L11n392

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,12,6,13 X₈₄₉₃ X_22,14,19,13 X_20,10,21,9 X_10,20,11,19 X_14,22,15,21 X_11,18,12,5 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -7, 9, -6}, {-4, -1, 2, -5, 7, -8, -10, 4, 6, -9, 11, -2, 3, 10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(3)-1)^{2}(t(3)t(2)-2t(2)-2t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-5}+3q^{4}+4q^{-4}-6q^{3}-9q^{-3}+13q^{2}+13q^{-2}-14q-16q^{-1}+17$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}+2a^{-4}z^{-2}+2a^{-4}+2a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{2}-3z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{2}-8a^{-2}-z^{6}-z^{4}+3z^{2}+4z^{-2}+8$ (db)
Kauffman polynomial	$a^{5}z^{5}-a^{5}z^{3}+4a^{4}z^{6}-5a^{4}z^{4}+6z^{4}a^{-4}+2a^{4}z^{2}-11z^{2}a^{-4}-2a^{-4}z^{-2}+8a^{-4}+8a^{3}z^{7}+3z^{7}a^{-3}-14a^{3}z^{5}-3z^{5}a^{-3}+9a^{3}z^{3}+10z^{3}a^{-3}-3a^{3}z-13za^{-3}+a^{3}z^{-1}+5a^{-3}z^{-1}+7a^{2}z^{8}+5z^{8}a^{-2}-6a^{2}z^{6}-8z^{6}a^{-2}-4a^{2}z^{4}+17z^{4}a^{-2}-24z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{2}+16a^{-2}+2az^{9}+2z^{9}a^{-1}+15az^{7}+10z^{7}a^{-1}-38az^{5}-26z^{5}a^{-1}+29az^{3}+29z^{3}a^{-1}-14az-24za^{-1}+5az^{-1}+9a^{-1}z^{-1}+12z^{8}-18z^{6}+12z^{4}-15z^{2}-4z^{-2}+11$ (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

χ

9

3

7

5

2

-3

5

8

1

7

3

6

5

-1

1

11

8

3

-1

7

8

1

-3

6

9

-3

-5

3

7

4

-7

1

6

-5

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$