10 84

10_83

10_85

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10 84 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X8,12,9,11 X20,15,1,16 X16,5,17,6 X12,18,13,17 X14,8,15,7 X18,14,19,13 X6,19,7,20 X2,10,3,9
Gauss code	1, -10, 2, -1, 5, -9, 7, -3, 10, -2, 3, -6, 8, -7, 4, -5, 6, -8, 9, -4
Dowker-Thistlethwaite code	4 10 16 14 2 8 18 20 12 6
Conway Notation	[.22.2]

Minimum Braid Representative: Length is 11, width is 4. Braid index is 4.

A Morse Link Presentation:

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 14.7099 A-Polynomial See Data:10 84/A-polynomial

[edit Notes for 10 84'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 84's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-9t^{2}+20t-25+20t^{-1}-9t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 87, 2 }
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+11q^{5}-14q^{4}+15q^{3}-13q^{2}+11q-6+3q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+3z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+5z^{2}a^{-2}-z^{2}a^{-6}-2z^{2}+4a^{-2}-2a^{-4}-1$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-3}+2z^{9}a^{-5}+4z^{8}a^{-2}+10z^{8}a^{-4}+6z^{8}a^{-6}+4z^{7}a^{-1}+5z^{7}a^{-3}+8z^{7}a^{-5}+7z^{7}a^{-7}-2z^{6}a^{-2}-17z^{6}a^{-4}-8z^{6}a^{-6}+4z^{6}a^{-8}+3z^{6}+az^{5}-4z^{5}a^{-1}-11z^{5}a^{-3}-20z^{5}a^{-5}-13z^{5}a^{-7}+z^{5}a^{-9}-5z^{4}a^{-2}+9z^{4}a^{-4}+2z^{4}a^{-6}-6z^{4}a^{-8}-6z^{4}-2az^{3}-2z^{3}a^{-1}+4z^{3}a^{-3}+11z^{3}a^{-5}+6z^{3}a^{-7}-z^{3}a^{-9}+7z^{2}a^{-2}+z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}+4z^{2}+az+2za^{-1}+2za^{-3}-za^{-7}-4a^{-2}-2a^{-4}-1$
The A2 invariant	$-q^{6}+q^{4}-q^{2}-1+4q^{-2}-q^{-4}+4q^{-6}+q^{-8}-q^{-10}+q^{-12}-4q^{-14}+2q^{-16}-q^{-18}-q^{-20}+2q^{-22}-q^{-24}$
The G2 invariant	$q^{32}-2q^{30}+5q^{28}-8q^{26}+8q^{24}-7q^{22}-2q^{20}+15q^{18}-30q^{16}+43q^{14}-49q^{12}+39q^{10}-14q^{8}-31q^{6}+88q^{4}-135q^{2}+155-130q^{-2}+45q^{-4}+73q^{-6}-192q^{-8}+270q^{-10}-257q^{-12}+158q^{-14}+9q^{-16}-171q^{-18}+266q^{-20}-245q^{-22}+128q^{-24}+45q^{-26}-178q^{-28}+213q^{-30}-130q^{-32}-29q^{-34}+208q^{-36}-308q^{-38}+285q^{-40}-137q^{-42}-84q^{-44}+291q^{-46}-414q^{-48}+398q^{-50}-255q^{-52}+33q^{-54}+188q^{-56}-340q^{-58}+366q^{-60}-262q^{-62}+73q^{-64}+112q^{-66}-230q^{-68}+224q^{-70}-106q^{-72}-61q^{-74}+203q^{-76}-245q^{-78}+172q^{-80}-12q^{-82}-172q^{-84}+292q^{-86}-302q^{-88}+206q^{-90}-47q^{-92}-113q^{-94}+215q^{-96}-231q^{-98}+180q^{-100}-86q^{-102}-8q^{-104}+69q^{-106}-96q^{-108}+82q^{-110}-51q^{-112}+23q^{-114}+2q^{-116}-13q^{-118}+15q^{-120}-13q^{-122}+7q^{-124}-3q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 10_84.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+2q^{3}-3q+5q^{-1}-2q^{-3}+2q^{-5}+q^{-7}-3q^{-9}+3q^{-11}-4q^{-13}+3q^{-15}-q^{-17}$
2	$q^{16}-2q^{14}-q^{12}+6q^{10}-8q^{8}-4q^{6}+22q^{4}-15q^{2}-19+40q^{-2}-7q^{-4}-35q^{-6}+33q^{-8}+10q^{-10}-30q^{-12}+6q^{-14}+19q^{-16}-7q^{-18}-23q^{-20}+19q^{-22}+19q^{-24}-39q^{-26}+8q^{-28}+35q^{-30}-33q^{-32}-8q^{-34}+31q^{-36}-12q^{-38}-12q^{-40}+11q^{-42}-3q^{-46}+q^{-48}$
