L11a142

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_10,4,11,3 X_2,17,3,18 X_12,5,13,6 X₆₇₁₈ X_16,10,17,9 X_20,16,21,15 X_22,14,7,13 X_14,22,15,21 X_4,20,5,19
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -5, 9, -10, 8, -7, 4, -2, 11, -8, 10, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-6u^{2}v^{3}+6u^{2}v^{2}-2u^{2}v-2uv^{4}+8uv^{3}-13uv^{2}+8uv-2u-2v^{3}+6v^{2}-6v+2}{uv^{2}}}$ (db)
Jones polynomial	$4q^{9/2}-{\frac {4}{q^{9/2}}}-8q^{7/2}+{\frac {8}{q^{7/2}}}+13q^{5/2}-{\frac {14}{q^{5/2}}}-18q^{3/2}+{\frac {18}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+20{\sqrt {q}}-{\frac {21}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+3z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+2az^{3}+2z^{3}a^{-1}-2z^{3}a^{-3}-az+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}-9az^{9}-15z^{9}a^{-1}-6z^{9}a^{-3}-13a^{2}z^{8}-2z^{8}a^{-2}-4z^{8}a^{-4}-11z^{8}-12a^{3}z^{7}+11az^{7}+44z^{7}a^{-1}+20z^{7}a^{-3}-z^{7}a^{-5}-8a^{4}z^{6}+23a^{2}z^{6}+31z^{6}a^{-2}+14z^{6}a^{-4}+48z^{6}-4a^{5}z^{5}+16a^{3}z^{5}+7az^{5}-35z^{5}a^{-1}-19z^{5}a^{-3}+3z^{5}a^{-5}-a^{6}z^{4}+6a^{4}z^{4}-14a^{2}z^{4}-37z^{4}a^{-2}-14z^{4}a^{-4}-44z^{4}+2a^{5}z^{3}-6a^{3}z^{3}-8az^{3}+8z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}+4a^{2}z^{2}+12z^{2}a^{-2}+4z^{2}a^{-4}+12z^{2}-a^{3}z-az-a^{2}+a^{3}z^{-1}+az^{-1}$ (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

12

1

10

3

-3

8

5

1

4

6

8

3

-5

4

10

5

2

10

8

-2

0

11

10

1

-2

8

11

3

-4

6

10

-4

-6

3

9

6

-8

1

5

-4

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$