10 69

10_68

10_70

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10 69 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X17,20,18,1 X9,19,10,18 X19,9,20,8
Gauss code	-1, 4, -3, 1, -6, 7, -2, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, 8
Dowker-Thistlethwaite code	4 10 14 12 18 2 16 6 20 8
Conway Notation	[211,21,21]

Minimum Braid Representative: Length is 12, width is 5. Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-2][-10] Hyperbolic Volume 14.1265 A-Polynomial See Data:10 69/A-polynomial

[edit Notes for 10 69'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 69's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-7t^{2}+21t-29+21t^{-1}-7t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 87, 2 }
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+11q^{5}-13q^{4}+15q^{3}-14q^{2}+11q-7+4q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+3z^{4}a^{-2}-3z^{4}a^{-4}-z^{4}+5z^{2}a^{-2}-5z^{2}a^{-4}+3z^{2}a^{-6}-z^{2}+2a^{-2}-2a^{-4}+2a^{-6}-a^{-8}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+4z^{8}a^{-2}+8z^{8}a^{-4}+4z^{8}a^{-6}+6z^{7}a^{-1}+14z^{7}a^{-3}+13z^{7}a^{-5}+5z^{7}a^{-7}+3z^{6}a^{-2}-4z^{6}a^{-4}+3z^{6}a^{-8}+4z^{6}+az^{5}-10z^{5}a^{-1}-32z^{5}a^{-3}-30z^{5}a^{-5}-8z^{5}a^{-7}+z^{5}a^{-9}-17z^{4}a^{-2}-14z^{4}a^{-4}-9z^{4}a^{-6}-5z^{4}a^{-8}-7z^{4}-az^{3}+5z^{3}a^{-1}+22z^{3}a^{-3}+23z^{3}a^{-5}+5z^{3}a^{-7}-2z^{3}a^{-9}+11z^{2}a^{-2}+12z^{2}a^{-4}+7z^{2}a^{-6}+3z^{2}a^{-8}+3z^{2}-za^{-1}-4za^{-3}-6za^{-5}-2za^{-7}+za^{-9}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}$
The A2 invariant	$-q^{6}+2q^{4}-q^{2}+4q^{-2}-3q^{-4}+2q^{-6}-q^{-8}+2q^{-12}-2q^{-14}+3q^{-16}-q^{-18}-q^{-20}+2q^{-22}-q^{-24}-q^{-26}$
The G2 invariant	$q^{32}-3q^{30}+7q^{28}-13q^{26}+14q^{24}-12q^{22}-q^{20}+26q^{18}-51q^{16}+77q^{14}-84q^{12}+57q^{10}-q^{8}-83q^{6}+165q^{4}-206q^{2}+191-107q^{-2}-23q^{-4}+170q^{-6}-269q^{-8}+282q^{-10}-199q^{-12}+45q^{-14}+111q^{-16}-215q^{-18}+217q^{-20}-120q^{-22}-17q^{-24}+156q^{-26}-210q^{-28}+145q^{-30}+6q^{-32}-193q^{-34}+324q^{-36}-341q^{-38}+228q^{-40}-18q^{-42}-207q^{-44}+382q^{-46}-429q^{-48}+337q^{-50}-147q^{-52}-86q^{-54}+260q^{-56}-320q^{-58}+260q^{-60}-108q^