3	$-q^{33}+2q^{31}+q^{29}-2q^{27}-3q^{25}+5q^{23}+4q^{21}-15q^{19}-7q^{17}+30q^{15}+22q^{13}-55q^{11}-56q^{9}+77q^{7}+116q^{5}-81q^{3}-185q+47q^{-1}+263q^{-3}+12q^{-5}-297q^{-7}-98q^{-9}+294q^{-11}+178q^{-13}-248q^{-15}-228q^{-17}+171q^{-19}+247q^{-21}-81q^{-23}-235q^{-25}-9q^{-27}+207q^{-29}+85q^{-31}-161q^{-33}-167q^{-35}+116q^{-37}+228q^{-39}-55q^{-41}-284q^{-43}-10q^{-45}+310q^{-47}+90q^{-49}-299q^{-51}-166q^{-53}+253q^{-55}+216q^{-57}-172q^{-59}-232q^{-61}+80q^{-63}+207q^{-65}-5q^{-67}-155q^{-69}-34q^{-71}+89q^{-73}+47q^{-75}-40q^{-77}-36q^{-79}+15q^{-81}+16q^{-83}-2q^{-85}-7q^{-87}+3q^{-91}-q^{-93}$
4	$q^{56}-2q^{54}-q^{52}+2q^{50}-q^{48}+6q^{46}-5q^{44}+9q^{40}-14q^{38}-q^{36}-23q^{34}+22q^{32}+77q^{30}-9q^{28}-67q^{26}-171q^{24}-2q^{22}+301q^{20}+235q^{18}-56q^{16}-607q^{14}-444q^{12}+443q^{10}+925q^{8}+579q^{6}-930q^{4}-1509q^{2}-245+1445q^{-2}+1983q^{-4}-232q^{-6}-2323q^{-8}-1791q^{-10}+793q^{-12}+3002q^{-14}+1352q^{-16}-1799q^{-18}-2874q^{-20}-729q^{-22}+2579q^{-24}+2432q^{-26}-351q^{-28}-2573q^{-30}-1800q^{-32}+1199q^{-34}+2311q^{-36}+854q^{-38}-1459q^{-40}-1961q^{-42}-114q^{-44}+1595q^{-46}+1507q^{-48}-353q^{-50}-1761q^{-52}-1176q^{-54}+880q^{-56}+2026q^{-58}+701q^{-60}-1513q^{-62}-2235q^{-64}-9q^{-66}+2367q^{-68}+1922q^{-70}-797q^{-72}-2995q^{-74}-1313q^{-76}+1876q^{-78}+2831q^{-80}+610q^{-82}-2624q^{-84}-2367q^{-86}+390q^{-88}+2514q^{-90}+1840q^{-92}-1100q^{-94}-2163q^{-96}-965q^{-98}+1077q^{-100}+1793q^{-102}+276q^{-104}-940q^{-106}-1105q^{-108}-115q^{-110}+828q^{-112}+531q^{-114}-8q^{-116}-475q^{-118}-325q^{-120}+130q^{-122}+200q^{-124}+148q^{-126}-69q^{-128}-119q^{-130}-12q^{-132}+15q^{-134}+46q^{-136}+2q^{-138}-19q^{-140}-2q^{-142}-2q^{-144}+7q^{-146}-3q^{-150}+q^{-152}$
5	$-q^{85}+2q^{83}+q^{81}-2q^{79}+q^{77}-2q^{75}-6q^{73}+q^{71}+6q^{69}+q^{67}+13q^{65}+13q^{63}-14q^{61}-35q^{59}-39q^{57}-13q^{55}+64q^{53}+142q^{51}+100q^{49}-98q^{47}-311q^{45}-330q^{43}-q^{41}+549q^{39}+850q^{37}+387q^{35}-741q^{33}-1659q^{31}-1355q^{29}+479q^{27}+2674q^{25}+3154q^{23}+695q^{21}-3376q^{19}-5677q^{17}-3348q^{15}+2943q^{13}+8388q^{11}+7598q^{9}-475q^{7}-10214q^{5}-12893q^{3}-4415q+9795q^{-1}+17931q^{-3}+11490q^{-5}-6365q^{-7}-21186q^{-9}-19189q^{-11}-93q^{-13}+21141q^{-15}+25898q^{-17}+8431q^{-19}-17601q^{-21}-29735q^{-23}-16708q^{-25}+11076q^{-27}+29878q^{-29}+23196q^{-31}-3242q^{-33}-26528q^{-35}-26612q^{-37}-4145q^{-39}+20728q^{-41}+26751q^{-43}+9861q^{-45}-14068q^{-47}-24316q^{-49}-13301q^{-51}+7817q^{-53}+20429q^{-55}+14786q^{-57}-2721q^{-59}-16324q^{-61}-15064q^{-63}-1184q^{-65}+12810q^{-67}+15086q^{-69}+4250q^{-71}-10052q^{-73}-15511q^{-75}-7310q^{-77}+7856q^{-79}+16703q^{-81}+10795q^{-83}-5503q^{-85}-18203q^{-87}-15247q^{-89}+2294q^{-91}+19488q^{-93}+20227q^{-95}+2311q^{-97}-19372q^{-99}-25126q^{-101}-8404q^{-103}+17105q^{-105}+28708q^{-107}+15262q^{-109}-12185q^{-111}-29724q^{-113}-21717q^{-115}+5021q^{-117}+27387q^{-119}+26228q^{-121}+3107q^{-123}-21689q^{-125}-27471q^{-127}-10525q^{-129}+13702q^{-131}+25036q^{-133}+15499q^{-135}-5220q^{-137}-19530q^{-139}-17038q^{-141}-1869q^{-143}+12466q^{-145}+15290q^{-147}+6291q^{-149}-5787q^{-151}-11362q^{-153}-7627q^{-155}+794q^{-157}+6798q^{-159}+6695q^{-161}+1905q^{-163}-3044q^{-165}-4579q^{-167}-2577q^{-169}+609q^{-171}+2454q^{-173}+2134q^{-175}+454q^{-177}-998q^{-179}-1275q^{-181}-623q^{-183}+198q^{-185}+588q^{-187}+465q^{-189}+54q^{-191}-220q^{-193}-215q^{-195}-82q^{-197}+45q^{-199}+86q^{-201}+55q^{-203}-12q^{-205}-33q^{-207}-12q^{-209}+2q^{-211}+5q^{-213}+6q^{-215}+2q^{-217}-7q^{-219}+3q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+q^{4}-q^{2}-1+4q^{-2}-q^{-4}+4q^{-6}+q^{-8}-q^{-10}+q^{-12}-4q^{-14}+2q^{-16}-q^{-18}-q^{-20}+2q^{-22}-q^{-24}$
