10 87

10_86

10_88

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10 87 Quick Notes

10 87 Further Notes and Views

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,4,11,3 X_14,6,15,5 X_20,16,1,15 X_16,7,17,8 X_6,19,7,20 X_8,12,9,11 X_18,13,19,14 X_12,17,13,18 X_2,10,3,9
Gauss code	1, -10, 2, -1, 3, -6, 5, -7, 10, -2, 7, -9, 8, -3, 4, -5, 9, -8, 6, -4
Dowker-Thistlethwaite code	4 10 14 16 2 8 18 20 12 6
Conway Notation	[.22.20]

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	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-5][-7]
Hyperbolic Volume	14.2736
A-Polynomial	See Data:10 87/A-polynomial

[edit Notes for 10 87's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$0$
Topological 4 genus	$0$
Concordance genus	$0$
Rasmussen s-Invariant	0

[edit Notes for 10 87's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+9t^{2}-18t+23-18t^{-1}+9t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 81, 0 }
Jones polynomial	$q^{6}-4q^{5}+7q^{4}-10q^{3}+13q^{2}-13q+13-10q^{-1}+6q^{-2}-3q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-3z^{4}+2a^{2}z^{2}+z^{2}a^{-2}+z^{2}a^{-4}-4z^{2}+a^{2}+3a^{-2}-a^{-4}-2$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-1}+2z^{9}a^{-3}+10z^{8}a^{-2}+5z^{8}a^{-4}+5z^{8}+6az^{7}+7z^{7}a^{-1}+5z^{7}a^{-3}+4z^{7}a^{-5}+5a^{2}z^{6}-21z^{6}a^{-2}-12z^{6}a^{-4}+z^{6}a^{-6}-3z^{6}+3a^{3}z^{5}-6az^{5}-21z^{5}a^{-1}-23z^{5}a^{-3}-11z^{5}a^{-5}+a^{4}z^{4}-5a^{2}z^{4}+8z^{4}a^{-2}+5z^{4}a^{-4}-2z^{4}a^{-6}-5z^{4}-3a^{3}z^{3}+2az^{3}+13z^{3}a^{-1}+15z^{3}a^{-3}+7z^{3}a^{-5}-a^{4}z^{2}+3a^{2}z^{2}+3z^{2}a^{-2}+z^{2}a^{-4}+z^{2}a^{-6}+7z^{2}+a^{3}z+az-za^{-1}-za^{-3}-a^{2}-3a^{-2}-a^{-4}-2$
The A2 invariant	$q^{12}-q^{10}+q^{8}+q^{6}-3q^{4}+2q^{2}-2+q^{-2}+2q^{-4}+4q^{-8}-2q^{-10}-2q^{-16}+q^{-18}$
The G2 invariant	$q^{66}-2q^{64}+4q^{62}-6q^{60}+5q^{58}-4q^{56}-2q^{54}+11q^{52}-19q^{50}+28q^{48}-31q^{46}+25q^{44}-8q^{42}-16q^{40}+46q^{38}-69q^{36}+84q^{34}-82q^{32}+51q^{30}+2q^{28}-68q^{26}+135q^{24}-165q^{22}+149q^{20}-83q^{18}-21q^{16}+122q^{14}-185q^{12}+174q^{10}-92q^{8}-27q^{6}+124q^{4}-159q^{2}+104+16q^{-2}-140q^{-4}+207q^{-6}-188q^{-8}+76q^{-10}+86q^{-12}-229q^{-14}+300q^{-16}-263q^{-18}+141q^{-20}+32q^{-22}-184q^{-24}+272q^{-26}-261q^{-28}+172q^{-30}-32q^{-32}-105q^{-34}+188q^{-36}-179q^{-38}+96q^{-40}+31q^{-42}-140q^{-44}+177q^{-46}-128q^{-48}+5q^{-50}+128q^{-52}-219q^{-54}+225q^{-56}-142q^{-58}+4q^{-60}+126q^{-62}-201q^{-64}+202q^{-66}-135q^{-68}+37q^{-70}+46q^{-72}-96q^{-74}+99q^{-76}-70q^{-78}+35q^{-80}-18q^{-84}+20q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}$

Further Quantum Invariants

Further quantum knot invariants for 10_87.

