8 1

7_7

8_2

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8 1 Quick Notes

8 1 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,16,6,1 X7,14,8,15 X13,8,14,9 X15,6,16,7
Gauss code	-1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5
Dowker-Thistlethwaite code	4 10 16 14 12 2 8 6
Conway Notation	[62]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-3] Hyperbolic Volume 3.42721 A-Polynomial See Data:8 1/A-polynomial

[edit Notes for 8 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for 8 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t+7-3t^{-1}$
Conway polynomial	$1-3z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 13, 0 }
Jones polynomial	$q^{2}-q+2-2q^{-1}+2q^{-2}-2q^{-3}+q^{-4}-q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-z^{2}a^{4}-a^{4}-z^{2}a^{2}-z^{2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{5}z^{7}+a^{3}z^{7}+a^{6}z^{6}+2a^{4}z^{6}+a^{2}z^{6}-5a^{5}z^{5}-4a^{3}z^{5}+az^{5}-5a^{6}z^{4}-8a^{4}z^{4}-2a^{2}z^{4}+z^{4}+7a^{5}z^{3}+5a^{3}z^{3}-az^{3}+z^{3}a^{-1}+6a^{6}z^{2}+7a^{4}z^{2}+z^{2}a^{-2}-3a^{5}z-3a^{3}z-a^{6}-a^{4}-a^{-2}$
The A2 invariant	$q^{20}+q^{18}-q^{12}-q^{10}+q^{-2}+q^{-6}+q^{-8}$
The G2 invariant	$q^{94}+q^{90}-q^{88}+2q^{80}-2q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^{8}+q^{-2}-q^{-6}+q^{-10}+q^{-14}+q^{-20}+q^{-24}+q^{-28}+q^{-34}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 8_1.

A1 Invariants.

