L10a87: Difference between revisions

Revision as of 13:17, 30 August 2005

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Other depiction

Mongolian ornament

Link Presentations

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

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

Polynomial invariants

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

2

1

-1

-8

4

-10

4

2

-2

-12

7

4

3

-14

5

4

-1

-16

6

7

-1

-18

4

6

2

-20

2

5

-3

-22

1

4

3

-24

2

-2

-26

1

Integral Khovanov Homology

(db, data source)

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