8 8

8_7

8_9

Visit 8 8's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 8's page at Knotilus! Visit 8 8's page at the original Knot Atlas!

8 8 Quick Notes

8 8 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,1,10,16 X15,11,16,10 X7283
Gauss code	-1, 8, -2, 1, -4, 5, -8, 2, -6, 7, -3, 4, -5, 3, -7, 6
Dowker-Thistlethwaite code	4 8 12 2 16 14 6 10
Conway Notation	[2312]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-6] Hyperbolic Volume 7.80134 A-Polynomial See Data:8 8/A-polynomial

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

Polynomial invariants

Alexander polynomial	$2t^{2}-6t+9-6t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$-q^{5}+2q^{4}-3q^{3}+4q^{2}-4q+5-3q^{-1}+2q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+2z^{2}-a^{2}+a^{-2}-a^{-4}+2$
Kauffman polynomial (db, data sources)	$z^{7}a^{-1}+z^{7}a^{-3}+4z^{6}a^{-2}+2z^{6}a^{-4}+2z^{6}+2az^{5}+z^{5}a^{-1}+z^{5}a^{-5}+2a^{2}z^{4}-9z^{4}a^{-2}-6z^{4}a^{-4}-z^{4}+a^{3}z^{3}-3z^{3}a^{-1}-5z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+5z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}-a^{3}z-az+za^{-1}+3za^{-3}+2za^{-5}+a^{2}-a^{-2}-a^{-4}+2$
The A2 invariant	$-q^{10}-q^{4}+2q^{2}+1+2q^{-2}+q^{-4}+q^{-8}-q^{-10}-q^{-16}$
The G2 invariant	$q^{52}-q^{50}+2q^{48}-2q^{46}-3q^{40}+4q^{38}-5q^{36}+4q^{34}-5q^{32}+q^{30}+3q^{28}-6q^{26}+8q^{24}-9q^{22}+7q^{20}-4q^{18}-3q^{16}+7q^{14}-8q^{12}+9q^{10}-q^{8}-3q^{6}+6q^{4}-3q^{2}+1+6q^{-2}-10q^{-4}+12q^{-6}-5q^{-8}+10q^{-12}-14q^{-14}+17q^{-16}-11q^{-18}+3q^{-20}+4q^{-22}-9q^{-24}+14q^{-26}-10q^{-28}+6q^{-30}+q^{-32}-5q^{-34}+7q^{-36}-5q^{-38}+4q^{-42}-8q^{-44}+7q^{-46}-3q^{-48}-4q^{-50}+10q^{-52}-13q^{-54}+10q^{-56}-5q^{-58}-4q^{-60}+7q^{-62}-10q^{-64}+9q^{-66}-5q^{-68}+2q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 8_8.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+q^{5}-q^{3}+2q+q^{-1}+q^{-5}-q^{-7}+q^{-9}-q^{-11}$
2	$q^{20}-q^{18}-q^{16}+2q^{14}-3q^{12}-q^{10}+5q^{8}-3q^{6}-2q^{4}+5q^{2}-q^{-2}+q^{-4}+3q^{-6}-2q^{-10}+3q^{-12}-5q^{-16}+2q^{-18}+2q^{-20}-4q^{-22}+q^{-24}+3q^{-26}-2q^{-28}-q^{-30}+q^{-32}$
3	$-q^{39}+q^{37}+q^{35}-q^{31}+q^{29}+q^{27}-4q^{25}-q^{23}+5q^{21}+2q^{19}-9q^{17}-4q^{15}+10q^{13}+7q^{11}-11q^{9}-7q^{7}+7q^{5}+11q^{3}-3q-7q^{-1}+q^{-3}+6q^{-5}+5q^{-7}-3q^{-9}-6q^{-11}+q^{-13}+9q^{-15}-q^{-17}-9q^{-19}-2q^{-21}+10q^{-23}+3q^{-25}-10q^{-27}-6q^{-29}+9q^{-31}+7q^{-33}-6q^{-35}-10q^{-37}+3q^{-39}+10q^{-41}-8q^{-45}-3q^{-47}+6q^{-49}+5q^{-51}-3q^{-53}-4q^{-55}+2q^{-59}+q^{-61}-q^{-63}$