A1 Invariants.

Weight	Invariant
1	$q^{9}-2q^{7}+3q^{5}-4q^{3}+3q+3q^{-5}-3q^{-7}+3q^{-9}-3q^{-11}+q^{-13}$
2	$q^{26}-2q^{24}+5q^{20}-8q^{18}+2q^{16}+14q^{14}-21q^{12}+29q^{8}-25q^{6}-12q^{4}+31q^{2}-7-18q^{-2}+12q^{-4}+14q^{-6}-14q^{-8}-11q^{-10}+25q^{-12}-2q^{-14}-28q^{-16}+23q^{-18}+12q^{-20}-32q^{-22}+9q^{-24}+20q^{-26}-18q^{-28}-4q^{-30}+12q^{-32}-2q^{-34}-3q^{-36}+q^{-38}$
3	$q^{51}-2q^{49}+2q^{45}+q^{43}-4q^{41}+8q^{37}-8q^{35}-13q^{33}+20q^{31}+30q^{29}-37q^{27}-63q^{25}+52q^{23}+112q^{21}-50q^{19}-165q^{17}+17q^{15}+209q^{13}+35q^{11}-221q^{9}-93q^{7}+184q^{5}+150q^{3}-125q-170q^{-1}+48q^{-3}+172q^{-5}+27q^{-7}-150q^{-9}-90q^{-11}+123q^{-13}+138q^{-15}-93q^{-17}-173q^{-19}+56q^{-21}+202q^{-23}-14q^{-25}-215q^{-27}-41q^{-29}+215q^{-31}+94q^{-33}-181q^{-35}-151q^{-37}+127q^{-39}+178q^{-41}-57q^{-43}-172q^{-45}-11q^{-47}+137q^{-49}+57q^{-51}-85q^{-53}-68q^{-55}+34q^{-57}+55q^{-59}-34q^{-63}-8q^{-65}+11q^{-67}+7q^{-69}-2q^{-71}-3q^{-73}+q^{-75}$
4	$q^{84}-2q^{82}+2q^{78}-2q^{76}+5q^{74}-6q^{72}+3q^{68}-13q^{66}+19q^{64}+4q^{62}+12q^{60}-15q^{58}-83q^{56}+22q^{54}+88q^{52}+132q^{50}-34q^{48}-331q^{46}-150q^{44}+224q^{42}+560q^{40}+211q^{38}-706q^{36}-781q^{34}+17q^{32}+1165q^{30}+1047q^{28}-620q^{26}-1580q^{24}-907q^{22}+1141q^{20}+1985q^{18}+322q^{16}-1569q^{14}-1876q^{12}+142q^{10}+1940q^{8}+1336q^{6}-512q^{4}-1853q^{2}-942+881q^{-2}+1504q^{-4}+615q^{-6}-987q^{-8}-1323q^{-10}-232q^{-12}+1059q^{-14}+1177q^{-16}-147q^{-18}-1272q^{-20}-906q^{-22}+660q^{-24}+1430q^{-26}+427q^{-28}-1216q^{-30}-1424q^{-32}+304q^{-34}+1636q^{-36}+1063q^{-38}-965q^{-40}-1898q^{-42}-392q^{-44}+1450q^{-46}+1759q^{-48}-130q^{-50}-1855q^{-52}-1277q^{-54}+510q^{-56}+1858q^{-58}+947q^{-60}-910q^{-62}-1516q^{-64}-680q^{-66}+983q^{-68}+1295q^{-70}+281q^{-72}-763q^{-74}-1064q^{-76}-96q^{-78}+660q^{-80}+664q^{-82}+112q^{-84}-544q^{-86}-408q^{-88}-24q^{-90}+293q^{-92}+297q^{-94}-35q^{-96}-153q^{-98}-144q^{-100}-5q^{-102}+102q^{-104}+45q^{-106}+6q^{-108}-35q^{-110}-24q^{-112}+7q^{-114}+6q^{-116}+7q^{-118}-2q^{-120}-3q^{-122}+q^{-124}$
