L11n201: Difference between revisions

Revision as of 13:17, 30 August 2005

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_5,14,6,15 X_16,7,17,8 X_15,21,16,20 X_18,14,19,13 X_21,6,22,7 X_22,18,9,17 X_4,19,5,20 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -10, 2, -9, -3, 7, 4, -11}, {10, -1, 11, -2, 6, 3, -5, -4, 8, -6, 9, 5, -7, -8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v+u^{2}v^{4}-u^{2}v^{3}+2u^{2}v^{2}-2u^{2}v+u^{2}+uv^{5}-2uv^{4}+2uv^{3}-uv^{2}+uv+v^{4}}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {5}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{3}+3a^{7}z+2a^{7}z^{-1}-a^{5}z^{5}-4a^{5}z^{3}-6a^{5}z-3a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z+a^{3}z^{-1}+az^{3}+az$ (db)
Kauffman polynomial	$-z^{7}a^{9}+6z^{5}a^{9}-11z^{3}a^{9}+6za^{9}-z^{8}a^{8}+4z^{6}a^{8}-3z^{4}a^{8}-z^{2}a^{8}-z^{9}a^{7}+4z^{7}a^{7}-5z^{5}a^{7}+6z^{3}a^{7}-6za^{7}+2a^{7}z^{-1}-3z^{8}a^{6}+12z^{6}a^{6}-14z^{4}a^{6}+9z^{2}a^{6}-3a^{6}-z^{9}a^{5}+3z^{7}a^{5}-6z^{5}a^{5}+15z^{3}a^{5}-12za^{5}+3a^{5}z^{-1}-2z^{8}a^{4}+7z^{6}a^{4}-11z^{4}a^{4}+11z^{2}a^{4}-3a^{4}-2z^{7}a^{3}+5z^{5}a^{3}-5z^{3}a^{3}+za^{3}+a^{3}z^{-1}-z^{6}a^{2}-a^{2}-3z^{3}a+za-z^{2}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

1

-1

0

2

-2

3

2

-1

-4

2

1

-6

3

0

-8

2

0

-10

2

3

1

-12

1

2

-1

-14

2

-16

1

0

-18

1

Integral Khovanov Homology

(db, data source)

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