L11n365

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_22,15,17,16 X_9,18,10,19 X_17,8,18,9 X_20,13,21,14 X_14,21,15,22 X_16,19,5,20 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {-6, 5, 9, -7, 8, -4}, {10, -1, 3, 6, -5, -2, 11, -3, 7, -8, 4, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(w-1)\left(-v^{2}w+vw^{3}-vw^{2}+vw-v+w^{2}\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-1+4q^{-1}-5q^{-2}+8q^{-3}-8q^{-4}+8q^{-5}-6q^{-6}+5q^{-7}-2q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-2a^{6}z^{4}-6a^{6}z^{2}-2a^{6}z^{-2}-6a^{6}+a^{4}z^{6}+4a^{4}z^{4}+6a^{4}z^{2}+a^{4}z^{-2}+3a^{4}-a^{2}z^{4}-a^{2}z^{2}+a^{2}$ (db)
Kauffman polynomial	$z^{6}a^{10}-4z^{4}a^{10}+5z^{2}a^{10}-2a^{10}+2z^{7}a^{9}-6z^{5}a^{9}+3z^{3}a^{9}+za^{9}+2z^{8}a^{8}-4z^{6}a^{8}-z^{4}a^{8}-z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+z^{9}a^{7}+z^{7}a^{7}-9z^{5}a^{7}+9z^{3}a^{7}-7za^{7}+2a^{7}z^{-1}+5z^{8}a^{6}-17z^{6}a^{6}+25z^{4}a^{6}-23z^{2}a^{6}-2a^{6}z^{-2}+11a^{6}+z^{9}a^{5}+z^{7}a^{5}-8z^{5}a^{5}+14z^{3}a^{5}-9za^{5}+2a^{5}z^{-1}+3z^{8}a^{4}-12z^{6}a^{4}+26z^{4}a^{4}-21z^{2}a^{4}-a^{4}z^{-2}+7a^{4}+2z^{7}a^{3}-5z^{5}a^{3}+9z^{3}a^{3}-2za^{3}+4z^{4}a^{2}-4z^{2}a^{2}+z^{3}a-za$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

1

-1

3

-3

4

3

-1

-5

4

1

3

-7

4

0

-9

4

0

-11

2

4

2

-13

3

4

-1

-15

1

4

3

-17

1

-1

-19

1

Integral Khovanov Homology

(db, data source)

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