5	$q^{125}-2q^{123}+2q^{119}-2q^{117}+2q^{115}+3q^{113}-6q^{111}-5q^{109}+4q^{107}+q^{105}+11q^{103}+18q^{101}-11q^{99}-41q^{97}-41q^{95}+6q^{93}+84q^{91}+130q^{89}+42q^{87}-186q^{85}-334q^{83}-161q^{81}+310q^{79}+708q^{77}+531q^{75}-384q^{73}-1380q^{71}-1309q^{69}+243q^{67}+2250q^{65}+2730q^{63}+523q^{61}-3118q^{59}-4910q^{57}-2293q^{55}+3477q^{53}+7555q^{51}+5356q^{49}-2612q^{47}-10011q^{45}-9571q^{43}+6q^{41}+11339q^{39}+14087q^{37}+4365q^{35}-10550q^{33}-17711q^{31}-9846q^{29}+7368q^{27}+19232q^{25}+15034q^{23}-2228q^{21}-17901q^{19}-18557q^{17}-3730q^{15}+13964q^{13}+19550q^{11}+8890q^{9}-8366q^{7}-17767q^{5}-12339q^{3}+2466q+14108q^{-1}+13564q^{-3}+2461q^{-5}-9526q^{-7}-12990q^{-9}-5920q^{-11}+5290q^{-13}+11443q^{-15}+7873q^{-17}-2014q^{-19}-9787q^{-21}-8885q^{-23}-157q^{-25}+8681q^{-27}+9624q^{-29}+1544q^{-31}-8304q^{-33}-10621q^{-35}-2745q^{-37}+8381q^{-39}+12159q^{-41}+4325q^{-43}-8384q^{-45}-14072q^{-47}-6688q^{-49}+7672q^{-51}+15854q^{-53}+9828q^{-55}-5647q^{-57}-16837q^{-59}-13368q^{-61}+2183q^{-63}+16255q^{-65}+16452q^{-67}+2547q^{-69}-13634q^{-71}-18265q^{-73}-7599q^{-75}+9072q^{-77}+17831q^{-79}+11923q^{-81}-3195q^{-83}-15007q^{-85}-14338q^{-87}-2660q^{-89}+10124q^{-91}+14150q^{-93}+7201q^{-95}-4357q^{-97}-11487q^{-99}-9419q^{-101}-818q^{-103}+7240q^{-105}+9053q^{-107}+4251q^{-109}-2750q^{-111}-6767q^{-113}-5465q^{-115}-676q^{-117}+3703q^{-119}+4749q^{-121}+2441q^{-123}-981q^{-125}-3064q^{-127}-2673q^{-129}-629q^{-131}+1324q^{-133}+1921q^{-135}+1164q^{-137}-116q^{-139}-1001q^{-141}-981q^{-143}-347q^{-145}+291q^{-147}+535q^{-149}+389q^{-151}+47q^{-153}-216q^{-155}-226q^{-157}-100q^{-159}+31q^{-161}+86q^{-163}+73q^{-165}+15q^{-167}-29q^{-169}-25q^{-171}-9q^{-173}+2q^{-175}+6q^{-177}+7q^{-179}-2q^{-181}-3q^{-183}+q^{-185}$

