L11a427

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_20,7,21,8 X_8,19,9,20 X_16,12,17,11 X_10,13,5,14 X_22,18,11,17 X_18,22,19,21 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 4, -5, 10, -7}, {6, -2, 7, -3, 11, -6, 8, -9, 5, -4, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-4t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-3t(1)t(3)^{2}t(2)+4t(3)^{2}t(2)-4t(1)t(2)+7t(1)t(3)t(2)-7t(3)t(2)+3t(2)-2t(3)^{2}+2t(1)-3t(1)t(3)+4t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{4}+4q^{3}-7q^{2}+11q-15+17q^{-1}-16q^{-2}+16q^{-3}-10q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{2}+a^{6}z^{-2}+a^{6}-2a^{4}z^{4}-4a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}+a^{2}z^{6}+2a^{2}z^{4}-z^{4}a^{-2}+3a^{2}z^{2}+a^{2}z^{-2}-z^{2}a^{-2}+4a^{2}+a^{-2}+z^{6}+2z^{4}-2$ (db)
Kauffman polynomial	$a^{8}z^{4}-a^{8}z^{2}+3a^{7}z^{5}-2a^{7}z^{3}+6a^{6}z^{6}-8a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+7a^{5}z^{7}-6a^{5}z^{5}-a^{5}z^{3}+4a^{5}z-2a^{5}z^{-1}+7a^{4}z^{8}-6a^{4}z^{6}-7a^{4}z^{4}+12a^{4}z^{2}+2a^{4}z^{-2}-6a^{4}+5a^{3}z^{9}-2a^{3}z^{7}+z^{7}a^{-3}-12a^{3}z^{5}-3z^{5}a^{-3}+9a^{3}z^{3}+2z^{3}a^{-3}-2a^{3}z^{-1}+2a^{2}z^{10}+6a^{2}z^{8}+4z^{8}a^{-2}-22a^{2}z^{6}-15z^{6}a^{-2}+9a^{2}z^{4}+15z^{4}a^{-2}+5a^{2}z^{2}-2z^{2}a^{-2}+a^{2}z^{-2}-3a^{2}-a^{-2}+10az^{9}+5z^{9}a^{-1}-27az^{7}-17z^{7}a^{-1}+15az^{5}+15z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}-6az-2za^{-1}+2z^{10}+3z^{8}-25z^{6}+22z^{4}-1$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

9

1

-1

7

3

5

4

1

-3

3

7

3

4

1

8

4

-4

-1

9

7

2

-3

9

10

1

-5

7

0

-7

4

10

6

-9

3

6

-3

-11

4

-13

1

3

-2

-15

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$