4	$q^{64}-q^{62}-q^{60}-q^{56}+3q^{54}-q^{52}+q^{50}+2q^{48}-4q^{46}+q^{44}-4q^{42}+5q^{40}+10q^{38}-6q^{36}-8q^{34}-14q^{32}+9q^{30}+26q^{28}+q^{26}-17q^{24}-34q^{22}+3q^{20}+40q^{18}+18q^{16}-11q^{14}-43q^{12}-12q^{10}+30q^{8}+27q^{6}+10q^{4}-27q^{2}-21+6q^{-2}+18q^{-4}+18q^{-6}-3q^{-8}-16q^{-10}-15q^{-12}+2q^{-14}+19q^{-16}+14q^{-18}-10q^{-20}-25q^{-22}-3q^{-24}+20q^{-26}+24q^{-28}-8q^{-30}-33q^{-32}-8q^{-34}+19q^{-36}+31q^{-38}-3q^{-40}-35q^{-42}-17q^{-44}+9q^{-46}+37q^{-48}+12q^{-50}-26q^{-52}-24q^{-54}-9q^{-56}+29q^{-58}+24q^{-60}-4q^{-62}-19q^{-64}-26q^{-66}+8q^{-68}+21q^{-70}+13q^{-72}-q^{-74}-22q^{-76}-8q^{-78}+4q^{-80}+12q^{-82}+11q^{-84}-7q^{-86}-7q^{-88}-5q^{-90}+q^{-92}+6q^{-94}+q^{-96}-2q^{-100}-q^{-102}+q^{-104}$
5	$-q^{95}+q^{93}+q^{91}+q^{87}-q^{85}-3q^{83}-q^{81}+q^{79}+4q^{75}+3q^{73}-2q^{71}-6q^{69}-6q^{67}+9q^{63}+15q^{61}+7q^{59}-16q^{57}-27q^{55}-13q^{53}+18q^{51}+43q^{49}+34q^{47}-19q^{45}-68q^{43}-55q^{41}+10q^{39}+83q^{37}+87q^{35}+7q^{33}-94q^{31}-118q^{29}-36q^{27}+85q^{25}+139q^{23}+63q^{21}-64q^{19}-131q^{17}-93q^{15}+31q^{13}+120q^{11}+99q^{9}+3q^{7}-78q^{5}-93q^{3}-33q+47q^{-1}+77q^{-3}+48q^{-5}-9q^{-7}-50q^{-9}-53q^{-11}-20q^{-13}+30q^{-15}+53q^{-17}+35q^{-19}-10q^{-21}-54q^{-23}-52q^{-25}+3q^{-27}+59q^{-29}+57q^{-31}+2q^{-33}-62q^{-35}-71q^{-37}-3q^{-39}+74q^{-41}+77q^{-43}+9q^{-45}-78q^{-47}-92q^{-49}-16q^{-51}+80q^{-53}+103q^{-55}+31q^{-57}-73q^{-59}-114q^{-61}-50q^{-63}+57q^{-65}+111q^{-67}+73q^{-69}-30q^{-71}-106q^{-73}-89q^{-75}+82q^{-79}+96q^{-81}+36q^{-83}-52q^{-85}-89q^{-87}-58q^{-89}+15q^{-91}+68q^{-93}+67q^{-95}+21q^{-97}-38q^{-99}-63q^{-101}-41q^{-103}+7q^{-105}+42q^{-107}+45q^{-109}+19q^{-111}-19q^{-113}-37q^{-115}-28q^{-117}-q^{-119}+20q^{-121}+25q^{-123}+15q^{-125}-6q^{-127}-17q^{-129}-14q^{-131}-3q^{-133}+5q^{-135}+9q^{-137}+7q^{-139}-q^{-141}-4q^{-143}-3q^{-145}-q^{-147}+2q^{-151}+q^{-153}-q^{-155}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}-q^{4}+2q^{2}+1+2q^{-2}+q^{-4}+q^{-8}-q^{-10}-q^{-16}$
1,1	$q^{28}-2q^{26}+4q^{24}-6q^{22}+9q^{20}-12q^{18}+14q^{16}-20q^{14}+19q^{12}-24q^{10}+22q^{8}-22q^{6}+18q^{4}-4q^{2}+20q^{-2}-27q^{-4}+44q^{-6}-48q^{-8}+56q^{-10}-52q^{-12}+48q^{-14}-40q^{-16}+24q^{-18}-15q^{-20}-4q^{-22}+14q^{-24}-24q^{-26}+31q^{-28}-32q^{-30}+30q^{-32}-24q^{-34}+17q^{-36}-12q^{-38}+6q^{-40}-2q^{-42}+q^{-44}$
2,0	$q^{26}-q^{22}+q^{18}-q^{16}-3q^{14}+2q^{10}-3q^{8}-3q^{6}+2q^{4}+2q^{2}+2+2q^{-2}+5q^{-4}+2q^{-6}+2q^{-8}+3q^{-10}+q^{-12}-2q^{-14}-4q^{-20}-3q^{-22}+q^{-26}-q^{-28}-q^{-30}+2q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-42}$
3,0	$-q^{48}+q^{44}+q^{42}-2q^{38}+2q^{36}+3q^{34}+q^{32}-4q^{30}-5q^{28}+4q^{26}+6q^{24}-q^{22}-10q^{20}-8q^{18}+8q^{16}+9q^{14}-3q^{12}-13q^{10}-8q^{8}+8q^{6}+6q^{4}-4+4q^{-2}+10q^{-4}+8q^{-6}+3q^{-8}+3q^{-10}+7q^{-12}+2q^{-14}-3q^{-16}-2q^{-18}+3q^{-20}+2q^{-22}-8q^{-24}-9q^{-26}+6q^{-30}+q^{-32}-10q^{-34}-7q^{-36}+4q^{-38}+8q^{-40}+2q^{-42}-8q^{-44}-5q^{-46}+3q^{-48}+7q^{-50}+4q^{-52}-5q^{-54}-4q^{-56}+5q^{-60}+4q^{-62}-2q^{-64}-3q^{-66}-3q^{-68}+q^{-70}+2q^{-72}+q^{-74}-q^{-78}$

