L11a267

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_22,12,9,11 X_2,9,3,10 X_16,14,17,13 X_14,5,15,6 X_4,15,5,16 X_20,18,21,17 X_18,7,19,8 X_6,19,7,20 X_8,22,1,21
Gauss code	{1, -4, 2, -7, 6, -10, 9, -11}, {4, -1, 3, -2, 5, -6, 7, -5, 8, -9, 10, -8, 11, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{3}-3u^{3}v^{2}+3u^{3}v-u^{3}-3u^{2}v^{3}+7u^{2}v^{2}-6u^{2}v+3u^{2}+3uv^{3}-6uv^{2}+7uv-3u-v^{3}+3v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+7q^{7/2}-11q^{5/2}+16q^{3/2}-17{\sqrt {q}}+{\frac {16}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+2z^{3}a^{-3}-6a^{3}z-a^{3}z^{-1}-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+9az^{3}-6z^{3}a^{-1}+6az-3za^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-4a^{5}z^{5}+4z^{5}a^{-5}+6a^{5}z^{3}-3z^{3}a^{-5}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+7z^{6}a^{-4}+12a^{4}z^{4}-7z^{4}a^{-4}-3a^{4}z^{2}+3z^{2}a^{-4}-a^{4}+3a^{3}z^{9}-5a^{3}z^{7}+8z^{7}a^{-3}-10a^{3}z^{5}-6z^{5}a^{-3}+21a^{3}z^{3}-z^{3}a^{-3}-10a^{3}z-za^{-3}+a^{3}z^{-1}+a^{2}z^{10}+8a^{2}z^{8}+7z^{8}a^{-2}-33a^{2}z^{6}-2z^{6}a^{-2}+30a^{2}z^{4}-14z^{4}a^{-2}-6a^{2}z^{2}+11z^{2}a^{-2}+7az^{9}+4z^{9}a^{-1}-8az^{7}+6z^{7}a^{-1}-24az^{5}-28z^{5}a^{-1}+36az^{3}+23z^{3}a^{-1}-13az-8za^{-1}+z^{10}+12z^{8}-31z^{6}+12z^{4}+5z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

4

1

-3

6

7

3

4

9

4

-5

2

8

7

1

0

9

10

1

-2

6

7

-1

-4

4

9

5

-6

3

6

-3

-8

1

5

4

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

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