A2 Invariants.

Weight	Invariant
1,0	$q^{12}-q^{10}+q^{8}+q^{6}-3q^{4}+2q^{2}-2+q^{-2}+2q^{-4}+4q^{-8}-2q^{-10}-2q^{-16}+q^{-18}$
1,1	$q^{36}-4q^{34}+10q^{32}-20q^{30}+38q^{28}-64q^{26}+100q^{24}-148q^{22}+207q^{20}-278q^{18}+362q^{16}-464q^{14}+572q^{12}-668q^{10}+742q^{8}-762q^{6}+697q^{4}-520q^{2}+238+136q^{-2}-566q^{-4}+996q^{-6}-1372q^{-8}+1646q^{-10}-1774q^{-12}+1752q^{-14}-1566q^{-16}+1260q^{-18}-851q^{-20}+388q^{-22}+70q^{-24}-474q^{-26}+771q^{-28}-952q^{-30}+994q^{-32}-922q^{-34}+773q^{-36}-586q^{-38}+402q^{-40}-244q^{-42}+133q^{-44}-62q^{-46}+22q^{-48}-6q^{-50}+q^{-52}$
2,0	$q^{32}-q^{30}-q^{28}+3q^{26}-5q^{22}+2q^{20}+7q^{18}-3q^{16}-11q^{14}+5q^{12}+14q^{10}-12q^{8}-9q^{6}+12q^{4}+7q^{2}-8-4q^{-2}+11q^{-4}-3q^{-6}-6q^{-8}+6q^{-10}+3q^{-12}-8q^{-14}+7q^{-16}+12q^{-18}-8q^{-20}-7q^{-22}+6q^{-24}+5q^{-26}-12q^{-28}-8q^{-30}+11q^{-32}+4q^{-34}-7q^{-36}-2q^{-38}+5q^{-40}+4q^{-42}-3q^{-44}-2q^{-46}+q^{-48}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{28}-2q^{26}+5q^{22}-7q^{20}-3q^{18}+15q^{16}-9q^{14}-10q^{12}+25q^{10}-8q^{8}-17q^{6}+23q^{4}-7q^{2}-16+10q^{-2}+2q^{-4}-4q^{-6}-3q^{-8}+13q^{-10}+10q^{-12}-17q^{-14}+8q^{-16}+14q^{-18}-24q^{-20}+2q^{-22}+16q^{-24}-19q^{-26}+5q^{-28}+10q^{-30}-11q^{-32}+4q^{-34}+2q^{-36}-3q^{-38}+q^{-40}$
1,0,0	$q^{15}-q^{13}+2q^{11}-q^{9}+2q^{7}-3q^{5}+2q^{3}-3q+q^{-5}+3q^{-7}+q^{-9}+4q^{-11}-2q^{-13}+q^{-15}-3q^{-17}+q^{-19}-2q^{-21}+q^{-23}$

A4 Invariants.

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

B2 Invariants.

Weight

Invariant

0,1

q^{28}-2q^{26}+4q^{24}-7q^{22}+11q^{20}-15q^{18}+21q^{16}-25q^{14}+28q^{12}-29q^{10}+24q^{8}-17q^{6}+5q^{4}+9q^{2}-24+38q^{-2}-48q^{-4}+56q^{-6}-57q^{-8}+55q^{-10}-44q^{-12}+33q^{-14}-16q^{-16}+2q^{-18}+12q^{-20}-22q^{-22}+28q^{-24}-31q^{-26}+29q^{-28}-26q^{-30}+19q^{-32}-14q^{-34}+8q^{-36}-3q^{-38}+q^{-40}

1,0

q^{46}-2q^{42}-2q^{40}+2q^{38}+6q^{36}+q^{34}-9q^{32}-9q^{30}+5q^{28}+18q^{26}+6q^{24}-18q^{22}-19q^{20}+9q^{18}+30q^{16}+6q^{14}-27q^{12}-19q^{10}+18q^{8}+26q^{6}-9q^{4}-28q^{2}-2+24q^{-2}+7q^{-4}-20q^{-6}-12q^{-8}+16q^{-10}+16q^{-12}-10q^{-14}-15q^{-16}+12q^{-18}+21q^{-20}-4q^{-22}-25q^{-24}-3q^{-26}+27q^{-28}+16q^{-30}-24q^{-32}-30q^{-34}+9q^{-36}+32q^{-38}+6q^{-40}-27q^{-42}-18q^{-44}+16q^{-46}+21q^{-48}-4q^{-50}-15q^{-52}-4q^{-54}+9q^{-56}+5q^{-58}-3q^{-60}-3q^{-62}+q^{-66}

D4 Invariants.