{-62}-54q^{-64}+173q^{-66}-190q^{-68}+100q^{-70}+42q^{-72}-183q^{-74}+249q^{-76}-204q^{-78}+69q^{-80}+100q^{-82}-232q^{-84}+292q^{-86}-249q^{-88}+132q^{-90}+7q^{-92}-133q^{-94}+191q^{-96}-182q^{-98}+127q^{-100}-49q^{-102}-15q^{-104}+55q^{-106}-69q^{-108}+56q^{-110}-35q^{-112}+14q^{-114}+q^{-116}-9q^{-118}+9q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 10_69.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+3q^{3}-3q+4q^{-1}-3q^{-3}+q^{-5}+2q^{-7}-2q^{-9}+4q^{-11}-4q^{-13}+2q^{-15}-q^{-17}$
2	$q^{16}-3q^{14}-q^{12}+11q^{10}-9q^{8}-11q^{6}+27q^{4}-10q^{2}-29+37q^{-2}+3q^{-4}-37q^{-6}+26q^{-8}+17q^{-10}-25q^{-12}-3q^{-14}+18q^{-16}+2q^{-18}-28q^{-20}+12q^{-22}+29q^{-24}-36q^{-26}-2q^{-28}+38q^{-30}-26q^{-32}-13q^{-34}+26q^{-36}-8q^{-38}-10q^{-40}+8q^{-42}-2q^{-46}+q^{-48}$
3	$-q^{33}+3q^{31}+q^{29}-7q^{27}-6q^{25}+13q^{23}+21q^{21}-24q^{19}-40q^{17}+27q^{15}+76q^{13}-24q^{11}-122q^{9}+2q^{7}+171q^{5}+40q^{3}-208q-101q^{-1}+227q^{-3}+170q^{-5}-212q^{-7}-227q^{-9}+166q^{-11}+266q^{-13}-105q^{-15}-268q^{-17}+31q^{-19}+242q^{-21}+48q^{-23}-195q^{-25}-109q^{-27}+128q^{-29}+167q^{-31}-57q^{-33}-212q^{-35}-22q^{-37}+238q^{-39}+98q^{-41}-245q^{-43}-165q^{-45}+220q^{-47}+223q^{-49}-175q^{-51}-250q^{-53}+110q^{-55}+251q^{-57}-48q^{-59}-215q^{-61}-11q^{-63}+166q^{-65}+42q^{-67}-110q^{-69}-47q^{-71}+57q^{-73}+41q^{-75}-25q^{-77}-26q^{-79}+9q^{-81}+12q^{-83}-2q^{-85}-4q^{-87}+2q^{-91}-q^{-93}$
4	$q^{56}-3q^{54}-q^{52}+7q^{50}+2q^{48}+2q^{46}-23q^{44}-13q^{42}+33q^{40}+29q^{38}+28q^{36}-89q^{34}-94q^{32}+65q^{30}+139q^{28}+164q^{26}-193q^{24}-350q^{22}-35q^{20}+333q^{18}+593q^{16}-131q^{14}-795q^{12}-538q^{10}+349q^{8}+1324q^{6}+445q^{4}-1054q^{2}-1455-251q^{-2}+1853q^{-4}+1509q^{-6}-591q^{-8}-2159q^{-10}-1370q^{-12}+1556q^{-14}+2328q^{-16}+480q^{-18}-1991q^{-20}-2216q^{-22}+544q^{-24}+2241q^{-26}+1394q^{-28}-1067q^{-30}-2215q^{-32}-522q^{-34}+1412q^{-36}+1715q^{-38}+15q^{-40}-1576q^{-42}-1258q^{-44}+372q^{-46}+1604q^{-48}+963q^{-50}-712q^{-52}-1757q^{-54}-672q^{-56}+1285q^{-58}+1781q^{-60}+265q^{-62}-1979q^{-64}-1672q^{-66}+632q^{-68}+2260q^{-70}+1348q^{-72}-1610q^{-74}-2315q^{-76}-389q^{-78}+1986q^{-80}+2133q^{-82}-609q^{-84}-2128q^{-86}-1280q^{-88