10 95

10_94

10_96

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10 95 Quick Notes

10 95 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,17,12,16 X15,9,16,8 X19,7,20,6 X5,15,6,14 X7,19,8,18 X13,1,14,20 X17,13,18,12 X9,2,10,3
Gauss code	-1, 10, -2, 1, -6, 5, -7, 4, -10, 2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 8
Dowker-Thistlethwaite code	4 10 14 18 2 16 20 8 12 6
Conway Notation	[.210.2.2]

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-10] Hyperbolic Volume 15.0479 A-Polynomial See Data:10 95/A-polynomial

[edit Notes for 10 95's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 10 95's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-9t^{2}+21t-27+21t^{-1}-9t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+3z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 91, 2 }
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+11q^{5}-14q^{4}+16q^{3}-14q^{2}+12q-8+4q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+z^{2}a^{-2}+5z^{2}a^{-4}-2z^{2}a^{-6}-z^{2}+3a^{-4}-2a^{-6}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-3}+2z^{9}a^{-5}+6z^{8}a^{-2}+11z^{8}a^{-4}+5z^{8}a^{-6}+7z^{7}a^{-1}+10z^{7}a^{-3}+8z^{7}a^{-5}+5z^{7}a^{-7}-6z^{6}a^{-2}-19z^{6}a^{-4}-6z^{6}a^{-6}+3z^{6}a^{-8}+4z^{6}+az^{5}-12z^{5}a^{-1}-25z^{5}a^{-3}-21z^{5}a^{-5}-8z^{5}a^{-7}+z^{5}a^{-9}-2z^{4}a^{-2}+13z^{4}a^{-4}+4z^{4}a^{-6}-5z^{4}a^{-8}-6z^{4}-az^{3}+5z^{3}a^{-1}+16z^{3}a^{-3}+17z^{3}a^{-5}+5z^{3}a^{-7}-2z^{3}a^{-9}+z^{2}a^{-2}-7z^{2}a^{-4}-4z^{2}a^{-6}+2z^{2}a^{-8}+2z^{2}-za^{-1}-3za^{-3}-5za^{-5}-2za^{-7}+za^{-9}+3a^{-4}+2a^{-6}$
The A2 invariant	$-q^{6}+2q^{4}-q^{2}-1+3q^{-2}-3q^{-4}+3q^{-6}+q^{-10}+3q^{-12}-2q^{-14}+3q^{-16}-2q^{-18}-2q^{-20}+q^{-22}-q^{-24}$
The G2 invariant	$q^{32}-3q^{30}+7q^{28}-13q^{26}+15q^{24}-14q^{22}+3q^{20}+22q^{18}-53q^{16}+87q^{14}-102q^{12}+76q^{10}-13q^{8}-85q^{6}+191q^{4}-252q^{2}+243-141q^{-2}-36q^{-4}+219q^{-6}-340q^{-8}+342q^{-10}-220q^{-12}+19q^{-14}+179q^{-16}-286q^{-18}+261q^{-20}-110q^{-22}-88q^{-24}+243q^{-26}-278q^{-28}+160q^{-30}+52q^{-32}-272q^{-34}+413q^{-36}-401q^{-38}+247q^{-40}+12q^{-42}-279q^{-44}+460q^{-46}-491q^{-48}+363q^{-50}-118q^{-52}-143q^{-54}+331q^{-56}-370q^{-58}+273q^{-60}-72q^{-62}-133q^{-64}+248q^{-66}-232q^{-68}+85q^{-70}+115q^{-72}-272q^{-74}+316q^{-76}-223q^{-78}+35q^{-80}+161q^{-82}-302q^{-84}+329q^{-86}-249q^{-88}+100q^{-90}+51q^{-92}-161q^{-94}+198q^{-96}-170q^{-98}+108q^{-100}-33q^{-102}-25q^{-104}+53q^{-106}-62q^{-108}+50q^{-110}-31q^{-112}+14q^{-114}+q^{-116}-8q^{-118}+9q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 10_95.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+3q^{3}-4q+4q^{-1}-2q^{-3}+2q^{-5}+2q^{-7}-3q^{-9}+4q^{-11}-4q^{-13}+2q^{-15}-q^{-17}$
2	$q^{16}-3q^{14}+11q^{10}-13q^{8}-12q^{6}+33q^{4}-10q^{2}-34+41q^{-2}+7q^{-4}-43q^{-6}+23q^{-8}+21q^{-10}-26q^{-12}-5q^{-14}+23q^{-16}+5q^{-18}-33q^{-20}+12q^{-22}+34q^{-24}-41q^{-26}-7q^{-28}+43q^{-30}-24q^{-32}-17q^{-34}+26q^{-36}-5q^{-38}-10q^{-40}+7q^{-42}-2q^{-46}+q^{-48}$
