 q + 6 4 3 2 q q q q q 5 6 7 8 9 10 11 12 23 q + 2 q - 24 q + 20 q + 3 q - 19 q + 12 q + 3 q - 13 14 15 16 17 9 q + 4 q + q - 2 q + q

See/edit the Rolfsen_Splice_Template.