L11n134

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_3,10,4,11 X_17,3,18,2 X_12,5,13,6 X₆₇₁₈ X_9,16,10,17 X_20,14,21,13 X_22,16,7,15 X_4,20,5,19 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 $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-uv^{4}+2uv^{3}-uv^{2}+2uv-u-v^{3}+v^{2}-v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{7/2}-2q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-az^{7}+a^{3}z^{5}-6az^{5}+z^{5}a^{-1}+4a^{3}z^{3}-12az^{3}+4z^{3}a^{-1}+4a^{3}z-9az+4za^{-1}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-3a^{2}z^{8}-z^{8}-a^{5}z^{7}+3a^{3}z^{7}+4az^{7}+10a^{4}z^{6}+12a^{2}z^{6}+2z^{6}+5a^{5}z^{5}-10az^{5}-5z^{5}a^{-1}-14a^{4}z^{4}-13a^{2}z^{4}-3z^{4}a^{-2}-2z^{4}-7a^{5}z^{3}-a^{3}z^{3}+19az^{3}+11z^{3}a^{-1}-2z^{3}a^{-3}+5a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+4z^{2}+2a^{5}z-3a^{3}z-11az-6za^{-1}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

8

1

-1

6

1

4

2

1

-1

2

3

1

2

0

2

3

1

-2

3

2

1

-4

2

3

1

-6

1

2

-1

-8

1

2

1

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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