Weight

Invariant

1,0,0,0

q^{38}-2q^{36}+2q^{34}-3q^{32}+6q^{30}-9q^{28}+8q^{26}-11q^{24}+18q^{22}-18q^{20}+18q^{18}-20q^{16}+27q^{14}-19q^{12}+16q^{10}-15q^{8}+8q^{6}+q^{4}-11q^{2}+13-29q^{-2}+33q^{-4}-37q^{-6}+42q^{-8}-44q^{-10}+48q^{-12}-36q^{-14}+41q^{-16}-29q^{-18}+24q^{-20}-13q^{-22}+6q^{-24}-q^{-26}-12q^{-28}+16q^{-30}-21q^{-32}+21q^{-34}-25q^{-36}+26q^{-38}-22q^{-40}+19q^{-42}-16q^{-44}+12q^{-46}-8q^{-48}+5q^{-50}-3q^{-52}+q^{-54}

G2 Invariants.

Weight

Invariant

1,0

q^{66}-2q^{64}+4q^{62}-6q^{60}+5q^{58}-4q^{56}-2q^{54}+11q^{52}-19q^{50}+28q^{48}-31q^{46}+25q^{44}-8q^{42}-16q^{40}+46q^{38}-69q^{36}+84q^{34}-82q^{32}+51q^{30}+2q^{28}-68q^{26}+135q^{24}-165q^{22}+149q^{20}-83q^{18}-21q^{16}+122q^{14}-185q^{12}+174q^{10}-92q^{8}-27q^{6}+124q^{4}-159q^{2}+104+16q^{-2}-140q^{-4}+207q^{-6}-188q^{-8}+76q^{-10}+86q^{-12}-229q^{-14}+300q^{-16}-263q^{-18}+141q^{-20}+32q^{-22}-184q^{-24}+272q^{-26}-261q^{-28}+172q^{-30}-32q^{-32}-105q^{-34}+188q^{-36}-179q^{-38}+96q^{-40}+31q^{-42}-140q^{-44}+177q^{-46}-128q^{-48}+5q^{-50}+128q^{-52}-219q^{-54}+225q^{-56}-142q^{-58}+4q^{-60}+126q^{-62}-201q^{-64}+202q^{-66}-135q^{-68}+37q^{-70}+46q^{-72}-96q^{-74}+99q^{-76}-70q^{-78}+35q^{-80}-18q^{-84}+20q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}

.

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.

In[3]:= K = Knot["10 87"];

In[4]:= Alexander[K][t]

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

Out[4]= $-2t^{3}+9t^{2}-18t+23-18t^{-1}+9t^{-2}-2t^{-3}$

In[5]:= Conway[K][z]

Out[5]= $-2z^{6}-3z^{4}+1$

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]= { 81, 0 }

In[8]:= Jones[K][q]

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

Out[8]= $q^{6}-4q^{5}+7q^{4}-10q^{3}+13q^{2}-13q+13-10q^{-1}+6q^{-2}-3q^{-3}+q^{-4}$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-3z^{4}+2a^{2}z^{2}+z^{2}a^{-2}+z^{2}a^{-4}-4z^{2}+a^{2}+3a^{-2}-a^{-4}-2$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $2z^{9}a^{-1}+2z^{9}a^{-3}+10z^{8}a^{-2}+5z^{8}a^{-4}+5z^{8}+6az^{7}+7z^{7}a^{-1}+5z^{7}a^{-3}+4z^{7}a^{-5}+5a^{2}z^{6}-21z^{6}a^{-2}-12z^{6}a^{-4}+z^{6}a^{-6}-3z^{6}+3a^{3}z^{5}-6az^{5}-21z^{5}a^{-1}-23z^{5}a^{-3}-11z^{5}a^{-5}+a^{4}z^{4}-5a^{2}z^{4}+8z^{4}a^{-2}+5z^{4}a^{-4}-2z^{4}a^{-6}-5z^{4}-3a^{3}z^{3}+2az^{3}+13z^{3}a^{-1}+15z^{3}a^{-3}+7z^{3}a^{-5}-a^{4}z^{2}+3a^{2}z^{2}+3z^{2}a^{-2}+z^{2}a^{-4}+z^{2}a^{-6}+7z^{2}+a^{3}z+az-za^{-1}-za^{-3}-a^{2}-3a^{-2}-a^{-4}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_98, K11a58, K11a165, K11n72, ...}