Weight	Invariant
1	$q^{13}-q^{7}+q^{-1}+q^{-5}$
2	$q^{38}-q^{34}-q^{28}+q^{24}+q^{12}+q^{10}-1+q^{-8}+q^{-14}$
3	$q^{75}-q^{71}-q^{69}+q^{65}-q^{61}+q^{57}+q^{55}-q^{51}+q^{37}+q^{35}-q^{31}-q^{25}-q^{23}-q^{17}+q^{13}+2q^{11}+q^{9}+q^{7}-q^{5}+q+q^{-1}-q^{-5}+q^{-9}-q^{-13}-q^{-15}+q^{-19}+q^{-27}$
4	$q^{124}-q^{120}-q^{118}-q^{116}+q^{114}+q^{112}+q^{110}-2q^{106}+q^{102}+q^{100}+q^{98}-q^{96}-q^{94}-q^{92}+q^{88}+q^{74}+q^{72}-q^{68}-2q^{66}+q^{62}-q^{58}-2q^{56}+2q^{52}+q^{50}-q^{46}+2q^{42}+q^{40}+q^{32}-q^{28}-q^{26}+q^{22}-q^{18}-q^{16}-q^{14}-q^{12}+q^{10}+2q^{8}+q^{6}-q^{4}-2q^{2}+2q^{-2}+4q^{-4}+q^{-6}-2q^{-8}-q^{-10}+q^{-12}+3q^{-14}-2q^{-18}-q^{-20}+2q^{-24}-q^{-28}-q^{-30}-q^{-32}+q^{-34}+q^{-44}$
5	$q^{185}-q^{181}-q^{179}-q^{177}+q^{173}+2q^{171}+q^{169}-q^{165}-2q^{163}-q^{161}+q^{159}+2q^{157}+q^{155}-q^{151}-2q^{149}-q^{147}+q^{143}+q^{141}+q^{139}-q^{135}+q^{123}+q^{121}-q^{117}-2q^{115}-2q^{113}+2q^{109}+2q^{107}-2q^{103}-2q^{101}-q^{99}+2q^{97}+4q^{95}+2q^{93}-q^{91}-2q^{89}-2q^{87}+2q^{83}+2q^{81}-2q^{77}-2q^{75}+q^{71}+2q^{69}+q^{67}-q^{65}-2q^{63}-q^{61}+q^{57}+q^{55}-q^{53}-2q^{51}-q^{49}+q^{45}-q^{41}+q^{37}+3q^{35}+2q^{33}-q^{25}+q^{23}+q^{21}+q^{19}+q^{17}-3q^{13}-3q^{11}-q^{9}+q^{7}+3q^{5}+2q^{3}-3q^{-1}-3q^{-3}+3q^{-7}+3q^{-9}-2q^{-13}-2q^{-15}+3q^{-19}+2q^{-21}-2q^{-25}+2q^{-29}+2q^{-31}+q^{-33}-q^{-35}-2q^{-37}-q^{-39}+q^{-41}+q^{-43}-q^{-49}-q^{-51}+q^{-65}$
6	$q^{258}-q^{254}-q^{252}-q^{250}+2q^{244}+2q^{242}+q^{240}-q^{236}-2q^{234}-3q^{232}+q^{228}+2q^{226}+2q^{224}+q^{222}-3q^{218}-2q^{216}-q^{214}+q^{210}+2q^{208}+2q^{206}-q^{200}-q^{198}-q^{196}+q^{192}+q^{184}+q^{182}-q^{178}-2q^{176}-2q^{174}-2q^{172}+q^{170}+3q^{168}+3q^{166}+2q^{164}-q^{162}-3q^{160}-4q^{158}-q^{156}+2q^{154}+4q^{152}+5q^{150}+2q^{148}-2q^{146}-5q^{144}-4q^{142}-2q^{140}+q^{138}+4q^{136}+3q^{134}+q^{132}-2q^{130}-3q^{128}-3q^{126}-q^{124}+3q^{122}+3q^{120}+3q^{118}+q^{116}-q^{114}-3q^{112}-4q^{110}-q^{108}+q^{106}+2q^{104}+2q^{102}+q^{100}-2q^{98}-4q^{96}-2q^{94}+q^{92}+3q^{90}+3q^{88}+2q^{86}-q^{84}-4q^{82}-q^{80}+2q^{78}+3q^{76}+3q^{74}+2q^{72}-q^{70}-3q^{68}-q^{66}+q^{64}+q^{62}-q^{58}-2q^{56}-2q^{54}+q^{50}-q^{46}-2q^{44}-2q^{42}-q^{40}+q^{38}+2q^{36}+3q^{34}+2q^{32}+q^{30}-q^{26}-3q^{24}-3q^{22}+3q^{18}+4q^{16}+4q^{14}+3q^{12}-2q^{10}-4q^{8}-4q^{6}-q^{4}+2q^{2}+5+6q^{-2}+q^{-4}-2q^{-6}-5q^{-8}-4q^{-10}-2q^{-12}+2q^{-14}+4q^{-16}+q^{-18}-2q^{-22}-q^{-24}-q^{-26}+q^{-30}-q^{-32}+q^{-34}+q^{-36}+2q^{-38}-q^{-42}-q^{-44}-2q^{-46}+q^{-48}+2q^{-50}+3q^{-52}+q^{-54}-q^{-58}-2q^{-60}+q^{-66}-q^{-74}-q^{-78}+q^{-90}$

A2 Invariants.

Weight	Invariant
1,0	$q^{20}+q^{18}-q^{12}-q^{10}+q^{-2}+q^{-6}+q^{-8}$
1,1	$q^{52}+2q^{48}-2q^{46}+2q^{44}-4q^{42}+2q^{40}-2q^{38}-2q^{32}+2q^{30}-3q^{28}+4q^{26}-2q^{24}+4q^{22}-2q^{20}+4q^{18}+2q^{14}+2q^{12}-2q^{6}-2q^{4}-2q^{2}-2+q^{-4}+2q^{-8}+2q^{-12}+2q^{-16}+q^{-20}$
2,0	$q^{52}+q^{50}+q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}+q^{18}+2q^{16}+q^{14}+q^{12}+q^{10}-q^{4}-2q^{2}-2-q^{-2}+q^{-4}+q^{-10}+q^{-12}+q^{-16}+q^{-18}+q^{-20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{40}+q^{36}-q^{32}-q^{30}-q^{28}+q^{24}+2q^{22}+q^{20}+2q^{18}-q^{14}-q^{12}-q^{10}-q^{8}-q^{6}+q^{-4}+q^{-8}+2q^{-10}+q^{-12}+q^{-16}$
1,0,0	$q^{27}+q^{25}+q^{23}-q^{17}-q^{15}-q^{13}+q^{-3}+q^{-7}+q^{-9}+q^{-11}$

B2 Invariants.