3	$-q^{33}+3q^{31}-7q^{27}-2q^{25}+17q^{23}+15q^{21}-39q^{19}-41q^{17}+53q^{15}+93q^{13}-47q^{11}-166q^{9}+13q^{7}+234q^{5}+60q^{3}-275q-165q^{-1}+279q^{-3}+262q^{-5}-228q^{-7}-331q^{-9}+150q^{-11}+358q^{-13}-52q^{-15}-342q^{-17}-33q^{-19}+288q^{-21}+112q^{-23}-216q^{-25}-178q^{-27}+131q^{-29}+234q^{-31}-37q^{-33}-282q^{-35}-59q^{-37}+298q^{-39}+168q^{-41}-293q^{-43}-259q^{-45}+242q^{-47}+324q^{-49}-158q^{-51}-345q^{-53}+57q^{-55}+318q^{-57}+25q^{-59}-244q^{-61}-79q^{-63}+160q^{-65}+92q^{-67}-86q^{-69}-72q^{-71}+33q^{-73}+46q^{-75}-10q^{-77}-23q^{-79}+3q^{-81}+9q^{-83}-q^{-85}-3q^{-87}+2q^{-91}-q^{-93}$
4	$q^{56}-3q^{54}+7q^{50}-2q^{48}-2q^{46}-20q^{44}+2q^{42}+49q^{40}+17q^{38}-15q^{36}-129q^{34}-69q^{32}+170q^{30}+217q^{28}+112q^{26}-389q^{24}-505q^{22}+75q^{20}+660q^{18}+855q^{16}-312q^{14}-1340q^{12}-897q^{10}+628q^{8}+2133q^{6}+913q^{4}-1538q^{2}-2513-775q^{-2}+2617q^{-4}+2803q^{-6}-195q^{-8}-3208q^{-10}-2770q^{-12}+1435q^{-14}+3634q^{-16}+1750q^{-18}-2218q^{-20}-3640q^{-22}-396q^{-24}+2807q^{-26}+2725q^{-28}-555q^{-30}-3004q^{-32}-1583q^{-34}+1303q^{-36}+2581q^{-38}+751q^{-40}-1791q^{-42}-2111q^{-44}-71q^{-46}+2095q^{-48}+1781q^{-50}-554q^{-52}-2525q^{-54}-1450q^{-56}+1476q^{-58}+2807q^{-60}+950q^{-62}-2587q^{-64}-2922q^{-66}+204q^{-68}+3264q^{-70}+2706q^{-72}-1554q^{-74}-3645q^{-76}-1619q^{-78}+2276q^{-80}+3619q^{-82}+351q^{-84}-2691q^{-86}-2685q^{-88}+301q^{-90}+2772q^{-92}+1572q^{-94}-772q^{-96}-2079q^{-98}-946q^{-100}+1047q^{-102}+1269q^{-104}+393q^{-106}-769q^{-108}-814q^{-110}+30q^{-112}+420q^{-114}+403q^{-116}-65q^{-118}-276q^{-120}-96q^{-122}+26q^{-124}+126q^{-126}+27q^{-128}-44q^{-130}-16q^{-132}-14q^{-134}+20q^{-136}+5q^{-138}-7q^{-140}+2q^{-142}-3q^{-144}+3q^{-146}-2q^{-150}+q^{-152}$
5	$-q^{85}+3q^{83}-7q^{79}+2q^{77}+6q^{75}+5q^{73}+3q^{71}-12q^{69}-36q^{67}-14q^{65}+57q^{63}+93q^{61}+51q^{59}-90q^{57}-240q^{55}-234q^{53}+86q^{51}+541q^{49}+638q^{47}+129q^{45}-824q^{43}-1462q^{41}-928q^{39}+882q^{37}+2657q^{35}+2557q^{33}-56q^{31}-3726q^{29}-5263q^{27}-2350q^{25}+3879q^{23}+8535q^{21}+6693q^{19}-1858q^{17}-11172q^{15}-12772q^{13}-3169q^{11}+11624q^{9}+19211q^{7}+11109q^{5}-8402q^{3}-23950q-20838q^{-1}+1133q^{-3}+25148q^{-5}+30006q^{-7}+9156q^{-9}-21625q^{-11}-36241q^{-13}-20461q^{-15}+13964q^{-17}+37948q^{-19}+30068q^{-21}-3791q^{-23}-34785q^{-25}-36085q^{-27}-6534q^{-29}+27940q^{-31}+37600q^{-33}+14968q^{-35}-19241q^{-37}-35145q^{-39}-20297q^{-41}+10569q^{-43}+30063q^{-45}+22581q^{-47}-3290q^{-49}-24034q^{-51}-22572q^{-53}-2186q^{-55}+18287q^{-57}+21612q^{-59}+6230q^{-61}-13611q^{-63}-20783q^{-65}-9643q^{-67}+9842q^{-69}+20739q^{-71}+13429q^{-73}-6337q^{-75}-21582q^{-77}-18138q^{-79}+2248q^{-81}+22434q^{-83}+23889q^{-85}+3420q^{-87}-22344q^{-89}-29999q^{-91}-10776q^{-93}+19913q^{-95}+35029q^{-97}+19587q^{-99}-14435q^{-101}-37440q^{-103}-28345q^{-105}+5937q^{-107}+35806q^{-109}+35172q^{-111}+4428q^{-113}-29726q^{-115}-38195q^{-117}-14591q^{-119}+20089q^{-121}+36389q^{-123}+22115q^{-125}-8823q^{-127}-30021q^{-129}-25468q^{-131}-1438q^{-133}+20855q^{-135}+24133q^{-137}+8550q^{-139}-11133q^{-141}-19273q^{-143}-11657q^{-145}+3213q^{-147}+12844q^{-149}+11108q^{-151}+1722q^{-153}-6768q^{-155}-8422q^{-157}-3705q^{-159}+2438q^{-161}+5199q^{-163}+3534q^{-165}-73q^{-167}-2533q^{-169}-2481q^{-171}-748q^{-173}+921q^{-175}+1365q^{-177}+714q^{-179}-169q^{-181}-590q^{-183}-443q^{-185}-65q^{-187}+206q^{-189}+212q^{-191}+64q^{-193}-52q^{-195}-70q^{-197}-38q^{-199}+q^{-201}+30q^{-203}+14q^{-205}-7q^{-207}-2q^{-209}-q^{-211}-4q^{-213}+2q^{-215}+3q^{-217}-3q^{-219}+2q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+2q^{4}-q^{2}-1+3q^{-2}-3q^{-4}+3q^{-6}+q^{-10}+3q^{-12}-2q^{-14}+3q^{-16}-2q^{-18}-2q^{-20}+q^{-22}-q^{-24}$
