L11n28

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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_5,12,6,13 X₈₄₉₃ X_13,22,14,5 X_21,14,22,15 X_11,18,12,19 X_9,20,10,21 X_19,10,20,11 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -9, 10, -8, 4, -6, 7, 11, -2, 3, 8, -10, 9, -7, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {1}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}(-z)-a^{9}z^{-1}+a^{7}z^{3}+3a^{7}z+a^{7}z^{-1}-a^{5}z^{3}+2a^{5}z^{-1}+a^{3}z^{5}+2a^{3}z^{3}-a^{3}z-2a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$a^{10}z^{8}-7a^{10}z^{6}+15a^{10}z^{4}-13a^{10}z^{2}+4a^{10}+a^{9}z^{9}-7a^{9}z^{7}+13a^{9}z^{5}-7a^{9}z^{3}-a^{9}z^{-1}+2a^{8}z^{8}-17a^{8}z^{6}+40a^{8}z^{4}-32a^{8}z^{2}+9a^{8}+a^{7}z^{9}-7a^{7}z^{7}+11a^{7}z^{5}+a^{7}z^{3}-a^{7}z-a^{7}z^{-1}+2a^{6}z^{8}-13a^{6}z^{6}+28a^{6}z^{4}-18a^{6}z^{2}+4a^{6}+3a^{5}z^{7}-10a^{5}z^{5}+12a^{5}z^{3}-5a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{4}+2a^{4}z^{2}-2a^{4}+3a^{3}z^{7}-7a^{3}z^{5}+2a^{3}z^{3}-3a^{3}z+2a^{3}z^{-1}+3a^{2}z^{6}-7a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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

1

2

χ

2

1

0

2

-2

3

1

2

-4

3

4

1

-6

1

4

1

2

-8

1

3

2

-10

1

4

-1

-12

1

2

-14

1

3

-2

-16

1

0

-18

0

-20

1

-1

Integral Khovanov Homology

(db, data source)

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