8 7

8_6

8_8

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8 7 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X7,15,8,14 X13,7,14,6 X15,9,16,8 X9,2,10,3
Gauss code	-1, 8, -2, 1, -4, 6, -5, 7, -8, 2, -3, 4, -6, 5, -7, 3
Dowker-Thistlethwaite code	4 10 12 14 2 16 6 8
Conway Notation	[4112]

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 1 3-genus 3 Bridge index 2 Super bridge index $\{3,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-8] Hyperbolic Volume 7.0222 A-Polynomial See Data:8 7/A-polynomial

[edit Notes for 8 7'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 8 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-3t^{2}+5t-5+5t^{-1}-3t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 23, 2 }
Jones polynomial	$-q^{6}+2q^{5}-3q^{4}+4q^{3}-4q^{2}+4q-2+2q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+5z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+8z^{2}a^{-2}-3z^{2}a^{-4}-3z^{2}+4a^{-2}-2a^{-4}-1$
Kauffman polynomial (db, data sources)	$z^{7}a^{-1}+z^{7}a^{-3}+4z^{6}a^{-2}+2z^{6}a^{-4}+2z^{6}+az^{5}-z^{5}a^{-1}+2z^{5}a^{-5}-12z^{4}a^{-2}-3z^{4}a^{-4}+2z^{4}a^{-6}-7z^{4}-3az^{3}-3z^{3}a^{-1}-2z^{3}a^{-3}-z^{3}a^{-5}+z^{3}a^{-7}+12z^{2}a^{-2}+4z^{2}a^{-4}-2z^{2}a^{-6}+6z^{2}+az+2za^{-1}+2za^{-3}-za^{-7}-4a^{-2}-2a^{-4}-1$
The A2 invariant	$-q^{6}+1+2q^{-2}+2q^{-6}+q^{-10}-q^{-14}-q^{-18}$
The G2 invariant	$q^{32}-q^{30}+2q^{28}-3q^{26}+q^{24}-q^{22}-3q^{20}+7q^{18}-8q^{16}+5q^{14}-3q^{12}-2q^{10}+6q^{8}-10q^{6}+7q^{4}-3q^{2}+6q^{-2}-6q^{-4}+4q^{-6}+3q^{-8}-q^{-10}+4q^{-12}-4q^{-14}+2q^{-16}+4q^{-18}-5q^{-20}+10q^{-22}-7q^{-24}+5q^{-26}+3q^{-28}-7q^{-30}+10q^{-32}-11q^{-34}+8q^{-36}-2q^{-38}-3q^{-40}+8q^{-42}-8q^{-44}+6q^{-46}-3q^{-50}+3q^{-52}-3q^{-54}-q^{-56}+4q^{-58}-5q^{-60}+5q^{-62}-3q^{-64}-2q^{-66}+4q^{-68}-7q^{-70}+5q^{-72}-5q^{-74}+2q^{-76}-q^{-78}-3q^{-80}+4q^{-82}-4q^{-84}+4q^{-86}-2q^{-88}+q^{-90}-2q^{-94}+2q^{-96}-q^{-98}+q^{-100}$