1,1	$q^{20}-6q^{18}+20q^{16}-50q^{14}+109q^{12}-214q^{10}+376q^{8}-600q^{6}+876q^{4}-1184q^{2}+1472-1672q^{-2}+1720q^{-4}-1564q^{-6}+1184q^{-8}-572q^{-10}-200q^{-12}+1064q^{-14}-1902q^{-16}+2622q^{-18}-3128q^{-20}+3366q^{-22}-3296q^{-24}+2936q^{-26}-2322q^{-28}+1538q^{-30}-692q^{-32}-134q^{-34}+827q^{-36}-1328q^{-38}+1598q^{-40}-1654q^{-42}+1540q^{-44}-1310q^{-46}+1030q^{-48}-756q^{-50}+518q^{-52}-332q^{-54}+200q^{-56}-114q^{-58}+60q^{-60}-28q^{-62}+12q^{-64}-4q^{-66}+q^{-68}$
2,0	$q^{18}-2q^{16}-2q^{14}+6q^{12}+3q^{10}-9q^{8}-8q^{6}+12q^{4}+10q^{2}-18-7q^{-2}+21q^{-4}+6q^{-6}-18q^{-8}+q^{-10}+17q^{-12}-4q^{-14}-8q^{-16}+8q^{-18}+5q^{-20}-12q^{-22}+8q^{-24}+10q^{-26}-15q^{-28}-4q^{-30}+18q^{-32}+3q^{-34}-21q^{-36}-q^{-38}+17q^{-40}-3q^{-42}-17q^{-44}+q^{-46}+11q^{-48}-q^{-50}-5q^{-52}+3q^{-56}-q^{-60}+q^{-62}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-3q^{12}+q^{10}+7q^{8}-13q^{6}+4q^{4}+18q^{2}-27+7q^{-2}+26q^{-4}-34q^{-6}+4q^{-8}+26q^{-10}-21q^{-12}-3q^{-14}+18q^{-16}+2q^{-18}-5q^{-20}-2q^{-22}+21q^{-24}-6q^{-26}-26q^{-28}+28q^{-30}-4q^{-32}-33q^{-34}+27q^{-36}+q^{-38}-22q^{-40}+16q^{-42}+2q^{-44}-9q^{-46}+5q^{-48}+q^{-50}-2q^{-52}+q^{-54}$
1,0,0	$-q^{7}+2q^{5}-2q^{3}+2q-2q^{-1}+3q^{-3}-3q^{-5}+2q^{-7}+q^{-11}+2q^{-13}+q^{-15}+4q^{-17}-2q^{-19}+3q^{-21}-3q^{-23}-3q^{-27}+q^{-29}-q^{-31}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+3q^{12}-7q^{10}+13q^{8}-21q^{6}+30q^{4}-36q^{2}+39-39q^{-2}+34q^{-4}-22q^{-6}+6q^{-8}+14q^{-10}-33q^{-12}+53q^{-14}-66q^{-16}+76q^{-18}-75q^{-20}+70q^{-22}-55q^{-24}+38q^{-26}-18q^{-28}-2q^{-30}+18q^{-32}-31q^{-34}+37q^{-36}-39q^{-38}+36q^{-40}-30q^{-42}+22q^{-44}-15q^{-46}+9q^{-48}-5q^{-50}+2q^{-52}-q^{-54}$
1,0	$q^{24}-3q^{20}-3q^{18}+4q^{16}+10q^{14}-17q^{10}-12q^{8}+18q^{6}+27q^{4}-7q^{2}-37-13q^{-2}+36q^{-4}+32q^{-6}-22q^{-8}-41q^{-10}+2q^{-12}+39q^{-14}+13q^{-16}-29q^{-18}-20q^{-20}+22q^{-22}+24q^{-24}-12q^{-26}-24q^{-28}+9q^{-30}+28q^{-32}-28q^{-36}-4q^{-38}+30q^{-40}+14q^{-42}-30q^{-44}-27q^{-46}+22q^{-48}+37q^{-50}-9q^{-52}-44q^{-54}-13q^{-56}+34q^{-58}+28q^{-60}-17q^{-62}-32q^{-64}-2q^{-66}+23q^{-68}+12q^{-70}-9q^{-72}-12q^{-74}+7q^{-78}+3q^{-80}-2q^{-82}-2q^{-84}+q^{-88}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{18}-3q^{16}+4q^{14}-6q^{12}+11q^{10}-17q^{8}+20q^{6}-23q^{4}+30q^{2}-32+30q^{-2}-29q^{-4}+29q^{-6}-21q^{-8}+7q^{-10}-2q^{-12}-6q^{-14}+22q^{-16}-35q^{-18}+41q^{-20}-45q^{-22}+62q^{-24}-55q^{-26}+59q^{-28}-52q^{-30}+56q^{-32}-37q^{-34}+28q^{-36}-25q^{-38}+7q^{-40}+3q^{-42}-16q^{-44}+14q^{-46}-29q^{-48}+31q^{-50}-29q^{-52}+28q^{-54}-29q^{-56}+25q^{-58}-17q^{-60}+14q^{-62}-12q^{-64}+8q^{-66}-4q^{-68}+3q^{-70}-2q^{-72}+q^{-74}$