}+975q^{-90}+2070q^{-92}+394q^{-94}-1173q^{-96}-1402q^{-98}-53q^{-100}+1241q^{-102}+719q^{-104}-216q^{-106}-843q^{-108}-422q^{-110}+403q^{-112}+434q^{-114}+157q^{-116}-265q^{-118}-269q^{-120}+37q^{-122}+118q^{-124}+117q^{-126}-33q^{-128}-80q^{-130}-9q^{-132}+8q^{-134}+31q^{-136}+q^{-138}-13q^{-140}-2q^{-144}+4q^{-146}-2q^{-150}+q^{-152}$
5	$-q^{85}+3q^{83}+q^{81}-7q^{79}-2q^{77}+2q^{75}+8q^{73}+15q^{71}+4q^{69}-33q^{67}-43q^{65}-q^{63}+58q^{61}+95q^{59}+42q^{57}-98q^{55}-226q^{53}-139q^{51}+167q^{49}+419q^{47}+351q^{45}-136q^{43}-745q^{41}-823q^{39}-7q^{37}+1144q^{35}+1555q^{33}+546q^{31}-1453q^{29}-2717q^{27}-1638q^{25}+1465q^{23}+4103q^{21}+3502q^{19}-723q^{17}-5462q^{15}-6166q^{13}-1115q^{11}+6247q^{9}+9339q^{7}+4252q^{5}-5844q^{3}-12391q-8571q^{-1}+3809q^{-3}+14563q^{-5}+13421q^{-7}-96q^{-9}-15055q^{-11}-17894q^{-13}-4872q^{-15}+13525q^{-17}+21057q^{-19}+10203q^{-21}-10166q^{-23}-22157q^{-25}-14879q^{-27}+5525q^{-29}+21093q^{-31}+18124q^{-33}-590q^{-35}-18204q^{-37}-19422q^{-39}-3902q^{-41}+14133q^{-43}+18999q^{-45}+7395q^{-47}-9700q^{-49}-17272q^{-51}-9699q^{-53}+5399q^{-55}+14814q^{-57}+11177q^{-59}-1575q^{-61}-12261q^{-63}-12095q^{-65}-1803q^{-67}+9744q^{-69}+12940q^{-71}+5020q^{-73}-7435q^{-75}-13829q^{-77}-8277q^{-79}+5020q^{-81}+14706q^{-83}+11762q^{-85}-2198q^{-87}-15282q^{-89}-15339q^{-91}-1250q^{-93}+15027q^{-95}+18637q^{-97}+5399q^{-99}-13533q^{-101}-21098q^{-103}-9887q^{-105}+10536q^{-107}+22050q^{-109}+14178q^{-111}-6236q^{-113}-21050q^{-115}-17417q^{-117}+1110q^{-119}+18082q^{-121}+18967q^{-123}+3828q^{-125}-13500q^{-127}-18371q^{-129}-7804q^{-131}+8192q^{-133}+15916q^{-135}+9953q^{-137}-3176q^{-139}-12057q^{-141}-10227q^{-143}-723q^{-145}+7853q^{-147}+8911q^{-149}+2955q^{-151}-4092q^{-153}-6647q^{-155}-3682q^{-157}+1358q^{-159}+4271q^{-161}+3311q^{-163}+145q^{-165}-2274q^{-167}-2393q^{-169}-747q^{-171}+955q^{-173}+1466q^{-175}+753q^{-177}-270q^{-179}-745q^{-181}-529q^{-183}-11q^{-185}+314q^{-187}+304q^{-189}+75q^{-191}-119q^{-193}-139q^{-195}-50q^{-197}+30q^{-199}+49q^{-201}+34q^{-203}-7q^{-205}-24q^{-207}-6q^{-209}+3q^{-211}+q^{-213}+4q^{-215}+2q^{-217}-4q^{-219}+2q^{-223}-q^{-225}$