1,1	$q^{20}-4q^{18}+12q^{16}-28q^{14}+56q^{12}-102q^{10}+168q^{8}-272q^{6}+409q^{4}-590q^{2}+804-1024q^{-2}+1229q^{-4}-1340q^{-6}+1342q^{-8}-1158q^{-10}+792q^{-12}-252q^{-14}-410q^{-16}+1118q^{-18}-1798q^{-20}+2348q^{-22}-2706q^{-24}+2824q^{-26}-2693q^{-28}+2326q^{-30}-1764q^{-32}+1078q^{-34}-351q^{-36}-330q^{-38}+882q^{-40}-1262q^{-42}+1445q^{-44}-1444q^{-46}+1300q^{-48}-1064q^{-50}+804q^{-52}-560q^{-54}+354q^{-56}-204q^{-58}+107q^{-60}-50q^{-62}+20q^{-64}-6q^{-66}+q^{-68}$
2,0	$q^{18}-q^{16}-2q^{14}+3q^{12}+2q^{10}-7q^{8}-5q^{6}+9q^{4}+6q^{2}-14-5q^{-2}+22q^{-4}+5q^{-6}-15q^{-8}+5q^{-10}+16q^{-12}-3q^{-14}-12q^{-16}+6q^{-18}+4q^{-20}-14q^{-22}+4q^{-24}+8q^{-26}-10q^{-28}-4q^{-30}+16q^{-32}-18q^{-36}+q^{-38}+15q^{-40}-q^{-42}-16q^{-44}+6q^{-46}+11q^{-48}-4q^{-50}-6q^{-52}+5q^{-56}-q^{-58}-2q^{-60}+q^{-62}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-2q^{12}+q^{10}+4q^{8}-9q^{6}+q^{4}+12q^{2}-20+27q^{-4}-25q^{-6}+3q^{-8}+34q^{-10}-17q^{-12}-3q^{-14}+18q^{-16}-7q^{-18}-12q^{-20}-4q^{-22}+11q^{-24}-5q^{-26}-19q^{-28}+24q^{-30}+5q^{-32}-30q^{-34}+23q^{-36}+7q^{-38}-26q^{-40}+16q^{-42}+4q^{-44}-12q^{-46}+7q^{-48}+q^{-50}-3q^{-52}+q^{-54}$
1,0,0	$-q^{7}+q^{5}-2q^{3}+q-2q^{-1}+4q^{-3}-q^{-5}+5q^{-7}+2q^{-9}+2q^{-11}-2q^{-15}-4q^{-19}+2q^{-21}-2q^{-23}+2q^{-25}-2q^{-27}+2q^{-29}-q^{-31}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{16}-q^{14}+3q^{10}-q^{8}-5q^{6}+q^{4}+4q^{2}-8-12q^{-2}+8q^{-4}+12q^{-6}-17q^{-8}-2q^{-10}+32q^{-12}+10q^{-14}-15q^{-16}+17q^{-18}+25q^{-20}-14q^{-22}-18q^{-24}+13q^{-26}-4q^{-28}-33q^{-30}+5q^{-32}+18q^{-34}-22q^{-36}-7q^{-38}+30q^{-40}-23q^{-44}+8q^{-46}+19q^{-48}-12q^{-50}-15q^{-52}+13q^{-54}+8q^{-56}-11q^{-58}-q^{-60}+7q^{-62}-2q^{-64}-2q^{-66}+q^{-68}$
1,0,0,0	$-q^{8}+q^{6}-2q^{4}-2q^{-2}+4q^{-4}-q^{-6}+5q^{-8}+3q^{-10}+3q^{-12}+2q^{-14}-q^{-18}-3q^{-20}-4q^{-24}+2q^{-26}-2q^{-28}+q^{-30}+q^{-32}-2q^{-34}+2q^{-36}-q^{-38}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+2q^{12}-5q^{10}+8q^{8}-13q^{6}+19q^{4}-26q^{2}+32-34q^{-2}+35q^{-4}-27q^{-6}+19q^{-8}-2q^{-10}-13q^{-12}+33q^{-14}-48q^{-16}+61q^{-18}-68q^{-20}+68q^{-22}-63q^{-24}+49q^{-26}-33q^{-28}+14q^{-30}+3q^{-32}-18q^{-34}+29q^{-36}-35q^{-38}+36q^{-40}-34q^{-42}+28q^{-44}-20q^{-46}+13q^{-48}-7q^{-50}+3q^{-52}-q^{-54}$
1,0	$q^{24}-2q^{20}-2q^{18}+3q^{16}+6q^{14}-q^{12}-11q^{10}-8q^{8}+10q^{6}+19q^{4}-2q^{2}-28-15q^{-2}+24q^{-4}+33q^{-6}-9q^{-8}-37q^{-10}-7q^{-12}+37q^{-14}+24q^{-16}-20q^{-18}-26q^{-20}+13q^{-22}+28q^{-24}-3q^{-26}-27q^{-28}-4q^{-30}+22q^{-32}+6q^{-34}-23q^{-36}-15q^{-38}+19q^{-40}+19q^{-42}-17q^{-44}-27q^{-46}+11q^{-48}+35q^{-50}+4q^{-52}-36q^{-54}-21q^{-56}+28q^{-58}+34q^{-60}-11q^{-62}-35q^{-64}-8q^{-66}+25q^{-68}+17q^{-70}-11q^{-72}-16q^{-74}+10q^{-78}+4q^{-80}-3q^{-82}-3q^{-84}+q^{-88}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{18}-2q^{16}+3q^{14}-4q^{12}+7q^{10}-11q^{8}+11q^{6}-16q^{4}+20q^{2}-27+23q^{-2}-26q^{-4}+30q^{-6}-23q^{-8}+20q^{-10}-7q^{-12}+13q^{-14}+13q^{-16}-15q^{-18}+26q^{-20}-33q^{-22}+45q^{-24}-53q^{-26}+45q^{-28}-57q^{-30}+51q^{-32}-46q^{-34}+37q^{-36}-34q^{-38}+22q^{-40}-6q^{-42}+q^{-44}+5q^{-46}-17q^{-48}+25q^{-50}-26q^{-52}+27q^{-54}-30q^{-56}+28q^{-58}-22q^{-60}+19q^{-62}-16q^{-64}+11q^{-66}-6q^{-68}+4q^{-70}-3q^{-72}+q^{-74}$