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

Vassiliev invariants

V₂ and V₃:

(0, 1)

V_2,1 through V_6,9:

V_2,1	V_3,1	V_4,1	V_4,2	V_4,3	V_5,1	V_5,2	V_5,3	V_5,4	V_6,1	V_6,2	V_6,3	V_6,4	V_6,5	V_6,6	V_6,7	V_6,8	V_6,9
$0$	$8$	$0$	$32$	$24$	$0$	${\frac {80}{3}}$	${\frac {32}{3}}$	$8$	$0$	$32$	$0$	$0$	$96$	${\frac {344}{3}}$	$-104$	$48$	$-32$

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=

0 is the signature of 10 87. 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

13

1

11

3

-3

9

4

1

3

7

6

3

-3

5

7

4

3

6

0

1

7

0

-1

4

7

3

-3

2

6

-4

-5

1

4

3

-7

2

-2

-9

1

Integral Khovanov Homology

(db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n+1

-dimensional representation of

sl(2)

)

$n$	$J_{n}$
2	$q^{18}-4q^{17}+q^{16}+15q^{15}-20q^{14}-13q^{13}+53q^{12}-31q^{11}-54q^{10}+97q^{9}-20q^{8}-105q^{7}+123q^{6}+7q^{5}-141q^{4}+120q^{3}+35q^{2}-143q+90+46q^{-1}-105q^{-2}+47q^{-3}+33q^{-4}-51q^{-5}+18q^{-6}+12q^{-7}-16q^{-8}+6q^{-9}+2q^{-10}-3q^{-11}+q^{-12}$
3	$q^{36}-4q^{35}+q^{34}+9q^{33}+5q^{32}-23q^{31}-25q^{30}+43q^{29}+60q^{28}-44q^{27}-127q^{26}+26q^{25}+202q^{24}+36q^{23}-275q^{22}-135q^{21}+317q^{20}+271q^{19}-326q^{18}-413q^{17}+287q^{16}+546q^{15}-205q^{14}-669q^{13}+113q^{12}+747q^{11}+11q^{10}-815q^{9}-116q^{8}+827q^{7}+242q^{6}-830q^{5}-329q^{4}+767q^{3}+419q^{2}-685q-453+549q^{-1}+464q^{-2}-410q^{-3}-419q^{-4}+272q^{-5}+336q^{-6}-154q^{-7}-245q^{-8}+80q^{-9}+154q^{-10}-39q^{-11}-83q^{-12}+20q^{-13}+39q^{-14}-13q^{-15}-16q^{-16}+10q^{-17}+6q^{-18}-8q^{-19}+2q^{-21}+2q^{-22}-3q^{-23}+q^{-24}$
4	$q^{60}-4q^{59}+q^{58}+9q^{57}-q^{56}+2q^{55}-35q^{54}-10q^{53}+50q^{52}+38q^{51}+59q^{50}-142q^{49}-149q^{48}+41q^{47}+156q^{46}+391q^{45}-146q^{44}-466q^{43}-343q^{42}+20q^{41}+1047q^{40}+406q^{39}-470q^{38}-1099q^{37}-948q^{36}+1348q^{35}+1450q^{34}+544q^{33}-1411q^{32}-2611q^{31}+512q^{30}+2056q^{29}+2401q^{28}-500q^{27}-3959q^{26}-1275q^{25}+1478q^{24}+4126q^{23}+1389q^{22}-4268q^{21}-3117q^{20}-28q^{19}+5059q^{18}+3417q^{17}-3695q^{16}-4449q^{15}-1756q^{14}+5267q^{13}+5060q^{12}-2692q^{11}-5219q^{10}-3322q^{9}+4901q^{8}+6185q^{7}-1368q^{6}-5337q^{5}-4613q^{4}+3810q^{3}+6521q^{2}+234q-4448-5236q^{-1}+1987q^{-2}+5610q^{-3}+1575q^{-4}-2600q^{-5}-4632q^{-6}+189q^{-7}+3592q^{-8}+1882q^{-9}-709q^{-10}-2969q^{-11}-655q^{-12}+1544q^{-13}+1209q^{-14}+251q^{-15}-1302q^{-16}-537q^{-17}+396q^{-18}+411q^{-19}+326q^{-20}-385q^{-21}-188q^{-22}+60q^{-23}+37q^{-24}+145q^{-25}-88q^{-26}-22q^{-27}+16q^{-28}-29q^{-29}+40q^{-30}-20q^{-31}+5q^{-32}+8q^{-33}-14q^{-34}+8q^{-35}-4q^{-36}+2q^{-37}+2q^{-38}-3q^{-39}+q^{-40}$