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-3q^{12}+q^{10}+6q^{8}-12q^{6}+5q^{4}+16q^{2}-23+6q^{-2}+25q^{-4}-30q^{-6}+2q^{-8}+27q^{-10}-21q^{-12}-4q^{-14}+16q^{-16}-q^{-18}-8q^{-20}-3q^{-22}+19q^{-24}-6q^{-26}-21q^{-28}+27q^{-30}-30q^{-34}+25q^{-36}+3q^{-38}-23q^{-40}+15q^{-42}+2q^{-44}-10q^{-46}+5q^{-48}+q^{-50}-2q^{-52}+q^{-54}$
1,0,0	$-q^{7}+2q^{5}-2q^{3}+2q-q^{-1}+4q^{-3}-2q^{-5}+2q^{-7}-q^{-15}+2q^{-17}-3q^{-19}+3q^{-21}-q^{-23}+2q^{-25}-q^{-27}+2q^{-29}-q^{-31}-q^{-33}-q^{-35}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+3q^{12}-7q^{10}+12q^{8}-18q^{6}+25q^{4}-30q^{2}+35-34q^{-2}+31q^{-4}-20q^{-6}+8q^{-8}+9q^{-10}-27q^{-12}+44q^{-14}-58q^{-16}+65q^{-18}-68q^{-20}+63q^{-22}-53q^{-24}+38q^{-26}-19q^{-28}+3q^{-30}+14q^{-32}-24q^{-34}+33q^{-36}-35q^{-38}+33q^{-40}-29q^{-42}+22q^{-44}-16q^{-46}+9q^{-48}-5q^{-50}+2q^{-52}-q^{-54}$
1,0	$q^{24}-3q^{20}-3q^{18}+4q^{16}+9q^{14}-q^{12}-15q^{10}-9q^{8}+17q^{6}+23q^{4}-6q^{2}-32-11q^{-2}+30q^{-4}+29q^{-6}-17q^{-8}-37q^{-10}-q^{-12}+35q^{-14}+15q^{-16}-25q^{-18}-20q^{-20}+17q^{-22}+23q^{-24}-10q^{-26}-23q^{-28}+5q^{-30}+24q^{-32}-24q^{-36}-5q^{-38}+25q^{-40}+14q^{-42}-24q^{-44}-23q^{-46}+18q^{-48}+34q^{-50}-5q^{-52}-38q^{-54}-15q^{-56}+30q^{-58}+28q^{-60}-14q^{-62}-31q^{-64}-4q^{-66}+22q^{-68}+13q^{-70}-9q^{-72}-13q^{-74}+7q^{-78}+3q^{-80}-2q^{-82}-2q^{-84}+q^{-88}$

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-3q^{30}+7q^{28}-13q^{26}+14q^{24}-12q^{22}-q^{20}+26q^{18}-51q^{16}+77q^{14}-84q^{12}+57q^{10}-q^{8}-83q^{6}+165q^{4}-206q^{2}+191-107q^{-2}-23q^{-4}+170q^{-6}-269q^{-8}+282q^{-10}-199q^{-12}+45q^{-14}+111q^{-16}-215q^{-18}+217q^{-20}-120q^{-22}-17q^{-24}+156q^{-26}-210q^{-28}+145q^{-30}+6q^{-32}-193q^{-34}+324q^{-36}-341q^{-38}+228q^{-40}-18q^{-42}-207q^{-44}+382q^{-46}-429q^{-48}+337q^{-50}-147q^{-52}-86q^{-54}+260q^{-56}-320q^{-58}+260q^{-60}-108q^{-62}-54q^{-64}+173q^{-66}-190q^{-68}+100q^{-70}+42q^{-72}-183q^{-74}+249q^{-76}-204q^{-78}+69q^{-80}+100q^{-82}-232q^{-84}+292q^{-86}-249q^{-88}+132q^{-90}+7q^{-92}-133q^{-94}+191q^{-96}-182q^{-98}+127q^{-100}-49q^{-102}-15q^{-104}+55q^{-106}-69q^{-108}+56q^{-110}-35q^{-112}+14q^{-114}+q^{-116}-9q^{-118}+9q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

.

