L11n409

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,16,8,17 X_22,17,19,18 X_20,12,21,11 X_10,20,11,19 X_18,21,5,22 X_9,14,10,15 X_15,8,16,9 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {6, -5, 7, -4}, {10, -1, -3, 9, -8, -6, 5, -2, 11, 8, -9, 3, 4, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(3)^{2}-3t(3)+1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+9q^{2}-11q+14-13q^{-1}+13q^{-2}-8q^{-3}+5q^{-4}-2q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}+3a^{2}z^{4}+z^{4}a^{-2}-3z^{4}-2a^{4}z^{2}+7a^{2}z^{2}+z^{2}a^{-2}-6z^{2}-2a^{4}+6a^{2}+2a^{-2}-6+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+6a^{2}z^{8}+5z^{8}+3a^{3}z^{7}+13az^{7}+10z^{7}a^{-1}+2a^{4}z^{6}-a^{2}z^{6}+9z^{6}a^{-2}+6z^{6}+3a^{5}z^{5}-5a^{3}z^{5}-25az^{5}-13z^{5}a^{-1}+4z^{5}a^{-3}-9a^{4}z^{4}-25a^{2}z^{4}-11z^{4}a^{-2}+z^{4}a^{-4}-28z^{4}-6a^{5}z^{3}-5a^{3}z^{3}+5az^{3}+3z^{3}a^{-1}-z^{3}a^{-3}+10a^{4}z^{2}+30a^{2}z^{2}+6z^{2}a^{-2}+26z^{2}+2a^{5}z+6a^{3}z+6az+2za^{-1}-4a^{4}-12a^{2}-4a^{-2}-11-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

χ

9

1

7

3

-3

5

6

1

5

3

5

3

-2

1

9

6

3

-1

8

9

1

-3

5

0

-5

3

8

5

-7

2

5

-3

-9

3

-11

2

-2

Integral Khovanov Homology

(db, data source)

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