L11n411

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,22,14,19 X_7,20,8,21 X_19,10,20,11 X_9,16,10,17 X_17,14,18,15 X_15,8,16,9 X_21,18,22,5 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3}, {10, -1, -4, 8, -6, 5, 11, -2, -3, 7, -8, 6, -7, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)(w+1)\left(2uvw^{2}-2uvw+uv+w^{3}-2w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{-13}+2q^{-12}-2q^{-11}+q^{-10}+q^{-8}+2q^{-7}-q^{-6}+2q^{-5}-q^{-4}+q^{-3}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+4a^{12}z^{-2}+5a^{12}-z^{4}a^{10}-8z^{2}a^{10}-5a^{10}z^{-2}-12a^{10}+z^{6}a^{8}+5z^{4}a^{8}+6z^{2}a^{8}+2a^{8}z^{-2}+5a^{8}+z^{6}a^{6}+5z^{4}a^{6}+6z^{2}a^{6}+2a^{6}$ (db)
Kauffman polynomial	$a^{15}z^{7}-5a^{15}z^{5}+6a^{15}z^{3}-4a^{15}z+a^{15}z^{-1}+2a^{14}z^{8}-11a^{14}z^{6}+15a^{14}z^{4}-8a^{14}z^{2}-a^{14}z^{-2}+3a^{14}+a^{13}z^{9}-3a^{13}z^{7}-10a^{13}z^{5}+30a^{13}z^{3}-21a^{13}z+5a^{13}z^{-1}+4a^{12}z^{8}-26a^{12}z^{6}+47a^{12}z^{4}-35a^{12}z^{2}-4a^{12}z^{-2}+16a^{12}+a^{11}z^{9}-2a^{11}z^{7}-20a^{11}z^{5}+54a^{11}z^{3}-39a^{11}z+9a^{11}z^{-1}+3a^{10}z^{8}-21a^{10}z^{6}+44a^{10}z^{4}-43a^{10}z^{2}-5a^{10}z^{-2}+21a^{10}+3a^{9}z^{7}-19a^{9}z^{5}+32a^{9}z^{3}-22a^{9}z+5a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+7a^{8}z^{4}-10a^{8}z^{2}-2a^{8}z^{-2}+7a^{8}+a^{7}z^{7}-4a^{7}z^{5}+2a^{7}z^{3}+a^{6}z^{6}-5a^{6}z^{4}+6a^{6}z^{2}-2a^{6}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

1

0

-9

1

2

1

-11

1

-13

2

5

2

1

-15

2

3

-17

2

5

2

1

-19

2

1

-21

1

3

-1

-23

1

0

-25

1

-27

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-7$	$i=-5$	$i=-3$
$r=-11$		${\mathbb {Z} }$
$r=-10$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-7$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$