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 69"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-7t^{2}+21t-29+21t^{-1}-7t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+4z^{8}a^{-2}+8z^{8}a^{-4}+4z^{8}a^{-6}+6z^{7}a^{-1}+14z^{7}a^{-3}+13z^{7}a^{-5}+5z^{7}a^{-7}+3z^{6}a^{-2}-4z^{6}a^{-4}+3z^{6}a^{-8}+4z^{6}+az^{5}-10z^{5}a^{-1}-32z^{5}a^{-3}-30z^{5}a^{-5}-8z^{5}a^{-7}+z^{5}a^{-9}-17z^{4}a^{-2}-14z^{4}a^{-4}-9z^{4}a^{-6}-5z^{4}a^{-8}-7z^{4}-az^{3}+5z^{3}a^{-1}+22z^{3}a^{-3}+23z^{3}a^{-5}+5z^{3}a^{-7}-2z^{3}a^{-9}+11z^{2}a^{-2}+12z^{2}a^{-4}+7z^{2}a^{-6}+3z^{2}a^{-8}+3z^{2}-za^{-1}-4za^{-3}-6za^{-5}-2za^{-7}+za^{-9}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}$

"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, 4)

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 $8$ $32$ $32$ ${\frac {412}{3}}$ ${\frac {68}{3}}$ $256$ ${\frac {1952}{3}}$ ${\frac {320}{3}}$ $128$ ${\frac {256}{3}}$ $512$ ${\frac {3296}{3}}$ ${\frac {544}{3}}$ ${\frac {45151}{15}}$ $-{\frac {4844}{15}}$ ${\frac {75004}{45}}$ ${\frac {689}{9}}$ ${\frac {3631}{15}}$

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 69. 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             7 5       -2 7           8 6         2 5         6 7           1 3       5 8             -3 1     3 7               4 -1   1 4                 -3 -3   3                   3 -5 1                     -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{23}-3q^{22}+2q^{21}+9q^{20}-21q^{19}+4q^{18}+43q^{17}-60q^{16}-9q^{15}+107q^{14}-100q^{13}-43q^{12}+172q^{11}-117q^{10}-83q^{9}+202q^{8}-101q^{7}-104q^{6}+180q^{5}-59q^{4}-95q^{3}+117q^{2}-19q-61+51q^{-1}-24q^{-3}+13q^{-4}+2q^{-5}-4q^{-6}+q^{-7}$
3	$-q^{45}+3q^{44}-2q^{43}-4q^{42}+q^{41}+17q^{40}-5q^{39}-39q^{38}+2q^{37}+83q^{36}+11q^{35}-143q^{34}-61q^{33}+235q^{32}+135q^{31}-320q^{30}-265q^{29}+402q^{28}+434q^{27}-461q^{26}-625q^{25}+477q^{24}+832q^{23}-464q^{22}-1010q^{21}+397q^{20}+1175q^{19}-324q^{18}-1270q^{17}+207q^{16}+1330q^{15}-100q^{14}-1309q^{13}-30q^{12}+1244q^{11}+143q^{10}-1115q^{9}-241q^{8}+945q^{7}+306q^{6}-744q^{5}-341q^{4}+552q^{3}+321q^{2}-362q-284+224q^{-1}+214q^{-2}-114q^{-3}-153q^{-4}+55q^{-5}+90q^{-6}-16q^{-7}-53q^{-8}+6q^{-9}+23q^{-10}-8q^{-12}-2q^{-13}+4q^{-14}-q^{-15}$