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-3q^{30}+7q^{28}-13q^{26}+15q^{24}-14q^{22}+3q^{20}+22q^{18}-53q^{16}+87q^{14}-102q^{12}+76q^{10}-13q^{8}-85q^{6}+191q^{4}-252q^{2}+243-141q^{-2}-36q^{-4}+219q^{-6}-340q^{-8}+342q^{-10}-220q^{-12}+19q^{-14}+179q^{-16}-286q^{-18}+261q^{-20}-110q^{-22}-88q^{-24}+243q^{-26}-278q^{-28}+160q^{-30}+52q^{-32}-272q^{-34}+413q^{-36}-401q^{-38}+247q^{-40}+12q^{-42}-279q^{-44}+460q^{-46}-491q^{-48}+363q^{-50}-118q^{-52}-143q^{-54}+331q^{-56}-370q^{-58}+273q^{-60}-72q^{-62}-133q^{-64}+248q^{-66}-232q^{-68}+85q^{-70}+115q^{-72}-272q^{-74}+316q^{-76}-223q^{-78}+35q^{-80}+161q^{-82}-302q^{-84}+329q^{-86}-249q^{-88}+100q^{-90}+51q^{-92}-161q^{-94}+198q^{-96}-170q^{-98}+108q^{-100}-33q^{-102}-25q^{-104}+53q^{-106}-62q^{-108}+50q^{-110}-31q^{-112}+14q^{-114}+q^{-116}-8q^{-118}+9q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

