L11n195

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,11,3,12 X_12,3,13,4 X_16,5,17,6 X₆₇₁₈ X_4,15,5,16 X_20,14,21,13 X_9,18,10,19 X_19,10,20,11 X_22,18,7,17 X_14,22,15,21
Gauss code	{1, -2, 3, -6, 4, -5}, {5, -1, -8, 9, 2, -3, 7, -11, 6, -4, 10, 8, -9, -7, 11, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-2t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-t(2)^{3}+3t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-2t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {1}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {10}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {8}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+3za^{7}+a^{7}z^{-1}-z^{7}a^{5}-5z^{5}a^{5}-9z^{3}a^{5}-6za^{5}-a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-6a^{10}z^{4}+2a^{10}z^{2}+4a^{9}z^{7}-7a^{9}z^{5}+2a^{9}z^{3}-a^{9}z+3a^{8}z^{8}-3a^{8}z^{6}-a^{8}z^{4}+a^{7}z^{9}+3a^{7}z^{7}-5a^{7}z^{5}-a^{7}z^{3}+4a^{7}z-a^{7}z^{-1}+4a^{6}z^{8}-6a^{6}z^{6}+6a^{6}z^{4}-3a^{6}z^{2}+a^{6}+a^{5}z^{9}-a^{5}z^{7}+7a^{5}z^{5}-10a^{5}z^{3}+6a^{5}z-a^{5}z^{-1}+a^{4}z^{8}+2a^{4}z^{4}-2a^{4}z^{2}+4a^{3}z^{5}-5a^{3}z^{3}+a^{2}z^{4}-a^{2}z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

0

1

-1

-2

3

-4

4

2

-2

-6

5

2

3

-8

5

4

-1

-10

5

0

-12

3

5

2

-14

3

5

-2

-16

1

4

3

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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