L11n427

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_11,18,12,19 X_3,10,4,11 X_19,2,20,3 X_7,16,8,17 X_20,9,21,10 X_17,12,18,7 X_22,16,13,15 X_14,6,15,5 X_4,14,5,13 X_6,21,1,22
Gauss code	{1, 4, -3, -10, 9, -11}, {-5, -1, 6, 3, -2, 7}, {10, -9, 8, 5, -7, 2, -4, -6, 11, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)t(3)^{3}-t(1)t(2)t(3)^{3}+t(1)^{2}t(2)^{2}t(3)^{2}-t(1)t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-2t(1)^{2}t(2)t(3)^{2}+4t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(3)^{2}-t(1)^{2}t(2)^{2}t(3)+2t(1)t(2)^{2}t(3)+t(1)t(3)+t(1)^{2}t(2)t(3)-4t(1)t(2)t(3)+2t(2)t(3)-t(3)+t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{-7}-3q^{-6}+7q^{-5}-8q^{-4}+10q^{-3}-q^{2}-9q^{-2}+3q+8q^{-1}-5$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+a^{8}-a^{6}z^{4}-4a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}+a^{4}z^{6}+4a^{4}z^{4}+6a^{4}z^{2}+a^{4}z^{-2}+4a^{4}+a^{2}z^{6}+3a^{2}z^{4}+2a^{2}z^{2}-z^{4}-2z^{2}$ (db)
Kauffman polynomial	$3a^{8}z^{4}-8a^{8}z^{2}-a^{8}z^{-2}+6a^{8}+a^{7}z^{7}-a^{7}z^{5}+3a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}-7a^{6}z^{6}+20a^{6}z^{4}-25a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+a^{5}z^{9}-a^{5}z^{5}+7a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+5a^{4}z^{8}-15a^{4}z^{6}+27a^{4}z^{4}-22a^{4}z^{2}-a^{4}z^{-2}+8a^{4}+a^{3}z^{9}+3a^{3}z^{7}-10a^{3}z^{5}+11a^{3}z^{3}-3a^{3}z+3a^{2}z^{8}-5a^{2}z^{6}+3a^{2}z^{4}-2a^{2}z^{2}+a^{2}+4az^{7}-9az^{5}+z^{5}a^{-1}+5az^{3}-2z^{3}a^{-1}-az+3z^{6}-7z^{4}+3z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

2

1

3

1

-2

-1

5

2

3

-3

5

4

-1

-5

5

4

1

-7

4

6

2

-9

3

4

-1

-11

1

5

4

-13

1

2

-1

-15

2

Integral Khovanov Homology

(db, data source)

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