Further Quantum Invariants

Further quantum knot invariants for 8_7.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+q^{3}+2q^{-1}+q^{-7}-q^{-9}+q^{-11}-q^{-13}$
2	$q^{16}-q^{14}-2q^{12}+2q^{10}-3q^{6}+2q^{4}+3q^{2}-2+2q^{-2}+3q^{-4}-2q^{-6}+2q^{-10}-2q^{-14}-q^{-16}+3q^{-18}-2q^{-20}-2q^{-22}+4q^{-24}-q^{-26}-2q^{-28}+2q^{-30}-q^{-32}-q^{-34}+q^{-36}$
3	$-q^{33}+q^{31}+2q^{29}-3q^{25}-2q^{23}+4q^{21}+3q^{19}-4q^{17}-6q^{15}+q^{13}+6q^{11}+2q^{9}-7q^{7}-2q^{5}+6q^{3}+7q-3q^{-1}-5q^{-3}+3q^{-5}+7q^{-7}-q^{-9}-6q^{-11}+2q^{-13}+5q^{-15}-6q^{-19}-2q^{-21}+3q^{-23}+3q^{-25}-q^{-27}-5q^{-29}-2q^{-31}+7q^{-33}+4q^{-35}-6q^{-37}-6q^{-39}+5q^{-41}+6q^{-43}-3q^{-45}-5q^{-47}+2q^{-49}+3q^{-51}-q^{-53}-2q^{-55}+q^{-57}-q^{-61}+q^{-65}+q^{-67}-q^{-69}$
4	$q^{56}-q^{54}-2q^{52}+q^{48}+5q^{46}-4q^{42}-4q^{40}-3q^{38}+10q^{36}+7q^{34}-2q^{32}-8q^{30}-13q^{28}+4q^{26}+10q^{24}+9q^{22}-18q^{18}-8q^{16}+2q^{14}+15q^{12}+14q^{10}-8q^{8}-13q^{6}-12q^{4}+10q^{2}+23+5q^{-2}-9q^{-4}-20q^{-6}+21q^{-10}+10q^{-12}-5q^{-14}-19q^{-16}-4q^{-18}+15q^{-20}+8q^{-22}-4q^{-24}-15q^{-26}-4q^{-28}+10q^{-30}+10q^{-32}-2q^{-34}-10q^{-36}-6q^{-38}+q^{-40}+11q^{-42}+7q^{-44}+q^{-46}-13q^{-48}-15q^{-50}+8q^{-52}+15q^{-54}+14q^{-56}-10q^{-58}-25q^{-60}-3q^{-62}+12q^{-64}+23q^{-66}+q^{-68}-21q^{-70}-9q^{-72}+16q^{-76}+7q^{-78}-8q^{-80}-4q^{-82}-6q^{-84}+5q^{-86}+4q^{-88}-q^{-90}+2q^{-92}-5q^{-94}+q^{-98}+3q^{-102}-q^{-104}-q^{-108}-q^{-110}+q^{-112}$
5	$-q^{85}+q^{83}+2q^{81}-q^{77}-3q^{75}-3q^{73}+6q^{69}+6q^{67}+q^{65}-5q^{63}-10q^{61}-8q^{59}+3q^{57}+16q^{55}+15q^{53}+3q^{51}-11q^{49}-22q^{47}-15q^{45}+4q^{43}+22q^{41}+23q^{39}+9q^{37}-12q^{35}-30q^{33}-24q^{31}-3q^{29}+22q^{27}+33q^{25}+21q^{23}-9q^{21}-34q^{19}-38q^{17}-14q^{15}+28q^{13}+51q^{11}+31q^{9}-8q^{7}-48q^{5}-50q^{3}-5q+48q^{-1}+59q^{-3}+23q^{-5}-33q^{-7}-63q^{-9}-34q^{-11}+25q^{-13}+60q^{-15}+39q^{-17}-15q^{-19}-52q^{-21}-40q^{-23}+7q^{-25}+45q^{-27}+34q^{-29}-7q^{-31}-36q^{-33}-30q^{-35}+4q^{-37}+33q^{-39}+24q^{-41}-7q^{-43}-26q^{-45}-20q^{-47}+2q^{-49}+23q^{-51}+20q^{-53}+5q^{-55}-15q^{-57}-23q^{-59}-15q^{-61}+3q^{-63}+25q^{-65}+32q^{-67}+14q^{-69}-24q^{-71}-47q^{-73}-31q^{-75}+