L11n440

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_18,11,19,12 X_3,11,4,10 X_9,1,10,4 X_7,15,8,14 X_13,5,14,8 X_19,13,20,22 X_15,21,16,20 X_21,17,22,16 X_12,17,9,18
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -11}, {-7, 6, -9, 10, 11, -3, -8, 9, -10, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvwx^{2}-uvwx-2uvx^{2}+2uvx-uv+uw+ux^{2}-ux-vwx^{2}+vwx+vx^{3}-wx^{3}+2wx^{2}-2wx-x^{2}+x}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$q^{15/2}-2q^{13/2}+5q^{11/2}-6q^{9/2}+5q^{7/2}-7q^{5/2}+3q^{3/2}-5{\sqrt {q}}-{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+2z^{5}a^{-3}+az^{3}-8z^{3}a^{-1}+10z^{3}a^{-3}-3z^{3}a^{-5}+4az-17za^{-1}+21za^{-3}-9za^{-5}+za^{-7}+5az^{-1}-17a^{-1}z^{-1}+21a^{-3}z^{-1}-11a^{-5}z^{-1}+2a^{-7}z^{-1}+2az^{-3}-7a^{-1}z^{-3}+9a^{-3}z^{-3}-5a^{-5}z^{-3}+a^{-7}z^{-3}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-4z^{4}a^{-8}+6z^{2}a^{-8}+a^{-8}z^{-2}-4a^{-8}+2z^{7}a^{-7}-6z^{5}a^{-7}+4z^{3}a^{-7}-a^{-7}z^{-3}-2za^{-7}+2a^{-7}z^{-1}+z^{8}a^{-6}+4z^{6}a^{-6}-26z^{4}a^{-6}+34z^{2}a^{-6}+7a^{-6}z^{-2}-24a^{-6}+7z^{7}a^{-5}-24z^{5}a^{-5}+23z^{3}a^{-5}-5a^{-5}z^{-3}-16za^{-5}+12a^{-5}z^{-1}+z^{8}a^{-4}+7z^{6}a^{-4}-43z^{4}a^{-4}+75z^{2}a^{-4}+18a^{-4}z^{-2}-58a^{-4}+5z^{7}a^{-3}-23z^{5}a^{-3}+41z^{3}a^{-3}-9a^{-3}z^{-3}-37za^{-3}+24a^{-3}z^{-1}+5z^{6}a^{-2}-32z^{4}a^{-2}+73z^{2}a^{-2}+19a^{-2}z^{-2}-60a^{-2}+az^{7}+z^{7}a^{-1}-7az^{5}-12z^{5}a^{-1}+15az^{3}+37z^{3}a^{-1}-2az^{-3}-7a^{-1}z^{-3}-16az-39za^{-1}+9az^{-1}+23a^{-1}z^{-1}+z^{6}-11z^{4}+26z^{2}+7z^{-2}-23$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

1

12

4

1

-3

10

2

1

8

4

5

1

6

1

4

1

2

4

2

1

7

4

2

0

6

-2

1

-4

1

-6

1

Integral Khovanov Homology

(db, data source)

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