L11n280

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_9,18,10,19 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_4,15,1,16 X_19,10,20,5
Gauss code	{1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11}, {-8, 2, 4, -5, 10, 9, -7, 6, -11, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)(w-1)\left(w^{2}-w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{3}+2q^{2}-5q+8-7q^{-1}+9q^{-2}-6q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-4a^{2}z^{4}+2z^{4}+2a^{4}z^{2}-7a^{2}z^{2}-z^{2}a^{-2}+6z^{2}+2a^{4}-8a^{2}-2a^{-2}+8+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-9a^{5}z^{5}+5a^{5}z^{3}+3a^{4}z^{8}-7a^{4}z^{6}-a^{4}z^{2}-2a^{4}z^{-2}+4a^{4}+a^{3}z^{9}+3a^{3}z^{7}-17a^{3}z^{5}+18a^{3}z^{3}+z^{3}a^{-3}-12a^{3}z-2za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+5a^{2}z^{8}-15a^{2}z^{6}+15a^{2}z^{4}+2z^{4}a^{-2}-11a^{2}z^{2}-2z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+10a^{2}+a^{-2}+az^{9}+az^{7}+z^{7}a^{-1}-11az^{5}-3z^{5}a^{-1}+23az^{3}+11z^{3}a^{-1}-21az-11za^{-1}+9az^{-1}+5a^{-1}z^{-1}+2z^{8}-7z^{6}+14z^{4}-9z^{2}-4z^{-2}+7$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

7

1

-1

5

1

3

4

1

-3

1

4

1

3

-1

5

6

1

-3

4

3

1

2

-5

2

5

3

-7

4

0

-9

1

4

3

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

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