L11a54

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_14,6,15,5 X_10,4,11,3 X_16,12,17,11 X_12,16,13,15 X_22,14,5,13 X_18,9,19,10 X_2,18,3,17 X_8,19,9,20
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 6, -7, 8, -4, 7, -6, 10, -9, 11, -2, 3, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2(u-1)(v-1)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-8q^{9/2}+{\frac {1}{q^{9/2}}}+12q^{7/2}-{\frac {3}{q^{7/2}}}-16q^{5/2}+{\frac {6}{q^{5/2}}}+18q^{3/2}-{\frac {11}{q^{3/2}}}-q^{13/2}+4q^{11/2}-18{\sqrt {q}}+{\frac {14}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}+2z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+az^{3}+3z^{3}a^{-1}-z^{3}a^{-5}-a^{3}z+az+za^{-1}-za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-6z^{4}a^{-6}+7z^{7}a^{-5}-12z^{5}a^{-5}+3z^{3}a^{-5}+8z^{8}a^{-4}+a^{4}z^{6}-16z^{6}a^{-4}-3a^{4}z^{4}+12z^{4}a^{-4}+2a^{4}z^{2}-4z^{2}a^{-4}+6z^{9}a^{-3}+3a^{3}z^{7}-11z^{7}a^{-3}-9a^{3}z^{5}+11z^{5}a^{-3}+7a^{3}z^{3}-6z^{3}a^{-3}-2a^{3}z+2za^{-3}+2z^{10}a^{-2}+4a^{2}z^{8}+6z^{8}a^{-2}-9a^{2}z^{6}-24z^{6}a^{-2}+3a^{2}z^{4}+29z^{4}a^{-2}-8z^{2}a^{-2}+4az^{9}+10z^{9}a^{-1}-8az^{7}-29z^{7}a^{-1}+4az^{5}+37z^{5}a^{-1}-az^{3}-18z^{3}a^{-1}-2az+2za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{10}+2z^{8}-14z^{6}+17z^{4}-6z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

5

1

4

8

7

3

-4

6

9

5

4

9

7

-2

2

9

0

7

11

4

-2

4

7

-3

-4

2

7

5

-6

1

4

-3

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$