L11a323

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{2}-2u^{2}v+u^{2}-3uv^{2}+5uv-3u+v^{2}-2v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+9q^{5/2}-16q^{3/2}+21{\sqrt {q}}-{\frac {25}{\sqrt {q}}}+{\frac {24}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {16}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-2a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-3a^{3}z+za^{-3}+az^{7}+3az^{5}-2z^{5}a^{-1}+5az^{3}-4z^{3}a^{-1}+4az-3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-7a^{3}z^{9}-13az^{9}-6z^{9}a^{-1}-10a^{4}z^{8}-19a^{2}z^{8}-7z^{8}a^{-2}-16z^{8}-8a^{5}z^{7}-3a^{3}z^{7}+11az^{7}+2z^{7}a^{-1}-4z^{7}a^{-3}-4a^{6}z^{6}+12a^{4}z^{6}+43a^{2}z^{6}+14z^{6}a^{-2}-z^{6}a^{-4}+42z^{6}-a^{7}z^{5}+11a^{5}z^{5}+23a^{3}z^{5}+20az^{5}+18z^{5}a^{-1}+9z^{5}a^{-3}+5a^{6}z^{4}-4a^{4}z^{4}-27a^{2}z^{4}-8z^{4}a^{-2}+2z^{4}a^{-4}-28z^{4}+a^{7}z^{3}-7a^{5}z^{3}-20a^{3}z^{3}-23az^{3}-18z^{3}a^{-1}-7z^{3}a^{-3}-2a^{6}z^{2}-a^{4}z^{2}+5a^{2}z^{2}+z^{2}a^{-2}-z^{2}a^{-4}+6z^{2}+2a^{5}z+6a^{3}z+8az+6za^{-1}+2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

1

-5

4

10

3

7

2

11

6

-5

0

14

10

4

-2

12

13

1

-4

10

12

-2

-6

6

12

6

-8

3

10

-7

-10

1

6

5

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$