L11n408

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,16,8,17 X_17,19,18,22 X_11,20,12,21 X_19,10,20,11 X_21,5,22,18 X_9,14,10,15 X_15,8,16,9 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {-6, 5, -7, 4}, {10, -1, -3, 9, -8, 6, -5, -2, 11, 8, -9, 3, -4, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}}}$ (db)
Jones polynomial	$q^{3}-q^{2}+q+2-q^{-1}+3q^{-2}-3q^{-3}+4q^{-4}-3q^{-5}+2q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}+a^{4}z^{4}+2a^{4}z^{2}+a^{4}+a^{2}z^{4}+3a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+3a^{2}+2a^{-2}-z^{4}-5z^{2}-2z^{-2}-5$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+3a^{4}z^{8}+a^{2}z^{8}+a^{7}z^{7}-3a^{5}z^{7}-5a^{3}z^{7}+z^{7}a^{-1}-10a^{6}z^{6}-15a^{4}z^{6}-5a^{2}z^{6}+z^{6}a^{-2}+z^{6}-5a^{7}z^{5}-2a^{5}z^{5}+7a^{3}z^{5}-az^{5}-5z^{5}a^{-1}+14a^{6}z^{4}+21a^{4}z^{4}+2a^{2}z^{4}-5z^{4}a^{-2}-10z^{4}+7a^{7}z^{3}+6a^{5}z^{3}-7a^{3}z^{3}-3az^{3}+3z^{3}a^{-1}-7a^{6}z^{2}-13a^{4}z^{2}+7a^{2}z^{2}+6z^{2}a^{-2}+19z^{2}-2a^{7}z-3a^{5}z+3a^{3}z+7az+3za^{-1}+2a^{6}+2a^{4}-6a^{2}-4a^{-2}-9-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

0

3

1

0

1

4

1

3

-1

2

4

1

-3

2

-5

2

0

-7

2

1

-9

1

2

1

-11

1

2

-1

-13

1

-15

1

-1

Integral Khovanov Homology

(db, data source)

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