12q^{-77}+54q^{-79}+54q^{-81}+2q^{-83}-57q^{-85}-69q^{-87}-21q^{-89}+46q^{-91}+77q^{-93}+39q^{-95}-30q^{-97}-71q^{-99}-49q^{-101}+13q^{-103}+56q^{-105}+49q^{-107}+3q^{-109}-37q^{-111}-43q^{-113}-12q^{-115}+22q^{-117}+29q^{-119}+14q^{-121}-7q^{-123}-20q^{-125}-14q^{-127}+2q^{-129}+11q^{-131}+10q^{-133}+3q^{-135}-5q^{-137}-8q^{-139}-4q^{-141}+3q^{-143}+5q^{-145}+2q^{-147}+q^{-149}-2q^{-151}-3q^{-153}-q^{-155}+q^{-157}+q^{-161}+q^{-163}-q^{-165}$
6	$q^{120}-q^{118}-2q^{116}+q^{112}+3q^{110}+q^{108}+3q^{106}-2q^{104}-8q^{102}-5q^{100}-q^{98}+6q^{96}+7q^{94}+14q^{92}+3q^{90}-12q^{88}-18q^{86}-17q^{84}-3q^{82}+6q^{80}+33q^{78}+30q^{76}+8q^{74}-15q^{72}-34q^{70}-34q^{68}-29q^{66}+17q^{64}+42q^{62}+47q^{60}+28q^{58}-q^{56}-31q^{54}-67q^{52}-42q^{50}-12q^{48}+30q^{46}+56q^{44}+66q^{42}+42q^{40}-27q^{38}-62q^{36}-86q^{34}-62q^{32}-8q^{30}+75q^{28}+117q^{26}+81q^{24}+18q^{22}-81q^{20}-134q^{18}-123q^{16}-12q^{14}+104q^{12}+154q^{10}+136q^{8}+15q^{6}-116q^{4}-190q^{2}-123+12q^{-2}+137q^{-4}+195q^{-6}+116q^{-8}-37q^{-10}-177q^{-12}-176q^{-14}-74q^{-16}+70q^{-18}+179q^{-20}+159q^{-22}+32q^{-24}-123q^{-26}-164q^{-28}-104q^{-30}+17q^{-32}+128q^{-34}+140q^{-36}+53q^{-38}-75q^{-40}-117q^{-42}-85q^{-44}+2q^{-46}+83q^{-48}+94q^{-50}+35q^{-52}-51q^{-54}-74q^{-56}-49q^{-58}+13q^{-60}+56q^{-62}+58q^{-64}+18q^{-66}-36q^{-68}-54q^{-70}-42q^{-72}+5q^{-74}+37q^{-76}+54q^{-78}+40q^{-80}+2q^{-82}-46q^{-84}-73q^{-86}-52q^{-88}-13q^{-90}+59q^{-92}+102q^{-94}+94q^{-96}+4q^{-98}-96q^{-100}-138q^{-102}-118q^{-104}+5q^{-106}+136q^{-108}+202q^{-110}+116q^{-112}-44q^{-114}-177q^{-116}-226q^{-118}-111q^{-120}+80q^{-122}+236q^{-124}+213q^{-126}+65q^{-128}-113q^{-130}-237q^{-132}-188q^{-134}-24q^{-136}+157q^{-138}+200q^{-140}+125q^{-142}-8q^{-144}-144q^{-146}-162q^{-148}-76q^{-150}+53q^{-152}+108q^{-154}+97q^{-156}+40q^{-158}-50q^{-160}-84q^{-162}-59q^{-164}+4q^{-166}+35q^{-168}+45q^{-170}+37q^{-172}-10q^{-174}-33q^{-176}-29q^{-178}-4q^{-180}+7q^{-182}+16q^{-184}+23q^{-186}-12q^{-190}-14q^{-192}-4q^{-194}-q^{-196}+5q^{-198}+13q^{-200}+2q^{-202}-3q^{-204}-6q^{-206}-2q^{-208}-3q^{-210}+5q^{-214}+q^{-216}+q^{-218}-q^{-220}-q^{-224}-q^{-226}+q^{-228}$

