L10a75: Difference between revisions

Revision as of 12:22, 30 August 2005

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_16,5,17,6 X_20,11,7,12 X_18,13,19,14 X_14,17,15,18 X_12,19,13,20 X₆₇₁₈ X_4,15,5,16
Gauss code	{1, -2, 3, -10, 4, -9}, {9, -1, 2, -3, 5, -8, 6, -7, 10, -4, 7, -6, 8, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {3u^{2}v^{2}-2u^{2}v-2uv^{2}+5uv-2u-2v+3}{uv}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {5}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {2}{q^{23/2}}}-{\frac {1}{q^{25/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{11}+2za^{11}-z^{5}a^{9}-3z^{3}a^{9}-2za^{9}-z^{5}a^{7}-2z^{3}a^{7}+za^{7}+a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-4za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{15}z^{5}-3a^{15}z^{3}+a^{15}z+2a^{14}z^{6}-6a^{14}z^{4}+3a^{14}z^{2}+2a^{13}z^{7}-5a^{13}z^{5}+2a^{13}z^{3}+2a^{12}z^{8}-7a^{12}z^{6}+10a^{12}z^{4}-5a^{12}z^{2}+a^{11}z^{9}-3a^{11}z^{7}+5a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{8}-12a^{10}z^{6}+20a^{10}z^{4}-8a^{10}z^{2}+a^{9}z^{9}-4a^{9}z^{7}+9a^{9}z^{5}-6a^{9}z^{3}+a^{9}z+a^{8}z^{8}-2a^{8}z^{6}+2a^{8}z^{4}-a^{8}z^{2}+a^{7}z^{7}-a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+a^{6}z^{6}-2a^{6}z^{4}-a^{6}z^{2}+a^{6}+a^{5}z^{5}-4a^{5}z^{3}+4a^{5}z-a^{5}z^{-1}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

1

0

-8

2

-10

2

1

-1

-12

4

2

-14

2

0

-16

4

0

-18

2

3

1

-20

1

3

-2

-22

1

2

1

-24

1

-1

-26

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-10$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$