L11n138

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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_17,3,18,2 X_5,13,6,12 X₆₇₁₈ X_9,16,10,17 X_15,20,16,21 X_13,22,14,7 X_21,14,22,15 X_19,4,20,5
Gauss code	{1, 4, -3, 11, -5, -6}, {6, -1, -7, 3, -2, 5, -9, 10, -8, 7, -4, 2, -11, 8, -10, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{3}+2a^{7}z-a^{5}z^{5}-4a^{5}z^{3}-4a^{5}z+a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}-az-az^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+5z^{5}a^{9}-6z^{3}a^{9}+za^{9}-2z^{8}a^{8}+11z^{6}a^{8}-17z^{4}a^{8}+8z^{2}a^{8}-z^{9}a^{7}+4z^{7}a^{7}-2z^{5}a^{7}-z^{3}a^{7}-za^{7}-3z^{8}a^{6}+16z^{6}a^{6}-24z^{4}a^{6}+12z^{2}a^{6}-z^{9}a^{5}+5z^{7}a^{5}-8z^{5}a^{5}+9z^{3}a^{5}-5za^{5}-z^{8}a^{4}+5z^{6}a^{4}-7z^{4}a^{4}+4z^{2}a^{4}-z^{5}a^{3}+4z^{3}a^{3}-4za^{3}+a^{3}z^{-1}-a^{2}-za+az^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

0

1

-2

1

0

-4

2

-6

1

2

1

-8

2

1

-10

1

0

-12

1

2

-1

-14

1

0

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

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