A2 Invariants.

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

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-q^{30}+2q^{28}-3q^{26}+q^{24}-q^{22}-3q^{20}+7q^{18}-8q^{16}+5q^{14}-3q^{12}-2q^{10}+6q^{8}-10q^{6}+7q^{4}-3q^{2}+6q^{-2}-6q^{-4}+4q^{-6}+3q^{-8}-q^{-10}+4q^{-12}-4q^{-14}+2q^{-16}+4q^{-18}-5q^{-20}+10q^{-22}-7q^{-24}+5q^{-26}+3q^{-28}-7q^{-30}+10q^{-32}-11q^{-34}+8q^{-36}-2q^{-38}-3q^{-40}+8q^{-42}-8q^{-44}+6q^{-46}-3q^{-50}+3q^{-52}-3q^{-54}-q^{-56}+4q^{-58}-5q^{-60}+5q^{-62}-3q^{-64}-2q^{-66}+4q^{-68}-7q^{-70}+5q^{-72}-5q^{-74}+2q^{-76}-q^{-78}-3q^{-80}+4q^{-82}-4q^{-84}+4q^{-86}-2q^{-88}+q^{-90}-2q^{-94}+2q^{-96}-q^{-98}+q^{-100}$

.

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 7"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-3t^{2}+5t-5+5t^{-1}-3t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 23, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+5z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+8z^{2}a^{-2}-3z^{2}a^{-4}-3z^{2}+4a^{-2}-2a^{-4}-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n24, ...}

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 $8$ $16$ $32$ ${\frac {124}{3}}$ $-{\frac {28}{3}}$ $128$ ${\frac {448}{3}}$ ${\frac {64}{3}}$ $-16$ ${\frac {256}{3}}$ $128$ ${\frac {992}{3}}$ $-{\frac {224}{3}}$ ${\frac {7951}{15}}$ ${\frac {1052}{5}}$ $-{\frac {6596}{45}}$ ${\frac {65}{9}}$ $-{\frac {929}{15}}$