.

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 95"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-9t^{2}+21t-27+21t^{-1}-9t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 91, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+3q^{7}-7q^{6}+11q^{5}-14q^{4}+16q^{3}-14q^{2}+12q-8+4q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-3}+2z^{9}a^{-5}+6z^{8}a^{-2}+11z^{8}a^{-4}+5z^{8}a^{-6}+7z^{7}a^{-1}+10z^{7}a^{-3}+8z^{7}a^{-5}+5z^{7}a^{-7}-6z^{6}a^{-2}-19z^{6}a^{-4}-6z^{6}a^{-6}+3z^{6}a^{-8}+4z^{6}+az^{5}-12z^{5}a^{-1}-25z^{5}a^{-3}-21z^{5}a^{-5}-8z^{5}a^{-7}+z^{5}a^{-9}-2z^{4}a^{-2}+13z^{4}a^{-4}+4z^{4}a^{-6}-5z^{4}a^{-8}-6z^{4}-az^{3}+5z^{3}a^{-1}+16z^{3}a^{-3}+17z^{3}a^{-5}+5z^{3}a^{-7}-2z^{3}a^{-9}+z^{2}a^{-2}-7z^{2}a^{-4}-4z^{2}a^{-6}+2z^{2}a^{-8}+2z^{2}-za^{-1}-3za^{-3}-5za^{-5}-2za^{-7}+za^{-9}+3a^{-4}+2a^{-6}$