5	$q^{90}-4q^{89}+q^{88}+9q^{87}-q^{86}-4q^{85}-10q^{84}-20q^{83}-3q^{82}+53q^{81}+57q^{80}+9q^{79}-65q^{78}-151q^{77}-129q^{76}+63q^{75}+320q^{74}+351q^{73}+81q^{72}-395q^{71}-767q^{70}-571q^{69}+300q^{68}+1236q^{67}+1361q^{66}+362q^{65}-1364q^{64}-2524q^{63}-1744q^{62}+845q^{61}+3444q^{60}+3784q^{59}+944q^{58}-3570q^{57}-6123q^{56}-3944q^{55}+2142q^{54}+7801q^{53}+7945q^{52}+1232q^{51}-7936q^{50}-12002q^{49}-6459q^{48}+5733q^{47}+15075q^{46}+12790q^{45}-987q^{44}-16028q^{43}-19243q^{42}-5945q^{41}+14406q^{40}+24602q^{39}+14131q^{38}-10120q^{37}-28002q^{36}-22616q^{35}+3740q^{34}+29233q^{33}+30312q^{32}+3785q^{31}-28199q^{30}-36688q^{29}-11811q^{28}+25764q^{27}+41502q^{26}+19260q^{25}-22173q^{24}-44870q^{23}-26171q^{22}+18380q^{21}+47190q^{20}+31969q^{19}-14339q^{18}-48648q^{17}-37297q^{16}+10504q^{15}+49507q^{14}+41817q^{13}-6259q^{12}-49591q^{11}-46135q^{10}+1776q^{9}+48605q^{8}+49590q^{7}+3628q^{6}-46021q^{5}-52288q^{4}-9434q^{3}+41535q^{2}+53054q+15615-34918q^{-1}-51744q^{-2}-21076q^{-3}+26730q^{-4}+47626q^{-5}+25016q^{-6}-17662q^{-7}-41084q^{-8}-26662q^{-9}+9036q^{-10}+32799q^{-11}+25672q^{-12}-1989q^{-13}-23822q^{-14}-22464q^{-15}-2828q^{-16}+15585q^{-17}+17807q^{-18}+5172q^{-19}-8907q^{-20}-12742q^{-21}-5576q^{-22}+4252q^{-23}+8230q^{-24}+4732q^{-25}-1508q^{-26}-4774q^{-27}-3377q^{-28}+174q^{-29}+2460q^{-30}+2115q^{-31}+284q^{-32}-1133q^{-33}-1170q^{-34}-306q^{-35}+453q^{-36}+563q^{-37}+213q^{-38}-137q^{-39}-255q^{-40}-129q^{-41}+55q^{-42}+92q^{-43}+40q^{-44}+11q^{-45}-27q^{-46}-41q^{-47}+9q^{-48}+14q^{-49}-7q^{-50}+11q^{-51}+3q^{-52}-12q^{-53}+2q^{-54}+4q^{-55}-4q^{-56}+2q^{-57}+2q^{-58}-3q^{-59}+q^{-60}$
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, 87]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 16, 1, 15], X[16, 7, 17, 8], X[6, 19, 7, 20], X[8, 12, 9, 11], X[18, 13, 19, 14], X[12, 17, 13, 18], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 87]]
Out[3]=	GaussCode[1, -10, 2, -1, 3, -6, 5, -7, 10, -2, 7, -9, 8, -3, 4, -5, 9, -8, 6, -4]
In[4]:=	DTCode[Knot[10, 87]]
Out[4]=	DTCode[4, 10, 14, 16, 2, 8, 18, 20, 12, 6]
In[5]:=	br = BR[Knot[10, 87]]
Out[5]=	BR[4, {1, 1, 1, 2, -1, -3, 2, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 87]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 87]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[10, 87]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 87]][t]