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 8 7. 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             2 1   -1 7           2 1     1 5         2 2       0 3       2 2         0 1     1 3           2 -1   1 1             0 -3   1               1 -5 1                 -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{17}-2q^{16}+4q^{14}-6q^{13}+q^{12}+9q^{11}-12q^{10}+q^{9}+14q^{8}-16q^{7}+16q^{5}-14q^{4}-2q^{3}+14q^{2}-9q-3+10q^{-1}-4q^{-2}-4q^{-3}+5q^{-4}-q^{-5}-2q^{-6}+q^{-7}$
3	$-q^{33}+2q^{32}-q^{30}-2q^{29}+3q^{28}+q^{27}-4q^{26}-q^{25}+7q^{24}-11q^{22}+q^{21}+16q^{20}-q^{19}-22q^{18}+q^{17}+26q^{16}+2q^{15}-31q^{14}-2q^{13}+30q^{12}+6q^{11}-31q^{10}-7q^{9}+26q^{8}+12q^{7}-26q^{6}-10q^{5}+18q^{4}+17q^{3}-18q^{2}-14q+10+19q^{-1}-8q^{-2}-15q^{-3}+2q^{-4}+14q^{-5}+q^{-6}-11q^{-7}-3q^{-8}+7q^{-9}+3q^{-10}-4q^{-11}-2q^{-12}+q^{-13}+2q^{-14}-q^{-15}$
4	$q^{54}-2q^{53}+q^{51}-q^{50}+5q^{49}-5q^{48}+q^{47}-6q^{45}+12q^{44}-8q^{43}+6q^{42}+q^{41}-17q^{40}+14q^{39}-12q^{38}+21q^{37}+10q^{36}-33q^{35}+5q^{34}-24q^{33}+43q^{32}+32q^{31}-44q^{30}-10q^{29}-46q^{28}+58q^{27}+56q^{26}-43q^{25}-17q^{24}-69q^{23}+60q^{22}+70q^{21}-37q^{20}-13q^{19}-79q^{18}+53q^{17}+66q^{16}-29q^{15}-q^{14}-79q^{13}+39q^{12}+55q^{11}-18q^{10}+11q^{9}-72q^{8}+20q^{7}+40q^{6}-4q^{5}+26q^{4}-61q^{3}-q^{2}+20q+7+40q^{-1}-43q^{-2}-14q^{-3}-2q^{-4}+6q^{-5}+45q^{-6}-21q^{-7}-13q^{-8}-15q^{-9}-4q^{-10}+35q^{-11}-3q^{-12}-4q^{-13}-14q^{-14}-10q^{-15}+18q^{-16}+2q^{-17}+2q^{-18}-5q^{-19}-7q^{-20}+5q^{-21}+q^{-22}+2q^{-23}-q^{-24}-2q^{-25}+q^{-26}$
5	$-q^{80}+2q^{79}-q^{77}+q^{76}-2q^{75}-3q^{74}+3q^{73}+3q^{72}+4q^{70}-4q^{69}-10q^{68}-q^{67}+6q^{66}+8q^{65}+11q^{64}-3q^{63}-19q^{62}-17q^{61}+21q^{59}+32q^{58}+12q^{57}-26q^{56}-51q^{55}-31q^{54}+27q^{53}+72q^{52}+58q^{51}-19q^{50}-94q^{49}-93q^{48}+5q^{47}+113q^{46}+127q^{45}+19q^{44}-125q^{43}-160q^{42}-43q^{41}+125q^{40}+186q^{39}+71q^{38}-125q^{37}-202q^{36}-86q^{35}+109q^{34}+209q^{33}+109q^{32}-107q^{31}-209q^{30}-108q^{29}+91q^{28}+201q^{27}+117q^{26}-87q^{25}-194q^{24}-105q^{23}+70q^{22}+179q^{21}+111q^{20}-68q^{19}-163q^{18}-96q^{17}+41q^{16}+145q^{15}+105q^{14}-39q^{13}-122q^{12}-85q^{11}+3q^{10}+98q^{9}+93q^{8}-2q^{7}-68q^{6}-64q^{5}-32q^{4}+39q^{3}+64q^{2}+28q-12-28q^{-1}-43q^{-2}-14q^{-3}+19q^{-4}+30q^{-5}+28q^{-6}+11q^{-7}-23q^{-8}-37q^{-9}-23q^{-10}+6q^{-11}+32q^{-12}+36q^{-13}+7q^{-14}-25q^{-15}-34q^{-16}-19q^{-17}+11q^{-18}+30q^{-19}+25q^{-20}-4q^{-21}-20q^{-22}-20q^{-23}-7q^{-24}+11q^{-25}+18q^{-26}+7q^{-27}-6q^{-28}-8q^{-29}-6q^{-30}-2q^{-31}+7q^{-32}+5q^{-33}-q^{-34}-2q^{-35}-q^{-36}-2q^{-37}+q^{-38}+2q^{-39}-q^{-40}$