Vassiliev invariants

V2 and V3:

(3, 5)

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 $12$ $40$ $72$ $158$ $10$ $480$ ${\frac {2320}{3}}$ ${\frac {352}{3}}$ $72$ $288$ $800$ $1896$ $120$ ${\frac {37871}{10}}$ ${\frac {1258}{5}}$ ${\frac {16502}{15}}$ ${\frac {209}{6}}$ ${\frac {911}{10}}$

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 10 95. 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
17											1	-1
15										2		2
13									5	1		-4
11								6	2			4
9							8	5				-3
7						8	6					2
5					6	8						2
3				6	8							-2
1			3	7								4
-1		1	5									-4
-3		3										3
-5	1											-1

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, 95]]
Out[2]=	10
In[3]:=	PD[Knot[10, 95]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 17, 12, 16], X[15, 9, 16, 8], X[19, 7, 20, 6], X[5, 15, 6, 14], X[7, 19, 8, 18], X[13, 1, 14, 20], X[17, 13, 18, 12], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[10, 95]]
Out[4]=	GaussCode[-1, 10, -2, 1, -6, 5, -7, 4, -10, 2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 8]
In[5]:=	BR[Knot[10, 95]]
Out[5]=	BR[4, {-1, -1, 2, 2, -3, 2, -1, 2, 3, 3, 2}]
In[6]:=	alex = Alexander[Knot[10, 95]][t]
Out[6]=	2 9 21 2 3 -27 + -- - -- + -- + 21 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 95]][z]
Out[7]=	2 4 6 1 + 3 z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 95]}
In[9]:=	{KnotDet[Knot[10, 95]], KnotSignature[Knot[10, 95]]}
Out[9]=	{91, 2}
In[10]:=	J=Jones[Knot[10, 95]][q]
Out[10]=	-2 4 2 3 4 5 6 7 8 -8 - q + - + 12 q - 14 q + 16 q - 14 q + 11 q - 7 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 95]}
In[12]:=	A2Invariant[Knot[10, 95]][q]
Out[12]=	-6 2 -2 2 4 6 10 12 14 -1 - q + -- - q + 3 q - 3 q + 3 q + q + 3 q - 2 q + 4 q 16 18 20 22 24 3 q - 2 q - 2 q + q - q
In[13]:=	Kauffman[Knot[10, 95]][a, z]
Out[13]=	2 2 2 2 2 3 z 2 z 5 z 3 z z 2 2 z 4 z 7 z z -- + -- + -- - --- - --- - --- - - + 2 z + ---- - ---- - ---- + -- - 6 4 9 7 5 3 a 8 6 4 2 a a a a a a a a a a 3 3 3 3 3 4 4 2 z 5 z 17 z 16 z 5 z 3 4 5 z 4 z ---- + ---- + ----- + ----- + ---- - a z - 6 z - ---- + ---- + 9 7 5 3 a 8 6 a a a a a a 4 4 5 5 5 5 5 13 z 2 z z 8 z 21 z 25 z 12 z 5 6 ----- - ---- + -- - ---- - ----- - ----- - ----- + a z + 4 z + 4 2 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 7 8 3 z 6 z 19 z 6 z 5 z 8 z 10 z 7 z 5 z ---- - ---- - ----- - ---- + ---- + ---- + ----- + ---- + ---- + 8 6 4 2 7 5 3 a 6 a a a a a a a a 8 8 9 9 11 z 6 z 2 z 2 z ----- + ---- + ---- + ---- 4 2 5 3 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 95]], Vassiliev[3][Knot[10, 95]]}
Out[14]=	{0, 5}
In[15]:=	Kh[Knot[10, 95]][q, t]
Out[15]=	3 1 3 1 5 3 q 3 5 7 q + 6 q + ----- + ----- + ---- + --- + --- + 8 q t + 6 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 8 q t + 8 q t + 6 q t + 8 q t + 5 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 2 q t + 5 q t + q t + 2 q t + q t