L11n302

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_20,13,21,14 X_22,19,11,20 X_15,5,16,10 X_17,9,18,8 X_7,17,8,16 X_9,19,10,18 X_14,21,15,22 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -7, 6, -8, 5}, {-11, 2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{2}w+uvw-uv-uw^{2}+uw-v^{2}w^{3}+v^{2}w^{2}+vw^{4}-vw^{3}+w^{3}}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{6}-q^{5}+2q^{4}-q^{3}+2q^{2}+1+q^{-1}-q^{-2}+q^{-3}-q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}+z^{6}-a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+7z^{4}-4a^{2}z^{2}-17z^{2}a^{-2}+4z^{2}a^{-4}+16z^{2}-4a^{2}-16a^{-2}+5a^{-4}+15-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{-4}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{2}z^{8}+z^{8}a^{-2}+z^{8}a^{-4}+z^{8}+a^{3}z^{7}+2az^{7}+2z^{7}a^{-1}+2z^{7}a^{-3}+z^{7}a^{-5}-6a^{2}z^{6}-8z^{6}a^{-2}-5z^{6}a^{-4}+z^{6}a^{-6}-8z^{6}-6a^{3}z^{5}-14az^{5}-16z^{5}a^{-1}-12z^{5}a^{-3}-4z^{5}a^{-5}+10a^{2}z^{4}+22z^{4}a^{-2}+7z^{4}a^{-4}-5z^{4}a^{-6}+20z^{4}+10a^{3}z^{3}+28az^{3}+37z^{3}a^{-1}+21z^{3}a^{-3}+2z^{3}a^{-5}-7a^{2}z^{2}-31z^{2}a^{-2}-7z^{2}a^{-4}+6z^{2}a^{-6}-25z^{2}-5a^{3}z-21az-33za^{-1}-16za^{-3}+za^{-5}+4a^{2}+20a^{-2}+6a^{-4}-2a^{-6}+17+a^{3}z^{-1}+5az^{-1}+9a^{-1}z^{-1}+5a^{-3}z^{-1}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{-4}z^{-2}-4z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

1

0

9

1

7

1

5

1

3

1

3

1

2

1

4

2

1

-1

1

2

-3

1

0

-5

1

0

-7

0

-9

1

-1

Integral Khovanov Homology

(db, data source)

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