Out[10]=	2 9 18 2 3 23 - -- + -- - -- - 18 t + 9 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 87]][z]
Out[11]=	4 6 1 - 3 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}
In[13]:=	{KnotDet[Knot[10, 87]], KnotSignature[Knot[10, 87]]}
Out[13]=	{81, 0}
In[14]:=	Jones[Knot[10, 87]][q]
Out[14]=	-4 3 6 10 2 3 4 5 6 13 + q - -- + -- - -- - 13 q + 13 q - 10 q + 7 q - 4 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 87]}
In[16]:=	A2Invariant[Knot[10, 87]][q]
Out[16]=	-12 -10 -8 -6 3 2 2 4 8 10 -2 + q - q + q + q - -- + -- + q + 2 q + 4 q - 2 q - 4 2 q q 16 18 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 87]][a, z]
Out[17]=	2 2 4 4 -4 3 2 2 z z 2 2 4 z 2 z -2 - a + -- + a - 4 z + -- + -- + 2 a z - 3 z + -- - ---- + 2 4 2 4 2 a a a a a 6 2 4 6 z a z - z - -- 2 a
In[18]:=	Kauffman[Knot[10, 87]][a, z]
Out[18]=	2 2 2 -4 3 2 z z 3 2 z z 3 z -2 - a - -- - a - -- - - + a z + a z + 7 z + -- + -- + ---- + 2 3 a 6 4 2 a a a a a 3 3 3 2 2 4 2 7 z 15 z 13 z 3 3 3 4 3 a z - a z + ---- + ----- + ----- + 2 a z - 3 a z - 5 z - 5 3 a a a 4 4 4 5 5 5 2 z 5 z 8 z 2 4 4 4 11 z 23 z 21 z ---- + ---- + ---- - 5 a z + a z - ----- - ----- - ----- - 6 4 2 5 3 a a a a a a 6 6 6 7 5 3 5 6 z 12 z 21 z 2 6 4 z 6 a z + 3 a z - 3 z + -- - ----- - ----- + 5 a z + ---- + 6 4 2 5 a a a a 7 7 8 8 9 9 5 z 7 z 7 8 5 z 10 z 2 z 2 z ---- + ---- + 6 a z + 5 z + ---- + ----- + ---- + ---- 3 a 4 2 3 a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 87]], Vassiliev[3][Knot[10, 87]]}
Out[19]=	{0, 1}
In[20]:=	Kh[Knot[10, 87]][q, t]
Out[20]=	7 1 2 1 4 2 6 4 - + 7 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 6 q t + 6 q t + 7 q t + 4 q t + 6 q t + 3 q t + 4 q t + 9 5 11 5 13 6 q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 87], 2][q]
Out[21]=	-12 3 2 6 16 12 18 51 33 47 105 46 90 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - --- + -- - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 143 q + 35 q + 120 q - 141 q + 7 q + 123 q - 105 q - 20 q + 9 10 11 12 13 14 15 16 97 q - 54 q - 31 q + 53 q - 13 q - 20 q + 15 q + q - 17 18 4 q + q

See/edit the Rolfsen_Splice_Template.

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10_86

10_88

10 87

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

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