L11n136

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_3,10,4,11 X_2,17,3,18 X_12,5,13,6 X₆₇₁₈ X_16,10,17,9 X_20,14,21,13 X_22,16,7,15 X_19,4,20,5 X_14,22,15,21
Gauss code	{1, -4, -3, 10, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -11, 9, -7, 4, 2, -10, -8, 11, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{6}-t(1)t(2)^{5}+t(2)^{5}+t(1)t(2)^{4}-t(1)t(2)^{3}+t(1)t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)+t(1)}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}+q^{7/2}-{\frac {2}{q^{7/2}}}-q^{5/2}+{\frac {2}{q^{5/2}}}+2q^{3/2}-{\frac {3}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-3{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+5a^{3}z^{3}+7a^{3}z+3a^{3}z^{-1}-az^{7}-7az^{5}+z^{5}a^{-1}-17az^{3}+5z^{3}a^{-1}-17az+7za^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-a^{4}z^{8}-2a^{2}z^{8}-z^{8}-a^{5}z^{7}+6a^{3}z^{7}+7az^{7}+5a^{4}z^{6}+11a^{2}z^{6}+6z^{6}+6a^{5}z^{5}-13a^{3}z^{5}-21az^{5}-2z^{5}a^{-1}-6a^{4}z^{4}-20a^{2}z^{4}-z^{4}a^{-2}-15z^{4}-10a^{5}z^{3}+16a^{3}z^{3}+33az^{3}+6z^{3}a^{-1}-z^{3}a^{-3}+a^{4}z^{2}+17a^{2}z^{2}-z^{2}a^{-4}+17z^{2}+4a^{5}z-11a^{3}z-22az-7za^{-1}-5a^{2}+a^{-4}-5+3a^{3}z^{-1}+5az^{-1}+2a^{-1}z^{-1}$ (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

χ

8

1

-1

6

0

4

2

1

-1

2

1

0

2

3

1

-2

1

-4

1

2

1

-6

1

0

-8

1

-10

1

0

-12

1

Integral Khovanov Homology

(db, data source)

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