4	$q^{74}-3q^{73}+2q^{72}+4q^{71}-6q^{70}+3q^{69}-16q^{68}+16q^{67}+34q^{66}-29q^{65}-14q^{64}-87q^{63}+63q^{62}+184q^{61}-28q^{60}-95q^{59}-393q^{58}+67q^{57}+606q^{56}+249q^{55}-126q^{54}-1218q^{53}-354q^{52}+1233q^{51}+1184q^{50}+396q^{49}-2512q^{48}-1703q^{47}+1462q^{46}+2751q^{45}+2072q^{44}-3607q^{43}-3958q^{42}+614q^{41}+4270q^{40}+4814q^{39}-3754q^{38}-6333q^{37}-1312q^{36}+4975q^{35}+7772q^{34}-2842q^{33}-7961q^{32}-3616q^{31}+4668q^{30}+10016q^{29}-1326q^{28}-8457q^{27}-5573q^{26}+3583q^{25}+11061q^{24}+349q^{23}-7816q^{22}-6805q^{21}+1953q^{20}+10743q^{19}+1940q^{18}-6116q^{17}-7108q^{16}+19q^{15}+9050q^{14}+3088q^{13}-3655q^{12}-6261q^{11}-1678q^{10}+6290q^{9}+3313q^{8}-1184q^{7}-4413q^{6}-2450q^{5}+3364q^{4}+2524q^{3}+384q^{2}-2313q-2106+1260q^{-1}+1320q^{-2}+785q^{-3}-814q^{-4}-1227q^{-5}+285q^{-6}+433q^{-7}+528q^{-8}-150q^{-9}-503q^{-10}+25q^{-11}+65q^{-12}+213q^{-13}+7q^{-14}-146q^{-15}-9q^{-17}+54q^{-18}+12q^{-19}-29q^{-20}+q^{-21}-5q^{-22}+8q^{-23}+2q^{-24}-4q^{-25}+q^{-26}$
5	$-q^{110}+3q^{109}-2q^{108}-4q^{107}+6q^{106}+2q^{105}-4q^{104}+5q^{103}-11q^{102}-22q^{101}+23q^{100}+43q^{99}+11q^{98}-14q^{97}-91q^{96}-111q^{95}+43q^{94}+237q^{93}+240q^{92}-4q^{91}-416q^{90}-629q^{89}-173q^{88}+712q^{87}+1263q^{86}+709q^{85}-927q^{84}-2331q^{83}-1819q^{82}+831q^{81}+3682q^{80}+3875q^{79}+33q^{78}-5244q^{77}-6859q^{76}-2134q^{75}+6237q^{74}+10922q^{73}+5989q^{72}-6302q^{71}-15435q^{70}-11638q^{69}+4407q^{68}+19803q^{67}+19118q^{66}-339q^{65}-23159q^{64}-27634q^{63}-6160q^{62}+24674q^{61}+36446q^{60}+14800q^{59}-24044q^{58}-44606q^{57}-24687q^{56}+21041q^{55}+51260q^{54}+35214q^{53}-16172q^{52}-56120q^{51}-45110q^{50}+9830q^{49}+58825q^{48}+54146q^{47}-2934q^{46}-59730q^{45}-61387q^{44}-4259q^{43}+58882q^{42}+67230q^{41}+11026q^{40}-56786q^{39}-71073q^{38}-17556q^{37}+53330q^{36}+73624q^{35}+23481q^{34}-48866q^{33}-74269q^{32}-29103q^{31}+43038q^{30}+73458q^{29}+34167q^{28}-36114q^{27}-70632q^{26}-38518q^{25}+27940q^{24}+65885q^{23}+41739q^{22}-19019q^{21}-59028q^{20}-43384q^{19}+9905q^{18}+50365q^{17}+42957q^{16}-1405q^{15}-40314q^{14}-40415q^{13}-5663q^{12}+29961q^{11}+35679q^{10}+10586q^{9}-19945q^{8}-29561q^{7}-13195q^{6}+11564q^{5}+22657q^{4}+13425q^{3}-4986q^{2}-16044q-12053+810q^{-1}+10277q^{-2}+9605q^{-3}+1561q^{-4}-5948q^{-5}-6966q^{-6}-2282q^{-7}+2915q^{-8}+4554q^{-9}+2265q^{-10}-1209q^{-11}-2741q^{-12}-1681q^{-13}+277q^{-14}+1451q^{-15}+1186q^{-16}+55q^{-17}-742q^{-18}-672q^{-19}-134q^{-20}+300q^{-21}+370q^{-22}+133q^{-23}-133q^{-24}-185q^{-25}-66q^{-26}+48q^{-27}+64q^{-28}+46q^{-29}-5q^{-30}-45q^{-31}-13q^{-32}+11q^{-33}+5q^{-34}+4q^{-35}+5q^{-36}-8q^{-37}-2q^{-38}+4q^{-39}-q^{-40}$
