L11n266

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_18,11,19,12 X_20,15,21,16 X_22,17,9,18 X_16,21,17,22 X_12,19,13,20 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -9, -4, 3, 6, -8, 7, -5, 9, -6, 8, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uvw^{3}+uvw^{2}-uv-uw^{4}+2uw^{3}-2uw^{2}+uw-vw^{4}+2vw^{3}-2vw^{2}+vw+w^{5}-w^{3}+w^{2}}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{-8}+2q^{-7}-5q^{-6}+5q^{-5}-5q^{-4}+6q^{-3}+q^{2}-3q^{-2}+3q^{-1}+1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-3z^{2}a^{6}-2a^{6}z^{-2}-5a^{6}+z^{6}a^{4}+6z^{4}a^{4}+16z^{2}a^{4}+7a^{4}z^{-2}+18a^{4}-z^{6}a^{2}-8z^{4}a^{2}-19z^{2}a^{2}-8a^{2}z^{-2}-20a^{2}+z^{4}+5z^{2}+3z^{-2}+7$ (db)
Kauffman polynomial	$z^{5}a^{9}-3z^{3}a^{9}+3za^{9}-a^{9}z^{-1}+2z^{6}a^{8}-4z^{4}a^{8}+a^{8}+2z^{7}a^{7}-2z^{5}a^{7}-5z^{3}a^{7}+3za^{7}-a^{7}z^{-1}+z^{8}a^{6}-z^{4}a^{6}-6z^{2}a^{6}-2a^{6}z^{-2}+7a^{6}+3z^{7}a^{5}-9z^{5}a^{5}+19z^{3}a^{5}-21za^{5}+7a^{5}z^{-1}+z^{8}a^{4}-5z^{6}a^{4}+18z^{4}a^{4}-27z^{2}a^{4}-7a^{4}z^{-2}+22a^{4}+2z^{7}a^{3}-16z^{5}a^{3}+47z^{3}a^{3}-45za^{3}+15a^{3}z^{-1}+z^{8}a^{2}-11z^{6}a^{2}+36z^{4}a^{2}-45z^{2}a^{2}-8a^{2}z^{-2}+28a^{2}+z^{7}a-10z^{5}a+26z^{3}a-24za+8az^{-1}+z^{8}-8z^{6}+21z^{4}-24z^{2}-3z^{-2}+13$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

5

1

3

1

-1

4

-3

3

4

1

0

-5

4

1

3

-7

1

3

1

-9

4

0

-11

1

0

-13

1

4

-3

-15

1

-17

1

-1

Integral Khovanov Homology

(db, data source)

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