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-2q^{30}+5q^{28}-8q^{26}+8q^{24}-7q^{22}-2q^{20}+15q^{18}-30q^{16}+43q^{14}-49q^{12}+39q^{10}-14q^{8}-31q^{6}+88q^{4}-135q^{2}+155-130q^{-2}+45q^{-4}+73q^{-6}-192q^{-8}+270q^{-10}-257q^{-12}+158q^{-14}+9q^{-16}-171q^{-18}+266q^{-20}-245q^{-22}+128q^{-24}+45q^{-26}-178q^{-28}+213q^{-30}-130q^{-32}-29q^{-34}+208q^{-36}-308q^{-38}+285q^{-40}-137q^{-42}-84q^{-44}+291q^{-46}-414q^{-48}+398q^{-50}-255q^{-52}+33q^{-54}+188q^{-56}-340q^{-58}+366q^{-60}-262q^{-62}+73q^{-64}+112q^{-66}-230q^{-68}+224q^{-70}-106q^{-72}-61q^{-74}+203q^{-76}-245q^{-78}+172q^{-80}-12q^{-82}-172q^{-84}+292q^{-86}-302q^{-88}+206q^{-90}-47q^{-92}-113q^{-94}+215q^{-96}-231q^{-98}+180q^{-100}-86q^{-102}-8q^{-104}+69q^{-106}-96q^{-108}+82q^{-110}-51q^{-112}+23q^{-114}+2q^{-116}-13q^{-118}+15q^{-120}-13q^{-122}+7q^{-124}-3q^{-126}+q^{-128}$

