L11n44

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,14,6,15 X₃₈₄₉ X_9,18,10,19 X_11,20,12,21 X_13,22,14,5 X_19,10,20,11 X_21,12,22,13 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, -11, -2, 3, 6, -9, 7, -10, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{7}-1}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-9 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-4za^{13}-3a^{13}z^{-1}+z^{7}a^{11}+8z^{5}a^{11}+21z^{3}a^{11}+21za^{11}+7a^{11}z^{-1}-z^{9}a^{9}-9z^{7}a^{9}-28z^{5}a^{9}-36z^{3}a^{9}-19za^{9}-4a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{18}+a^{13}z^{5}-6a^{13}z^{3}+9a^{13}z-3a^{13}z^{-1}+a^{12}z^{8}-8a^{12}z^{6}+21a^{12}z^{4}-21a^{12}z^{2}+7a^{12}+a^{11}z^{9}-9a^{11}z^{7}+29a^{11}z^{5}-42a^{11}z^{3}+28a^{11}z-7a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+21a^{10}z^{4}-21a^{10}z^{2}+7a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-36a^{9}z^{3}+19a^{9}z-4a^{9}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-8

1

-10

1

-12

1

-14

1

-16

2

1

-18

1

-20

1

0

-22

1

0

-24

1

0

-26

1

-1

-28

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-10$	$i=-8$	$i=-6$
$r=-9$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$