L11a429

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-5t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-2t(2)^{2}-4t(1)t(3)^{2}t(2)+5t(3)^{2}t(2)-5t(1)t(2)+9t(1)t(3)t(2)-9t(3)t(2)+4t(2)+2t(1)t(3)^{2}-2t(3)^{2}+2t(1)-4t(1)t(3)+5t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{9}-4q^{8}+9q^{7}-14q^{6}+20q^{5}-22q^{4}+23q^{3}-19q^{2}-q^{-2}+15q+4q^{-1}-8$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}-2z^{4}a^{-6}-z^{4}+4z^{2}a^{-2}-4z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+4a^{-2}-6a^{-4}+2a^{-6}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-2z^{4}a^{-10}+z^{2}a^{-10}+4z^{7}a^{-9}-9z^{5}a^{-9}+5z^{3}a^{-9}+7z^{8}a^{-8}-16z^{6}a^{-8}+11z^{4}a^{-8}-4z^{2}a^{-8}+a^{-8}+6z^{9}a^{-7}-7z^{7}a^{-7}-4z^{5}a^{-7}+3z^{3}a^{-7}+2z^{10}a^{-6}+12z^{8}a^{-6}-34z^{6}a^{-6}+23z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+12z^{9}a^{-5}-18z^{7}a^{-5}+6z^{5}a^{-5}-5z^{3}a^{-5}+6za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+13z^{8}a^{-4}-28z^{6}a^{-4}+12z^{4}a^{-4}+8z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+6z^{9}a^{-3}-9z^{5}a^{-3}+z^{3}a^{-3}+6za^{-3}-2a^{-3}z^{-1}+8z^{8}a^{-2}-7z^{6}a^{-2}-4z^{4}a^{-2}+8z^{2}a^{-2}+a^{-2}z^{-2}-5a^{-2}+7z^{7}a^{-1}+az^{5}-9z^{5}a^{-1}-az^{3}+3z^{3}a^{-1}+4z^{6}-6z^{4}+3z^{2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

19

1

17

3

-3

15

6

1

5

13

9

4

-5

11

5

6

9

11

9

-2

7

12

11

1

5

8

12

4

3

7

11

-4

1

3

10

7

-1

1

5

-4

-3

3

-5

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$