L11a15

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,10,19,9 X₈₄₉₃ X_14,6,15,5 X_22,16,5,15 X_20,18,21,17 X_16,22,17,21 X_10,20,11,19 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 11, -2, 3, -6, 7, -9, 8, -4, 10, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(3v^{2}-5v+3\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-12q^{9/2}+13q^{7/2}-14q^{5/2}+{\frac {1}{q^{5/2}}}+11q^{3/2}-{\frac {3}{q^{3/2}}}-q^{17/2}+4q^{15/2}-6q^{13/2}+9q^{11/2}-9{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-7}+a^{-7}z^{-1}+z^{5}a^{-5}+z^{3}a^{-5}-za^{-5}-2a^{-5}z^{-1}+z^{5}a^{-3}-za^{-3}+z^{5}a^{-1}-az^{3}+2z^{3}a^{-1}-az+3za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-9}-3z^{5}a^{-9}+z^{3}a^{-9}+4z^{8}a^{-8}-16z^{6}a^{-8}+15z^{4}a^{-8}-z^{2}a^{-8}-2a^{-8}+5z^{9}a^{-7}-20z^{7}a^{-7}+21z^{5}a^{-7}-6z^{3}a^{-7}+a^{-7}z^{-1}+2z^{10}a^{-6}-21z^{6}a^{-6}+24z^{4}a^{-6}+z^{2}a^{-6}-5a^{-6}+9z^{9}a^{-5}-33z^{7}a^{-5}+36z^{5}a^{-5}-13z^{3}a^{-5}-za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}-11z^{6}a^{-4}+9z^{4}a^{-4}+2z^{2}a^{-4}-3a^{-4}+4z^{9}a^{-3}-8z^{7}a^{-3}+9z^{5}a^{-3}-9z^{3}a^{-3}+2za^{-3}+4z^{8}a^{-2}-2z^{6}a^{-2}+a^{2}z^{4}-4z^{4}a^{-2}-a^{2}z^{2}+z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}+3az^{5}-4az^{3}-7z^{3}a^{-1}+2az+5za^{-1}-a^{-1}z^{-1}+4z^{6}-3z^{4}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

18

1

16

3

-3

14

3

1

2

12

6

3

-3

10

6

3

8

7

6

-1

6

7

6

1

4

7

3

2

5

7

-2

0

2

6

4

-2

1

3

-2

-4

2

-6

1

-1

Integral Khovanov Homology

(db, data source)

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