L11a149

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,17,15,18 X_16,7,17,8 X_22,15,7,16 X_18,13,19,14 X_6,20,1,19 X_20,12,21,11 X_12,6,13,5 X_4,21,5,22
Gauss code	{1, -2, 3, -11, 10, -8}, {5, -1, 2, -3, 9, -10, 7, -4, 6, -5, 4, -7, 8, -9, 11, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{4}-5u^{2}v^{3}+5u^{2}v^{2}-2u^{2}v-2uv^{4}+8uv^{3}-11uv^{2}+8uv-2u-2v^{3}+5v^{2}-5v+2}{uv^{2}}}$ (db)
Jones polynomial	${\frac {19}{q^{9/2}}}-{\frac {19}{q^{7/2}}}+{\frac {15}{q^{5/2}}}+q^{3/2}-{\frac {12}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {8}{q^{15/2}}}+{\frac {13}{q^{13/2}}}-{\frac {17}{q^{11/2}}}-3{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+2a^{7}z^{3}-a^{7}z^{-1}-a^{5}z^{7}-3a^{5}z^{5}-a^{5}z^{3}+4a^{5}z+3a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-7a^{3}z^{3}-7a^{3}z-2a^{3}z^{-1}+az^{5}+3az^{3}+2az$ (db)
Kauffman polynomial	$a^{11}z^{5}-a^{11}z^{3}+4a^{10}z^{6}-6a^{10}z^{4}+a^{10}z^{2}+7a^{9}z^{7}-11a^{9}z^{5}+3a^{9}z^{3}+8a^{8}z^{8}-13a^{8}z^{6}+6a^{8}z^{4}-a^{8}+6a^{7}z^{9}-8a^{7}z^{7}+5a^{7}z^{5}-2a^{7}z^{3}+a^{7}z^{-1}+2a^{6}z^{10}+7a^{6}z^{8}-21a^{6}z^{6}+18a^{6}z^{4}-3a^{6}+10a^{5}z^{9}-24a^{5}z^{7}+24a^{5}z^{5}-6a^{5}z^{3}-5a^{5}z+3a^{5}z^{-1}+2a^{4}z^{10}+3a^{4}z^{8}-13a^{4}z^{6}+10a^{4}z^{4}+a^{4}z^{2}-3a^{4}+4a^{3}z^{9}-6a^{3}z^{7}-2a^{3}z^{5}+8a^{3}z^{3}-8a^{3}z+2a^{3}z^{-1}+4a^{2}z^{8}-8a^{2}z^{6}+a^{2}z^{4}+2a^{2}z^{2}+3az^{7}-9az^{5}+8az^{3}-3az+z^{6}-3z^{4}+2z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

4

1

-3

-2

8

2

6

-4

8

5

-3

-6

11

7

4

-8

9

0

-10

8

10

-2

-12

5

9

4

-14

3

8

-5

-16

1

5

4

-18

3

-3

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$