.

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 84"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-9t^{2}+20t-25+20t^{-1}-9t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 87, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-8q^{6}+11q^{5}-14q^{4}+15q^{3}-13q^{2}+11q-6+3q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-3}+2z^{9}a^{-5}+4z^{8}a^{-2}+10z^{8}a^{-4}+6z^{8}a^{-6}+4z^{7}a^{-1}+5z^{7}a^{-3}+8z^{7}a^{-5}+7z^{7}a^{-7}-2z^{6}a^{-2}-17z^{6}a^{-4}-8z^{6}a^{-6}+4z^{6}a^{-8}+3z^{6}+az^{5}-4z^{5}a^{-1}-11z^{5}a^{-3}-20z^{5}a^{-5}-13z^{5}a^{-7}+z^{5}a^{-9}-5z^{4}a^{-2}+9z^{4}a^{-4}+2z^{4}a^{-6}-6z^{4}a^{-8}-6z^{4}-2az^{3}-2z^{3}a^{-1}+4z^{3}a^{-3}+11z^{3}a^{-5}+6z^{3}a^{-7}-z^{3}a^{-9}+7z^{2}a^{-2}+z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}+4z^{2}+az+2za^{-1}+2za^{-3}-za^{-7}-4a^{-2}-2a^{-4}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a46, K11n184, ...}

