L11n27

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_5,12,6,13 X₃₈₄₉ 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 $u$ , $v$ , $w$ , ...)	${\frac {3(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {1}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{9}-a^{9}z^{-1}+z^{3}a^{7}+za^{7}+a^{7}z^{-1}+2z^{3}a^{5}+4za^{5}+2a^{5}z^{-1}-3za^{3}-2a^{3}z^{-1}-za$ (db)
Kauffman polynomial	$-z^{8}a^{10}+7z^{6}a^{10}-17z^{4}a^{10}+16z^{2}a^{10}-4a^{10}-z^{9}a^{9}+5z^{7}a^{9}-6z^{5}a^{9}+a^{9}z^{-1}-4z^{8}a^{8}+23z^{6}a^{8}-42z^{4}a^{8}+31z^{2}a^{8}-9a^{8}-z^{9}a^{7}+z^{7}a^{7}+13z^{5}a^{7}-22z^{3}a^{7}+5za^{7}+a^{7}z^{-1}-3z^{8}a^{6}+14z^{6}a^{6}-18z^{4}a^{6}+11z^{2}a^{6}-4a^{6}-4z^{7}a^{5}+19z^{5}a^{5}-25z^{3}a^{5}+13za^{5}-2a^{5}z^{-1}-2z^{6}a^{4}+7z^{4}a^{4}-5z^{2}a^{4}+2a^{4}-3z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}-z^{2}a^{2}-za$ (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

0

1

-2

2

0

-4

1

2

-6

3

0

-8

2

1

-10

1

3

1

-12

3

2

1

-14

1

-16

1

3

-2

-18

0

-20

1

-1

Integral Khovanov Homology

(db, data source)

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