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{22}+q^{20}-2q^{18}+2q^{16}-3q^{14}+3q^{12}-4q^{10}+3q^{8}-q^{6}+q^{4}+3q^{2}-2+6q^{-2}-5q^{-4}+7q^{-6}-6q^{-8}+5q^{-10}-4q^{-12}+3q^{-14}-q^{-16}-q^{-18}+3q^{-20}-3q^{-22}+3q^{-24}-3q^{-26}+3q^{-28}-3q^{-30}+q^{-32}-q^{-34}$
1,0	$q^{36}-q^{32}-q^{30}+q^{28}+2q^{26}-3q^{22}-3q^{20}+3q^{16}-4q^{12}-2q^{10}+2q^{8}+4q^{6}-q^{4}-q^{2}+2+5q^{-2}+q^{-4}-q^{-6}+3q^{-10}+2q^{-12}-q^{-14}-q^{-16}+2q^{-18}+2q^{-20}-q^{-22}-3q^{-24}+3q^{-28}+q^{-30}-4q^{-32}-4q^{-34}+q^{-36}+3q^{-38}-3q^{-42}-2q^{-44}+2q^{-46}+2q^{-48}-q^{-50}-q^{-52}+q^{-56}$

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

$q^{52}-q^{50}+2q^{48}-2q^{46}-3q^{40}+4q^{38}-5q^{36}+4q^{34}-5q^{32}+q^{30}+3q^{28}-6q^{26}+8q^{24}-9q^{22}+7q^{20}-4q^{18}-3q^{16}+7q^{14}-8q^{12}+9q^{10}-q^{8}-3q^{6}+6q^{4}-3q^{2}+1+6q^{-2}-10q^{-4}+12q^{-6}-5q^{-8}+10q^{-12}-14q^{-14}+17q^{-16}-11q^{-18}+3q^{-20}+4q^{-22}-9q^{-24}+14q^{-26}-10q^{-28}+6q^{-30}+q^{-32}-5q^{-34}+7q^{-36}-5q^{-38}+4q^{-42}-8q^{-44}+7q^{-46}-3q^{-48}-4q^{-50}+10q^{-52}-13q^{-54}+10q^{-56}-5q^{-58}-4q^{-60}+7q^{-62}-10q^{-64}+9q^{-66}-5q^{-68}+2q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

.

