L11a326

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-v+1\right)\left(u^{2}v^{2}-uv^{2}+2uv-u+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+9q^{5/2}-15q^{3/2}+19{\sqrt {q}}-{\frac {24}{\sqrt {q}}}+{\frac {23}{q^{3/2}}}-{\frac {20}{q^{5/2}}}+{\frac {15}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+4a^{3}z^{5}+5a^{3}z^{3}+2a^{3}z-az^{9}-6az^{7}+z^{7}a^{-1}-13az^{5}+4z^{5}a^{-1}-12az^{3}+5z^{3}a^{-1}-4az+2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+8a^{5}z^{7}-12a^{5}z^{5}+5a^{5}z^{3}-a^{5}z+10a^{4}z^{8}-16a^{4}z^{6}+z^{6}a^{-4}+10a^{4}z^{4}-2z^{4}a^{-4}-3a^{4}z^{2}+z^{2}a^{-4}+8a^{3}z^{9}-9a^{3}z^{7}+4z^{7}a^{-3}+2a^{3}z^{5}-9z^{5}a^{-3}+a^{3}z^{3}+5z^{3}a^{-3}-za^{-3}+3a^{2}z^{10}+10a^{2}z^{8}+7z^{8}a^{-2}-31a^{2}z^{6}-15z^{6}a^{-2}+28a^{2}z^{4}+7z^{4}a^{-2}-8a^{2}z^{2}-z^{2}a^{-2}+15az^{9}+7z^{9}a^{-1}-33az^{7}-12z^{7}a^{-1}+27az^{5}+3z^{5}a^{-1}-10az^{3}+2az+az^{-1}+a^{-1}z^{-1}+3z^{10}+7z^{8}-27z^{6}+22z^{4}-6z^{2}-1$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

1

-5

4

9

3

6

2

10

6

-4

0

14

9

5

-2

11

12

1

-4

9

12

-3

-6

6

11

5

-8

3

9

-6

-10

1

6

5

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$