L11a165

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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_14,5,15,6 X_20,18,21,17 X_18,12,19,11 X_12,20,13,19 X_22,16,7,15 X_16,22,17,21 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -7, 11, -4, 8, -9, 5, -6, 7, -5, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{4}-4u^{2}v^{3}+3u^{2}v^{2}+3uv^{3}-5uv^{2}+3uv+3v^{2}-4v+2}{uv^{2}}}$ (db)
Jones polynomial	${\frac {6}{q^{9/2}}}+q^{7/2}-{\frac {8}{q^{7/2}}}-2q^{5/2}+{\frac {8}{q^{5/2}}}+4q^{3/2}-{\frac {9}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {4}{q^{11/2}}}-6{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+4a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-8a^{3}z^{3}-5a^{3}z-az^{7}-5az^{5}+z^{5}a^{-1}-8az^{3}+4z^{3}a^{-1}-6az+4za^{-1}-2az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-5az^{9}-2z^{9}a^{-1}-4a^{4}z^{8}-2a^{2}z^{8}-z^{8}a^{-2}+z^{8}-5a^{5}z^{7}+7a^{3}z^{7}+23az^{7}+11z^{7}a^{-1}-4a^{6}z^{6}+8a^{4}z^{6}+20a^{2}z^{6}+6z^{6}a^{-2}+14z^{6}-3a^{7}z^{5}+12a^{5}z^{5}+2a^{3}z^{5}-32az^{5}-19z^{5}a^{-1}-2a^{8}z^{4}+5a^{6}z^{4}+2a^{4}z^{4}-26a^{2}z^{4}-12z^{4}a^{-2}-33z^{4}-a^{9}z^{3}+2a^{7}z^{3}-11a^{5}z^{3}-7a^{3}z^{3}+19az^{3}+12z^{3}a^{-1}+a^{8}z^{2}-2a^{6}z^{2}-5a^{4}z^{2}+12a^{2}z^{2}+9z^{2}a^{-2}+23z^{2}+a^{9}z-a^{7}z+6a^{5}z+4a^{3}z-8az-4za^{-1}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

8

1

-1

6

1

4

3

1

-2

2

3

1

2

0

4

3

-1

-2

5

3

2

-4

4

5

1

-6

4

0

-8

2

4

2

-10

2

4

-2

-12

2

-14

1

2

-1

-16

1

Integral Khovanov Homology

(db, data source)

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