6	$q^{111}-2q^{110}+q^{108}-q^{107}+2q^{106}+5q^{104}-7q^{103}-3q^{102}+2q^{101}-5q^{100}+5q^{99}+5q^{98}+16q^{97}-15q^{96}-9q^{95}-q^{94}-15q^{93}+7q^{92}+17q^{91}+39q^{90}-22q^{89}-18q^{88}-12q^{87}-40q^{86}+3q^{85}+40q^{84}+86q^{83}-14q^{82}-28q^{81}-43q^{80}-103q^{79}-22q^{78}+74q^{77}+176q^{76}+43q^{75}-17q^{74}-98q^{73}-232q^{72}-108q^{71}+92q^{70}+312q^{69}+176q^{68}+58q^{67}-141q^{66}-413q^{65}-272q^{64}+43q^{63}+436q^{62}+354q^{61}+206q^{60}-118q^{59}-569q^{58}-463q^{57}-72q^{56}+485q^{55}+487q^{54}+366q^{53}-32q^{52}-635q^{51}-594q^{50}-195q^{49}+465q^{48}+529q^{47}+466q^{46}+58q^{45}-627q^{44}-637q^{43}-267q^{42}+426q^{41}+508q^{40}+493q^{39}+106q^{38}-589q^{37}-623q^{36}-284q^{35}+394q^{34}+461q^{33}+481q^{32}+124q^{31}-535q^{30}-583q^{29}-286q^{28}+351q^{27}+399q^{26}+456q^{25}+147q^{24}-449q^{23}-524q^{22}-297q^{21}+270q^{20}+312q^{19}+424q^{18}+189q^{17}-321q^{16}-437q^{15}-309q^{14}+159q^{13}+191q^{12}+364q^{11}+230q^{10}-166q^{9}-310q^{8}-289q^{7}+50q^{6}+47q^{5}+262q^{4}+229q^{3}-26q^{2}-157q-210-8q^{-1}-78q^{-2}+127q^{-3}+162q^{-4}+48q^{-5}-26q^{-6}-89q^{-7}+10q^{-8}-128q^{-9}+11q^{-10}+51q^{-11}+37q^{-12}+27q^{-13}+10q^{-14}+73q^{-15}-92q^{-16}-31q^{-17}-32q^{-18}-17q^{-19}+3q^{-20}+34q^{-21}+108q^{-22}-23q^{-23}-7q^{-24}-42q^{-25}-43q^{-26}-39q^{-27}+4q^{-28}+83q^{-29}+13q^{-30}+23q^{-31}-13q^{-32}-24q^{-33}-44q^{-34}-21q^{-35}+37q^{-36}+8q^{-37}+23q^{-38}+6q^{-39}-q^{-40}-22q^{-41}-18q^{-42}+10q^{-43}-q^{-44}+9q^{-45}+5q^{-46}+5q^{-47}-7q^{-48}-7q^{-49}+3q^{-50}-2q^{-51}+2q^{-52}+q^{-53}+2q^{-54}-q^{-55}-2q^{-56}+q^{-57}$
7	$-q^{147}+2q^{146}-q^{144}+q^{143}-2q^{142}-2q^{140}-q^{139}+7q^{138}+q^{137}-q^{136}+4q^{135}-7q^{134}-3q^{133}-7q^{132}-6q^{131}+18q^{130}+6q^{129}+2q^{128}+9q^{127}-13q^{126}-8q^{125}-18q^{124}-17q^{123}+32q^{122}+16q^{121}+11q^{120}+19q^{119}-26q^{118}-19q^{117}-35q^{116}-33q^{115}+50q^{114}+42q^{113}+44q^{112}+36q^{111}-59q^{110}-64q^{109}-82q^{108}-65q^{107}+84q^{106}+117q^{105}+145q^{104}+100q^{103}-110q^{102}-182q^{101}-239q^{100}-177q^{99}+112q^{98}+273q^{97}+387q^{96}+302q^{95}-98q^{94}-371q^{93}-566q^{92}-481q^{91}+35q^{90}+446q^{89}+775q^{88}+721q^{87}+87q^{86}-499q^{85}-990q^{84}-979q^{83}-256q^{82}+491q^{81}+1163q^{80}+1255q^{79}+467q^{78}-436q^{77}-1292q^{76}-1498q^{75}-682q^{74}+333q^{73}+1361q^{72}+1686q^{71}+877q^{70}-205q^{69}-1364q^{68}-1814q^{67}-1047q^{66}+81q^{65}+1343q^{64}+1884q^{63}+1141q^{62}+27q^{61}-1276q^{60}-1903q^{59}-1218q^{58}-113q^{57}+1238q^{56}+1897q^{55}+1230q^{54}+155q^{53}-1174q^{52}-1863q^{51}-1238q^{50}-193q^{49}+1136q^{48}+1838q^{47}+1215q^{46}+200q^{45}-1082q^{44}-1784q^{43}-1203q^{42}-233q^{41}+1039q^{40}+1741q^{39}+1174q^{38}+253q^{37}-952q^{36}-1669q^{35}-1174q^{34}-305q^{33}+876q^{32}+1589q^{31}+1138q^{30}+369q^{29}-731q^{28}-1485q^{27}-1145q^{26}-447q^{25}+610q^{24}+1354q^{23}+1087q^{22}+539q^{21}-412q^{20}-1199q^{19}-1075q^{18}-619q^{17}+259q^{16}+1017q^{15}+966q^{14}+687q^{13}-38q^{12}-808q^{11}-897q^{10}-728q^{9}-101q^{8}+599q^{7}+719q^{6}+716q^{5}+273q^{4}-369q^{3}-582q^{2}-668q-345+191q^{-1}+370q^{-2}+554q^{-3}+398q^{-4}-24q^{-5}-204q^{-6}-425q^{-7}-363q^{-8}-60q^{-9}+41q^{-10}+264q^{-11}+294q^{-12}+103q^{-13}+60q^{-14}-126