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 $8$ $16$ $32$ ${\frac {124}{3}}$ $-{\frac {28}{3}}$ $128$ ${\frac {448}{3}}$ $-{\frac {32}{3}}$ $16$ ${\frac {256}{3}}$ $128$ ${\frac {992}{3}}$ $-{\frac {224}{3}}$ ${\frac {7951}{15}}$ ${\frac {92}{5}}$ ${\frac {2044}{45}}$ ${\frac {65}{9}}$ $-{\frac {449}{15}}$

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 84. 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                   3   3 13                 5 1   -4 11               6 3     3 9             8 5       -3 7           7 6         1 5         6 8           2 3       5 7             -2 1     2 7               5 -1   1 4                 -3 -3   2                   2 -5 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{23}-4q^{22}+3q^{21}+12q^{20}-27q^{19}+3q^{18}+55q^{17}-66q^{16}-22q^{15}+123q^{14}-93q^{13}-69q^{12}+181q^{11}-93q^{10}-111q^{9}+197q^{8}-67q^{7}-124q^{6}+161q^{5}-27q^{4}-101q^{3}+93q^{2}+q-54+34q^{-1}+5q^{-2}-17q^{-3}+8q^{-4}+q^{-5}-3q^{-6}+q^{-7}$
3	$-q^{45}+4q^{44}-3q^{43}-7q^{42}+4q^{41}+22q^{40}-4q^{39}-58q^{38}+109q^{36}+38q^{35}-181q^{34}-121q^{33}+259q^{32}+250q^{31}-308q^{30}-433q^{29}+319q^{28}+638q^{27}-271q^{26}-852q^{25}+186q^{24}+1027q^{23}-51q^{22}-1172q^{21}-88q^{20}+1256q^{19}+232q^{18}-1284q^{17}-371q^{16}+1262q^{15}+478q^{14}-1162q^{13}-587q^{12}+1036q^{11}+632q^{10}-834q^{9}-663q^{8}+637q^{7}+612q^{6}-408q^{5}-547q^{4}+245q^{3}+413q^{2}-99q-296+29q^{-1}+181q^{-2}+5q^{-3}-99q^{-4}-10q^{-5}+48q^{-6}+6q^{-7}-22q^{-8}-2q^{-9}+11q^{-10}-2q^{-11}-3q^{-12}-q^{-13}+3q^{-14}-q^{-15}$
4	$q^{74}-4q^{73}+3q^{72}+7q^{71}-9q^{70}+q^{69}-21q^{68}+24q^{67}+51q^{66}-40q^{65}-26q^{64}-128q^{63}+74q^{62}+268q^{61}+12q^{60}-96q^{59}-583q^{58}-76q^{57}+735q^{56}+551q^{55}+201q^{54}-1526q^{53}-1066q^{52}+900q^{51}+1767q^{50}+1718q^{49}-2242q^{48}-3108q^{47}-298q^{46}+2830q^{45}+4658q^{44}-1568q^{43}-5232q^{42}-3055q^{41}+2573q^{40}+7892q^{39}+653q^{38}-6187q^{37}-6244q^{36}+891q^{35}+10090q^{34}+3372q^{33}-5742q^{32}-8620q^{31}-1335q^{30}+10812q^{29}+5586q^{28}-4417q^{27}-9766q^{26}-3391q^{25}+10227q^{24}+6994q^{23}-2557q^{22}-9678q^{21}-5100q^{20}+8380q^{19}+7496q^{18}-244q^{17}-8221q^{16}-6212q^{15}+5381q^{14}+6723q^{13}+1978q^{12}-5438q^{11}-6065q^{10}+2073q^{9}+4578q^{8}+3053q^{7}-2287q^{6}-4415q^{5}-136q^{4}+1994q^{3}+2521q^{2}-196q-2200-674q^{-1}+304q^{-2}+1257q^{-3}+383q^{-4}-691q^{-5}-328q^{-6}-178q^{-7}+370q^{-8}+220q^{-9}-140q^{-10}-37q^{-11}-112q^{-12}+67q^{-13}+51q^{-14}-36q^{-15}+21q^{-16}-26q^{-17}+12q^{-18}+6q^{-19}-14q^{-20}+8q^{-21}-3q^{-22}+3q^{-23}+q^{-24}-3q^{-25}+q^{-26}$
5	Not Available
6	Not Available
7	Not Available

