L11n137

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_10,4,11,3 X_2,17,3,18 X_12,5,13,6 X₆₇₁₈ X_16,10,17,9 X_13,20,14,21 X_15,22,16,7 X_4,20,5,19 X_21,14,22,15
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 $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-3u^{2}v^{3}+3u^{2}v^{2}-u^{2}v-uv^{4}+5uv^{3}-7uv^{2}+5uv-u-v^{3}+3v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	${\frac {12}{q^{9/2}}}-{\frac {11}{q^{7/2}}}+{\frac {7}{q^{5/2}}}-{\frac {5}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {10}{q^{13/2}}}-{\frac {12}{q^{11/2}}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+2a^{7}z^{3}-a^{7}z^{-1}-a^{5}z^{7}-4a^{5}z^{5}-4a^{5}z^{3}+2a^{5}z+3a^{5}z^{-1}+a^{3}z^{5}+a^{3}z^{3}-3a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{11}z^{5}-a^{11}z^{3}+4a^{10}z^{6}-7a^{10}z^{4}+2a^{10}z^{2}+6a^{9}z^{7}-11a^{9}z^{5}+4a^{9}z^{3}+4a^{8}z^{8}-a^{8}z^{6}-10a^{8}z^{4}+7a^{8}z^{2}-a^{8}+a^{7}z^{9}+8a^{7}z^{7}-16a^{7}z^{5}+7a^{7}z^{3}-a^{7}z+a^{7}z^{-1}+5a^{6}z^{8}-2a^{6}z^{6}-8a^{6}z^{4}+9a^{6}z^{2}-3a^{6}+a^{5}z^{9}+2a^{5}z^{7}+a^{5}z^{5}-2a^{5}z^{3}-4a^{5}z+3a^{5}z^{-1}+a^{4}z^{8}+3a^{4}z^{6}-4a^{4}z^{4}+4a^{4}z^{2}-3a^{4}+5a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z+2a^{3}z^{-1}+a^{2}z^{4}$ (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

χ

0

1

-1

-2

4

-4

4

2

-2

-6

7

3

4

-8

6

5

-1

-10

6

0

-12

4

6

2

-14

3

6

-3

-16

1

4

3

-18

3

-3

-20

1

Integral Khovanov Homology

(db, data source)

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