6	$q^{153}-3q^{152}+2q^{151}+4q^{150}-6q^{149}-2q^{148}-q^{147}+15q^{146}-10q^{145}-q^{144}+28q^{143}-37q^{142}-30q^{141}-13q^{140}+77q^{139}+25q^{138}+16q^{137}+95q^{136}-172q^{135}-228q^{134}-159q^{133}+263q^{132}+308q^{131}+360q^{130}+480q^{129}-561q^{128}-1153q^{127}-1254q^{126}+185q^{125}+1187q^{124}+2222q^{123}+2830q^{122}-345q^{121}-3582q^{120}-5840q^{119}-3119q^{118}+999q^{117}+7049q^{116}+11702q^{115}+5902q^{114}-4632q^{113}-16430q^{112}-17035q^{111}-9338q^{110}+9969q^{109}+30667q^{108}+29753q^{107}+9643q^{106}-25339q^{105}-46128q^{104}-46371q^{103}-9077q^{102}+48021q^{101}+76748q^{100}+61052q^{99}-5489q^{98}-73901q^{97}-115855q^{96}-76895q^{95}+29385q^{94}+125127q^{93}+154418q^{92}+73981q^{91}-59584q^{90}-190230q^{89}-196007q^{88}-58377q^{87}+128868q^{86}+256318q^{85}+212139q^{84}+29782q^{83}-219573q^{82}-327261q^{81}-209585q^{80}+56438q^{79}+314426q^{78}+365081q^{77}+183183q^{76}-174563q^{75}-417104q^{74}-376572q^{73}-77438q^{72}+301775q^{71}+479632q^{70}+351254q^{69}-71583q^{68}-440451q^{67}-507943q^{66}-225474q^{65}+234221q^{64}+532645q^{63}+486687q^{62}+47351q^{61}-410722q^{60}-582966q^{59}-347730q^{58}+146413q^{57}+534670q^{56}+572713q^{55}+151582q^{54}-354261q^{53}-609988q^{52}-433462q^{51}+59356q^{50}+504456q^{49}+616966q^{48}+236940q^{47}-282948q^{46}-601618q^{45}-491012q^{44}-28635q^{43}+446200q^{42}+627244q^{41}+312203q^{40}-190207q^{39}-555571q^{38}-523805q^{37}-125675q^{36}+349536q^{35}+595214q^{34}+375256q^{33}-69685q^{32}-458164q^{31}-516598q^{30}-222600q^{29}+210606q^{28}+503619q^{27}+402191q^{26}+61232q^{25}-308130q^{24}-447067q^{23}-285746q^{22}+54973q^{21}+353221q^{20}+364485q^{19}+159174q^{18}-138309q^{17}-316308q^{16}-280859q^{15}-66294q^{14}+182376q^{13}+263001q^{12}+185928q^{11}-4633q^{10}-165324q^{9}-209517q^{8}-114346q^{7}+48543q^{6}+139926q^{5}+145060q^{4}+56360q^{3}-49406q^{2}-114531q-96180-16455q^{-1}+46574q^{-2}+79144q^{-3}+55332q^{-4}+5971q^{-5}-42476q^{-6}-53117q^{-7}-26252q^{-8}+2555q^{-9}+29120q^{-10}+30777q^{-11}+16168q^{-12}-8288q^{-13}-19992q^{-14}-15023q^{-15}-7012q^{-16}+6080q^{-17}+11202q^{-18}+9900q^{-19}+788q^{-20}-4943q^{-21}-5119q^{-22}-4545q^{-23}-53q^{-24}+2616q^{-25}+3823q^{-26}+1091q^{-27}-702q^{-28}-1015q^{-29}-1647q^{-30}-516q^{-31}+305q^{-32}+1102q^{-33}+354q^{-34}-35q^{-35}-55q^{-36}-407q^{-37}-197q^{-38}-30q^{-39}+268q^{-40}+56q^{-41}-3q^{-42}+32q^{-43}-74q^{-44}-40q^{-45}-25q^{-46}+59q^{-47}+4q^{-48}-10q^{-49}+13q^{-50}-10q^{-51}-4q^{-52}-5q^{-53}+8q^{-54}+2q^{-55}-4q^{-56}+q^{-57}$
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, 69]]
Out[2]=	PD[X[1, 4, 2, 5], X[7, 12, 8, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[13, 17, 14, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[17, 20, 18, 1], X[9, 19, 10, 18], X[19, 9, 20, 8]]
In[3]:=	GaussCode[Knot[10, 69]]
Out[3]=	GaussCode[-1, 4, -3, 1, -6, 7, -2, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, 8]