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, 84]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[8, 12, 9, 11], X[20, 15, 1, 16], X[16, 5, 17, 6], X[12, 18, 13, 17], X[14, 8, 15, 7], X[18, 14, 19, 13], X[6, 19, 7, 20], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 84]]
Out[3]=	GaussCode[1, -10, 2, -1, 5, -9, 7, -3, 10, -2, 3, -6, 8, -7, 4, -5, 6, -8, 9, -4]
In[4]:=	DTCode[Knot[10, 84]]
Out[4]=	DTCode[4, 10, 16, 14, 2, 8, 18, 20, 12, 6]
In[5]:=	br = BR[Knot[10, 84]]
Out[5]=	BR[4, {1, 1, 1, 2, -1, -3, 2, 2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 84]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 84]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 84]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 84]][t]
Out[10]=	2 9 20 2 3 -25 + -- - -- + -- + 20 t - 9 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 84]][z]
Out[11]=	2 4 6 1 + 2 z + 3 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 84], Knot[11, Alternating, 46], Knot[11, NonAlternating, 184]}
In[13]:=	{KnotDet[Knot[10, 84]], KnotSignature[Knot[10, 84]]}
Out[13]=	{87, 2}
In[14]:=	Jones[Knot[10, 84]][q]
Out[14]=	-2 3 2 3 4 5 6 7 8 -6 - q + - + 11 q - 13 q + 15 q - 14 q + 11 q - 8 q + 4 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 84]}
In[16]:=	A2Invariant[Knot[10, 84]][q]
Out[16]=	-6 -4 -2 2 4 6 8 10 12 14 -1 - q + q - q + 4 q - q + 4 q + q - q + q - 4 q + 16 18 20 22 24 2 q - q - q + 2 q - q
In[17]:=	HOMFLYPT[Knot[10, 84]][a, z]
Out[17]=	2 2 4 4 4 6 6 2 4 2 z 5 z 4 z 2 z 3 z z z -1 - -- + -- - 2 z - -- + ---- - z - -- + ---- + ---- + -- + -- 4 2 6 2 6 4 2 4 2 a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 84]][a, z]
Out[18]=	2 2 2 2 3 2 4 z 2 z 2 z 2 z z z 7 z z -1 - -- - -- - -- + --- + --- + a z + 4 z + -- - -- + -- + ---- - -- + 4 2 7 3 a 8 6 4 2 9 a a a a a a a a a 3 3 3 3 4 4 4 6 z 11 z 4 z 2 z 3 4 6 z 2 z 9 z ---- + ----- + ---- - ---- - 2 a z - 6 z - ---- + ---- + ---- - 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 5 5 6 5 z z 13 z 20 z 11 z 4 z 5 6 4 z ---- + -- - ----- - ----- - ----- - ---- + a z + 3 z + ---- - 2 9 7 5 3 a 8 a a a a a a 6 6 6 7 7 7 7 8 8 8 z 17 z 2 z 7 z 8 z 5 z 4 z 6 z 10 z ---- - ----- - ---- + ---- + ---- + ---- + ---- + ---- + ----- + 6 4 2 7 5 3 a 6 4 a a a a a a a a 8 9 9 4 z 2 z 2 z ---- + ---- + ---- 2 5 3 a a a
In[19]:=	{Vassiliev[2][Knot[10, 84]], Vassiliev[3][Knot[10, 84]]}
Out[19]=	{2, 2}
In[20]:=	Kh[Knot[10, 84]][q, t]
Out[20]=	3 1 2 1 4 2 q 3 5 7 q + 5 q + ----- + ----- + ---- + --- + --- + 7 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 + 7 q t + 6 q t + 8 q t + 5 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 3 q t + 5 q t + q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 84], 2][q]
Out[21]=	-7 3 -5 8 17 5 34 2 3 4 -54 + q - -- + q + -- - -- + -- + -- + q + 93 q - 101 q - 27 q + 6 4 3 2 q q q q q 5 6 7 8 9 10 11 161 q - 124 q - 67 q + 197 q - 111 q - 93 q + 181 q - 12 13 14 15 16 17 18 69 q - 93 q + 123 q - 22 q - 66 q + 55 q + 3 q - 19 20 21 22 23 27 q + 12 q + 3 q - 4 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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