L11n352

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_11,20,12,21 X_7,18,8,19 X_9,13,10,22 X_21,17,22,16 X_17,8,18,9 X_15,11,16,10 X_19,12,20,5 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 7, -5, 8, -3, 9}, {11, -2, -8, 6, -7, 4, -9, 3, -6, 5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{3}-2uv^{2}w^{2}+2uv^{2}w-uvw^{3}+2uvw^{2}-uvw+uw^{2}-uw+v^{3}w^{2}-v^{3}w+v^{2}w^{2}-2v^{2}w+v^{2}-2vw^{2}+2vw-v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-8}-q^{-7}+q^{-6}+2q^{-5}-3q^{-4}+6q^{-3}-q^{2}-5q^{-2}+3q+6q^{-1}-5$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+a^{8}-2a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}+2a^{4}z^{2}+a^{4}z^{-2}+3a^{4}+a^{2}z^{6}+4a^{2}z^{4}+5a^{2}z^{2}+2a^{2}-z^{4}-2z^{2}-1$ (db)
Kauffman polynomial	$a^{7}z^{9}+a^{5}z^{9}+a^{8}z^{8}+2a^{6}z^{8}+2a^{4}z^{8}+a^{2}z^{8}-7a^{7}z^{7}-7a^{5}z^{7}+3a^{3}z^{7}+3az^{7}-7a^{8}z^{6}-18a^{6}z^{6}-12a^{4}z^{6}+2a^{2}z^{6}+3z^{6}+12a^{7}z^{5}+11a^{5}z^{5}-7a^{3}z^{5}-5az^{5}+z^{5}a^{-1}+15a^{8}z^{4}+45a^{6}z^{4}+28a^{4}z^{4}-9a^{2}z^{4}-7z^{4}-3a^{7}z^{3}+5a^{5}z^{3}+7a^{3}z^{3}-3az^{3}-2z^{3}a^{-1}-14a^{8}z^{2}-38a^{6}z^{2}-23a^{4}z^{2}+4a^{2}z^{2}+3z^{2}-5a^{7}z-8a^{5}z-3a^{3}z+az+za^{-1}+6a^{8}+13a^{6}+9a^{4}-1+2a^{7}z^{-1}+2a^{5}z^{-1}-a^{8}z^{-2}-2a^{6}z^{-2}-a^{4}z^{-2}$ (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

2

3

χ

5

1

-1

3

2

1

3

1

-2

-1

3

2

1

-3

1

4

1

-5

3

2

1

-7

1

2

4

3

-9

1

3

-1

-11

1

4

3

-13

1

0

-15

0

-17

1

Integral Khovanov Homology

(db, data source)

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