L11a5

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(3v^{2}-5v+3\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-11q^{9/2}+13q^{7/2}-14q^{5/2}+{\frac {1}{q^{5/2}}}+12q^{3/2}-{\frac {4}{q^{3/2}}}-q^{17/2}+3q^{15/2}-5q^{13/2}+8q^{11/2}-10{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}+z^{5}a^{-5}-az^{3}+z^{3}a^{-1}+2z^{3}a^{-5}-z^{3}a^{-7}-za^{-3}+2za^{-5}-za^{-7}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-4}-2z^{10}a^{-6}-5z^{9}a^{-3}-9z^{9}a^{-5}-4z^{9}a^{-7}-6z^{8}a^{-2}-z^{8}a^{-4}+2z^{8}a^{-6}-3z^{8}a^{-8}-6z^{7}a^{-1}+12z^{7}a^{-3}+36z^{7}a^{-5}+17z^{7}a^{-7}-z^{7}a^{-9}+9z^{6}a^{-2}+18z^{6}a^{-4}+16z^{6}a^{-6}+13z^{6}a^{-8}-6z^{6}-4az^{5}+4z^{5}a^{-1}-13z^{5}a^{-3}-46z^{5}a^{-5}-21z^{5}a^{-7}+4z^{5}a^{-9}-a^{2}z^{4}-4z^{4}a^{-2}-23z^{4}a^{-4}-27z^{4}a^{-6}-15z^{4}a^{-8}+6z^{4}+4az^{3}+4z^{3}a^{-1}+8z^{3}a^{-3}+22z^{3}a^{-5}+11z^{3}a^{-7}-3z^{3}a^{-9}+z^{2}a^{-2}+6z^{2}a^{-4}+11z^{2}a^{-6}+6z^{2}a^{-8}-2za^{-3}-4za^{-5}-2za^{-7}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

18

1

16

2

-2

14

3

1

2

12

5

2

-3

10

6

3

8

7

5

-2

6

7

6

1

4

5

7

2

5

7

-2

0

3

7

4

-2

1

3

-2

-4

3

-6

1

-1

Integral Khovanov Homology

(db, data source)

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