L11n135

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_3,10,4,11 X_17,3,18,2 X_12,5,13,6 X₆₇₁₈ X_16,10,17,9 X_20,14,21,13 X_22,16,7,15 X_4,20,5,19 X_14,22,15,21
Gauss code	{1, 4, -3, -10, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -11, 9, -7, -4, 2, 10, -8, 11, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)^{2}t(2)^{3}+t(1)t(2)^{3}-t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+t(1)t(2)-t(2)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-3q^{9/2}+3q^{7/2}-4q^{5/2}+3q^{3/2}-{\frac {1}{q^{3/2}}}+q^{15/2}-q^{13/2}+2q^{11/2}-3{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+5z^{3}a^{-5}+7za^{-5}+2a^{-5}z^{-1}-z^{7}a^{-3}-7z^{5}a^{-3}-17z^{3}a^{-3}-16za^{-3}-5a^{-3}z^{-1}+z^{5}a^{-1}+5z^{3}a^{-1}+8za^{-1}+3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-3}-z^{9}a^{-5}-z^{8}a^{-2}-2z^{8}a^{-4}-z^{8}a^{-6}+7z^{7}a^{-3}+6z^{7}a^{-5}-z^{7}a^{-7}+6z^{6}a^{-2}+11z^{6}a^{-4}+4z^{6}a^{-6}-z^{6}a^{-8}-2z^{5}a^{-1}-21z^{5}a^{-3}-15z^{5}a^{-5}+4z^{5}a^{-7}-15z^{4}a^{-2}-23z^{4}a^{-4}-4z^{4}a^{-6}+5z^{4}a^{-8}-z^{4}-az^{3}+6z^{3}a^{-1}+29z^{3}a^{-3}+19z^{3}a^{-5}-3z^{3}a^{-7}+13z^{2}a^{-2}+20z^{2}a^{-4}+2z^{2}a^{-6}-6z^{2}a^{-8}+z^{2}+2az-9za^{-1}-20za^{-3}-9za^{-5}-5a^{-2}-5a^{-4}+a^{-8}+3a^{-1}z^{-1}+5a^{-3}z^{-1}+2a^{-5}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

0

12

2

1

-1

10

1

8

2

0

6

2

1

4

1

2

1

2

0

2

-2

1

0

-4

1

Integral Khovanov Homology

(db, data source)

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