Weight	Invariant
0,1	$q^{40}+q^{36}+q^{32}-q^{30}+q^{28}-q^{24}-q^{20}-2q^{16}+q^{14}-q^{12}+q^{10}-q^{8}+q^{6}+q^{-4}+q^{-8}+q^{-12}+q^{-16}$
1,0	$q^{66}+q^{58}-q^{54}-q^{52}-q^{46}-q^{44}+q^{40}+q^{38}+q^{36}+q^{32}+q^{30}+q^{28}-q^{18}-q^{16}-q^{10}-q^{8}+q^{-6}+q^{-14}+q^{-16}+q^{-18}+q^{-26}$

G2 Invariants.

Weight	Invariant
1,0	$q^{94}+q^{90}-q^{88}+2q^{80}-2q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^{8}+q^{-2}-q^{-6}+q^{-10}+q^{-14}+q^{-20}+q^{-24}+q^{-28}+q^{-34}+q^{-38}$

.

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 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t+7-3t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$1-3z^{2}$

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]=

{ 13, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{5}z^{7}+a^{3}z^{7}+a^{6}z^{6}+2a^{4}z^{6}+a^{2}z^{6}-5a^{5}z^{5}-4a^{3}z^{5}+az^{5}-5a^{6}z^{4}-8a^{4}z^{4}-2a^{2}z^{4}+z^{4}+7a^{5}z^{3}+5a^{3}z^{3}-az^{3}+z^{3}a^{-1}+6a^{6}z^{2}+7a^{4}z^{2}+z^{2}a^{-2}-3a^{5}z-3a^{3}z-a^{6}-a^{4}-a^{-2}$

Vassiliev invariants

V2 and V3:

(-3, 3)

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 $-12$ $24$ $72$ $66$ $38$ $-288$ $-432$ $-96$ $-104$ $-288$ $288$ $-792$ $-456$ ${\frac {1009}{10}}$ ${\frac {2626}{15}}$ $-{\frac {6182}{15}}$ ${\frac {943}{6}}$ $-{\frac {1551}{10}}$

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 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-6	-5	-4	-3	-2	-1	0	1	2	χ
5									1	1
3										0
1							2	1		1
-1						1	1			0
-3					1	1				0
-5				1	1					0
-7				1						-1
-9		1	1							0
-11										0
-13	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, 1]]
Out[2]=	8
In[3]:=	PD[Knot[8, 1]]
Out[3]=	PD[X[1, 4, 2, 5], X[9, 12, 10, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[5, 16, 6, 1], X[7, 14, 8, 15], X[13, 8, 14, 9], X[15, 6, 16, 7]]
In[4]:=	GaussCode[Knot[8, 1]]
Out[4]=	GaussCode[-1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5]
In[5]:=	BR[Knot[8, 1]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -3, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[8, 1]][t]
Out[6]=	3 7 - - - 3 t t
In[7]:=	Conway[Knot[8, 1]][z]
Out[7]=	2 1 - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 1]}
In[9]:=	{KnotDet[Knot[8, 1]], KnotSignature[Knot[8, 1]]}
Out[9]=	{13, 0}
In[10]:=	J=Jones[Knot[8, 1]][q]
Out[10]=	-6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 1], Knot[11, NonAlternating, 70]}
In[12]:=	A2Invariant[Knot[8, 1]][q]
Out[12]=	-20 -18 -12 -10 2 6 8 q + q - q - q + q + q + q
In[13]:=	Kauffman[Knot[8, 1]][a, z]
Out[13]=	2 3 -2 4 6 3 5 z 4 2 6 2 z 3 -a - a - a - 3 a z - 3 a z + -- + 7 a z + 6 a z + -- - a z + 2 a a 3 3 5 3 4 2 4 4 4 6 4 5 5 a z + 7 a z + z - 2 a z - 8 a z - 5 a z + a z - 3 5 5 5 2 6 4 6 6 6 3 7 5 7 4 a z - 5 a z + a z + 2 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[8, 1]], Vassiliev[3][Knot[8, 1]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[8, 1]][q, t]
Out[15]=	1 1 1 1 1 1 1 1 - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 9 5 9 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 1 1 5 2 ---- + --- + q t + q t 3 q t q t