In[4]:=	DTCode[Knot[10, 69]]
Out[4]=	DTCode[4, 10, 14, 12, 18, 2, 16, 6, 20, 8]
In[5]:=	br = BR[Knot[10, 69]]
Out[5]=	BR[5, {1, 1, 2, -1, -3, 2, 1, 4, -3, 2, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 69]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 69]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 69]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 69]][t]
Out[10]=	-3 7 21 2 3 -29 + t - -- + -- + 21 t - 7 t + t 2 t t
In[11]:=	Conway[Knot[10, 69]][z]
Out[11]=	2 4 6 1 + 2 z - z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 69]}
In[13]:=	{KnotDet[Knot[10, 69]], KnotSignature[Knot[10, 69]]}
Out[13]=	{87, 2}
In[14]:=	Jones[Knot[10, 69]][q]
Out[14]=	-2 4 2 3 4 5 6 7 8 -7 - q + - + 11 q - 14 q + 15 q - 13 q + 11 q - 7 q + 3 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 69]}
In[16]:=	A2Invariant[Knot[10, 69]][q]
Out[16]=	-6 2 -2 2 4 6 8 12 14 16 -q + -- - q + 4 q - 3 q + 2 q - q + 2 q - 2 q + 3 q - 4 q 18 20 22 24 26 q - q + 2 q - q - q
In[17]:=	HOMFLYPT[Knot[10, 69]][a, z]
Out[17]=	2 2 2 4 4 6 -8 2 2 2 2 3 z 5 z 5 z 4 3 z 3 z z -a + -- - -- + -- - z + ---- - ---- + ---- - z - ---- + ---- + -- 6 4 2 6 4 2 4 2 2 a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 69]][a, z]
Out[18]=	2 2 -8 2 2 2 z 2 z 6 z 4 z z 2 3 z 7 z -a - -- - -- - -- + -- - --- - --- - --- - - + 3 z + ---- + ---- + 6 4 2 9 7 5 3 a 8 6 a a a a a a a a a 2 2 3 3 3 3 3 12 z 11 z 2 z 5 z 23 z 22 z 5 z 3 4 ----- + ----- - ---- + ---- + ----- + ----- + ---- - a z - 7 z - 4 2 9 7 5 3 a a a a a a a 4 4 4 4 5 5 5 5 5 5 z 9 z 14 z 17 z z 8 z 30 z 32 z 10 z ---- - ---- - ----- - ----- + -- - ---- - ----- - ----- - ----- + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 7 7 7 7 5 6 3 z 4 z 3 z 5 z 13 z 14 z 6 z a z + 4 z + ---- - ---- + ---- + ---- + ----- + ----- + ---- + 8 4 2 7 5 3 a a a a a a a 8 8 8 9 9 4 z 8 z 4 z z z ---- + ---- + ---- + -- + -- 6 4 2 5 3 a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 69]], Vassiliev[3][Knot[10, 69]]}
Out[19]=	{2, 4}
In[20]:=	Kh[Knot[10, 69]][q, t]
Out[20]=	3 1 3 1 4 3 q 3 5 7 q + 5 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 7 q t + 8 q t + 6 q t + 7 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
In[21]:=	ColouredJones[Knot[10, 69], 2][q]
Out[21]=	-7 4 2 13 24 51 2 3 4 -61 + q - -- + -- + -- - -- + -- - 19 q + 117 q - 95 q - 59 q + 6 5 4 3 q q q q q 5 6 7 8 9 10 11 180 q - 104 q - 101 q + 202 q - 83 q - 117 q + 172 q - 12 13 14 15 16 17 18 43 q - 100 q + 107 q - 9 q - 60 q + 43 q + 4 q - 19 20 21 22 23 21 q + 9 q + 2 q - 3 q + q

See/edit the Rolfsen_Splice_Template.