L11n99

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,21,14,20 X_7,17,8,16 X_19,9,20,8 X_9,19,10,18 X_17,11,18,10 X_15,5,16,22 X_21,15,22,14 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, -6, 7, 11, -2, -3, 9, -8, 4, -7, 6, -5, 3, -9, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{3}-uv^{2}-uv+2u+2v^{5}-v^{4}-v^{3}+v^{2}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{15/2}-q^{13/2}+q^{11/2}-2q^{3/2}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}-z^{5}a^{-3}+az^{3}-4z^{3}a^{-1}-5z^{3}a^{-3}+3az-2za^{-1}-6za^{-3}+2za^{-5}+za^{-7}+az^{-1}+a^{-1}z^{-1}-4a^{-3}z^{-1}+2a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-2z^{8}a^{-2}-z^{8}a^{-4}-z^{8}-az^{7}+6z^{7}a^{-1}+8z^{7}a^{-3}-z^{7}a^{-7}+13z^{6}a^{-2}+8z^{6}a^{-4}-z^{6}a^{-6}-z^{6}a^{-8}+5z^{6}+6az^{5}-11z^{5}a^{-1}-22z^{5}a^{-3}+5z^{5}a^{-7}-25z^{4}a^{-2}-19z^{4}a^{-4}+6z^{4}a^{-6}+5z^{4}a^{-8}-5z^{4}-10az^{3}+10z^{3}a^{-1}+29z^{3}a^{-3}+4z^{3}a^{-5}-5z^{3}a^{-7}+19z^{2}a^{-2}+19z^{2}a^{-4}-7z^{2}a^{-6}-6z^{2}a^{-8}-z^{2}+5az-5za^{-1}-16za^{-3}-6za^{-5}-5a^{-2}-6a^{-4}+a^{-6}+2a^{-8}+1-az^{-1}+a^{-1}z^{-1}+4a^{-3}z^{-1}+2a^{-5}z^{-1}$ (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

0

12

1

0

10

2

1

-1

8

1

0

6

2

3

1

0

4

1

2

1

3

1

0

1

-2

1

-4

1

0

-6

1

Integral Khovanov Homology

(db, data source)

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