q^{-15}-191q^{-16}-75q^{-17}-113q^{-18}+15q^{-19}+86q^{-20}+25q^{-21}+109q^{-22}+35q^{-23}-q^{-24}+48q^{-25}-67q^{-26}-60q^{-27}-43q^{-28}-89q^{-29}+14q^{-30}+23q^{-31}+49q^{-32}+128q^{-33}+34q^{-34}-34q^{-36}-108q^{-37}-52q^{-38}-45q^{-39}-4q^{-40}+91q^{-41}+66q^{-42}+49q^{-43}+20q^{-44}-52q^{-45}-37q^{-46}-53q^{-47}-45q^{-48}+25q^{-49}+29q^{-50}+42q^{-51}+33q^{-52}-9q^{-53}-5q^{-54}-20q^{-55}-33q^{-56}-6q^{-57}+2q^{-58}+16q^{-59}+19q^{-60}-2q^{-61}+4q^{-62}-q^{-63}-11q^{-64}-5q^{-65}-4q^{-66}+4q^{-67}+7q^{-68}-q^{-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, 7]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 16], X[5, 13, 6, 12], X[7, 15, 8, 14], X[13, 7, 14, 6], X[15, 9, 16, 8], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[8, 7]]
Out[3]=	GaussCode[-1, 8, -2, 1, -4, 6, -5, 7, -8, 2, -3, 4, -6, 5, -7, 3]
In[4]:=	DTCode[Knot[8, 7]]
Out[4]=	DTCode[4, 10, 12, 14, 2, 16, 6, 8]
In[5]:=	br = BR[Knot[8, 7]]
Out[5]=	BR[3, {1, 1, 1, 1, -2, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 8}
In[7]:=	BraidIndex[Knot[8, 7]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[8, 7]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[8, 7]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 2, {3, 6}, 1}
In[10]:=	alex = Alexander[Knot[8, 7]][t]
Out[10]=	-3 3 5 2 3 -5 + t - -- + - + 5 t - 3 t + t 2 t t
In[11]:=	Conway[Knot[8, 7]][z]
Out[11]=	2 4 6 1 + 2 z + 3 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 7], Knot[11, NonAlternating, 24]}
In[13]:=	{KnotDet[Knot[8, 7]], KnotSignature[Knot[8, 7]]}
Out[13]=	{23, 2}
In[14]:=	Jones[Knot[8, 7]][q]
Out[14]=	-2 2 2 3 4 5 6 -2 - q + - + 4 q - 4 q + 4 q - 3 q + 2 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 7]}
In[16]:=	A2Invariant[Knot[8, 7]][q]
Out[16]=	-6 2 6 10 14 18 1 - q + 2 q + 2 q + q - q - q
In[17]:=	HOMFLYPT[Knot[8, 7]][a, z]
Out[17]=	2 2 4 4 6 2 4 2 3 z 8 z 4 z 5 z z -1 - -- + -- - 3 z - ---- + ---- - z - -- + ---- + -- 4 2 4 2 4 2 2 a a a a a a a
In[18]:=	Kauffman[Knot[8, 7]][a, z]
Out[18]=	2 2 2 3 2 4 z 2 z 2 z 2 2 z 4 z 12 z z -1 - -- - -- - -- + --- + --- + a z + 6 z - ---- + ---- + ----- + -- - 4 2 7 3 a 6 4 2 7 a a a a a a a a 3 3 3 4 4 4 5 5 z 2 z 3 z 3 4 2 z 3 z 12 z 2 z z -- - ---- - ---- - 3 a z - 7 z + ---- - ---- - ----- + ---- - -- + 5 3 a 6 4 2 5 a a a a a a a 6 6 7 7 5 6 2 z 4 z z z a z + 2 z + ---- + ---- + -- + -- 4 2 3 a a a a
In[19]:=	{Vassiliev[2][Knot[8, 7]], Vassiliev[3][Knot[8, 7]]}
Out[19]=	{2, 2}
In[20]:=	Kh[Knot[8, 7]][q, t]
Out[20]=	3 1 1 1 1 q 3 5 3 q + 2 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 2 q t + 2 q t + q t + 2 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[8, 7], 2][q]
Out[21]=	-7 2 -5 5 4 4 10 2 3 4 -3 + q - -- - q + -- - -- - -- + -- - 9 q + 14 q - 2 q - 14 q + 6 4 3 2 q q q q q 5 7 8 9 10 11 12 13 14 16 q - 16 q + 14 q + q - 12 q + 9 q + q - 6 q + 4 q - 16 17 2 q + q

See/edit the Rolfsen_Splice_Template.