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 8"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{2}-6t+9-6t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+2z^{2}-a^{2}+a^{-2}-a^{-4}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{-1}+z^{7}a^{-3}+4z^{6}a^{-2}+2z^{6}a^{-4}+2z^{6}+2az^{5}+z^{5}a^{-1}+z^{5}a^{-5}+2a^{2}z^{4}-9z^{4}a^{-2}-6z^{4}a^{-4}-z^{4}+a^{3}z^{3}-3z^{3}a^{-1}-5z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+5z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}-a^{3}z-az+za^{-1}+3za^{-3}+2za^{-5}+a^{2}-a^{-2}-a^{-4}+2$

Vassiliev invariants

V2 and V3:

(2, 1)

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 $8$ $8$ $32$ ${\frac {124}{3}}$ $-{\frac {4}{3}}$ $64$ ${\frac {272}{3}}$ ${\frac {32}{3}}$ $8$ ${\frac {256}{3}}$ $32$ ${\frac {992}{3}}$ $-{\frac {32}{3}}$ ${\frac {6271}{15}}$ ${\frac {1516}{15}}$ ${\frac {484}{45}}$ ${\frac {113}{9}}$ $-{\frac {449}{15}}$

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=$0 is the signature of 8 8. 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
11									1	-1
9								1		1
7							2	1		-1
5						2	1			1
3					2	2				0
1				3	2					1
-1			1	3						2
-3		1	2							-1
-5		1								1
-7	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[8, 8]]
Out[2]=	8
In[3]:=	PD[Knot[8, 8]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[11, 15, 12, 14], X[5, 13, 6, 12], X[13, 7, 14, 6], X[9, 1, 10, 16], X[15, 11, 16, 10], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[8, 8]]
Out[4]=	GaussCode[-1, 8, -2, 1, -4, 5, -8, 2, -6, 7, -3, 4, -5, 3, -7, 6]
In[5]:=	BR[Knot[8, 8]]
Out[5]=	BR[4, {1, 1, 1, 2, -1, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[8, 8]][t]
Out[6]=	2 6 2 9 + -- - - - 6 t + 2 t 2 t t
In[7]:=	Conway[Knot[8, 8]][z]
Out[7]=	2 4 1 + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 8], Knot[10, 129], Knot[11, NonAlternating, 39], Knot[11, NonAlternating, 45], Knot[11, NonAlternating, 50], Knot[11, NonAlternating, 132]}
In[9]:=	{KnotDet[Knot[8, 8]], KnotSignature[Knot[8, 8]]}
Out[9]=	{25, 0}
In[10]:=	J=Jones[Knot[8, 8]][q]
Out[10]=	-3 2 3 2 3 4 5 5 - q + -- - - - 4 q + 4 q - 3 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 8], Knot[10, 129]}
In[12]:=	A2Invariant[Knot[8, 8]][q]
Out[12]=	-10 -4 2 2 4 8 10 16 1 - q - q + -- + 2 q + q + q - q - q 2 q
In[13]:=	Kauffman[Knot[8, 8]][a, z]
Out[13]=	2 2 -4 -2 2 2 z 3 z z 3 2 4 z 5 z 2 - a - a + a + --- + --- + - - a z - a z - z + ---- + ---- - 5 3 a 4 2 a a a a 3 3 3 4 4 2 2 3 z 5 z 3 z 3 3 4 6 z 9 z 2 4 2 a z - ---- - ---- - ---- + a z - z - ---- - ---- + 2 a z + 5 3 a 4 2 a a a a 5 5 6 6 7 7 z z 5 6 2 z 4 z z z -- + -- + 2 a z + 2 z + ---- + ---- + -- + -- 5 a 4 2 3 a a a a a
In[14]:=	{Vassiliev[2][Knot[8, 8]], Vassiliev[3][Knot[8, 8]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[8, 8]][q, t]
Out[15]=	3 1 1 1 2 1 3 - + 3 q + ----- + ----- + ----- + ---- + --- + 2 q t + 2 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 2 q t + 2 q t + q t + 2 q t + q t + q t + q t