L11n276

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_9,18,10,19 X_22,17,11,18 X_20,11,21,12 X_16,21,17,22 X_4,15,1,16 X_19,10,20,5
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11}, {8, 2, -4, -5, 10, -9, 7, 6, -11, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{4}-uvw^{3}+uvw^{2}-vw^{2}+vw-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-12}+q^{-11}-q^{-10}+2q^{-9}+q^{-7}+q^{-6}+q^{-4}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+2z^{2}a^{12}+4a^{12}z^{-2}+6a^{12}-z^{6}a^{10}-8z^{4}a^{10}-19z^{2}a^{10}-5a^{10}z^{-2}-16a^{10}+z^{8}a^{8}+8z^{6}a^{8}+21z^{4}a^{8}+22z^{2}a^{8}+2a^{8}z^{-2}+10a^{8}$ (db)
Kauffman polynomial	$a^{15}z^{3}-2a^{15}z+a^{15}z^{-1}+a^{14}z^{4}-2a^{14}z^{2}-a^{14}z^{-2}+2a^{14}-a^{13}z^{5}+8a^{13}z^{3}-13a^{13}z+5a^{13}z^{-1}-a^{12}z^{6}+7a^{12}z^{4}-14a^{12}z^{2}-4a^{12}z^{-2}+13a^{12}+a^{11}z^{7}-9a^{11}z^{5}+26a^{11}z^{3}-27a^{11}z+9a^{11}z^{-1}+a^{10}z^{8}-9a^{10}z^{6}+27a^{10}z^{4}-34a^{10}z^{2}-5a^{10}z^{-2}+20a^{10}+a^{9}z^{7}-8a^{9}z^{5}+19a^{9}z^{3}-16a^{9}z+5a^{9}z^{-1}+a^{8}z^{8}-8a^{8}z^{6}+21a^{8}z^{4}-22a^{8}z^{2}-2a^{8}z^{-2}+10a^{8}$ (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

-7

1

-9

1

-11

1

-13

2

-15

2

1

-17

3

1

2

-19

2

1

-21

1

0

-23

0

-25

1

-1

Integral Khovanov Homology

(db, data source)

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