L11a356

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{4}v^{3}-u^{4}v^{2}+u^{3}v^{4}-3u^{3}v^{3}+4u^{3}v^{2}-2u^{3}v-u^{2}v^{4}+4u^{2}v^{3}-7u^{2}v^{2}+4u^{2}v-u^{2}-2uv^{3}+4uv^{2}-3uv+u-v^{2}+v}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}-q^{7/2}+{\frac {10}{q^{7/2}}}+3q^{5/2}-{\frac {13}{q^{5/2}}}-6q^{3/2}+{\frac {13}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+9{\sqrt {q}}-{\frac {12}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+5a^{3}z^{5}+4az^{5}-z^{5}a^{-1}-4a^{5}z^{3}+9a^{3}z^{3}+4az^{3}-3z^{3}a^{-1}-4a^{5}z+6a^{3}z-az-2za^{-1}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+4a^{8}z^{2}+2a^{7}z^{7}-7a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+2a^{6}z^{8}-4a^{6}z^{6}+a^{6}z^{2}+2a^{5}z^{9}-5a^{5}z^{7}+9a^{5}z^{5}-12a^{5}z^{3}+3a^{5}z+a^{4}z^{10}-a^{4}z^{6}+2a^{4}z^{4}-4a^{4}z^{2}+5a^{3}z^{9}-16a^{3}z^{7}+30a^{3}z^{5}+z^{5}a^{-3}-27a^{3}z^{3}-2z^{3}a^{-3}+10a^{3}z-a^{3}z^{-1}+a^{2}z^{10}+3a^{2}z^{8}-11a^{2}z^{6}+3z^{6}a^{-2}+16a^{2}z^{4}-6z^{4}a^{-2}-6a^{2}z^{2}+z^{2}a^{-2}+a^{2}+3az^{9}-4az^{7}+5z^{7}a^{-1}+az^{5}-12z^{5}a^{-1}+3az^{3}+9z^{3}a^{-1}+2az-3za^{-1}-az^{-1}+5z^{8}-12z^{6}+12z^{4}-4z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

1

3

2

5

2

-3

0

7

4

3

-2

7

6

-1

-4

6

0

-6

5

8

3

-8

3

5

-2

-10

1

5

4

-12

1

3

-2

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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