L11a444

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_20,12,21,11 X_16,8,17,7 X_18,10,19,9 X_10,20,11,19 X_22,16,13,15 X_12,18,5,17 X_8,22,9,21 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 5, -6, 3, -8}, {11, -2, 7, -4, 8, -5, 6, -3, 9, -7}

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

Polynomial invariants

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

17

1

-1

15

3

13

5

1

-4

11

8

3

5

9

7

-2

7

9

6

3

5

8

9

1

3

6

9

-3

1

4

9

5

-1

2

5

-3

4

-5

1

2

-1

-7

1

Integral Khovanov Homology

(db, data source)

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