L11a399

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,6,13,5 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_4,15,1,16 X_18,22,19,21 X_20,9,21,10 X_10,19,5,20 X_16,12,17,11 X_22,18,11,17
Gauss code	{1, -4, 3, -6}, {2, -1, 5, -3, 8, -9}, {10, -2, 4, -5, 6, -10, 11, -7, 9, -8, 7, -11}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2(t(1)-1)(t(2)-1)^{2}(t(3)-1)^{2}}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{6}-4q^{5}-q^{-5}+8q^{4}+4q^{-4}-13q^{3}-7q^{-3}+18q^{2}+14q^{-2}-20q-17q^{-1}+21$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}-a^{4}z^{2}+z^{2}a^{-4}+a^{4}z^{-2}-z^{6}a^{-2}+2a^{2}z^{4}-2z^{4}a^{-2}+2a^{2}z^{2}-2z^{2}a^{-2}-2a^{2}z^{-2}-a^{2}-z^{6}-z^{4}+z^{-2}+1$ (db)
Kauffman polynomial	$z^{6}a^{-6}-2z^{4}a^{-6}+4z^{7}a^{-5}+a^{5}z^{5}-10z^{5}a^{-5}-a^{5}z^{3}+5z^{3}a^{-5}+7z^{8}a^{-4}+4a^{4}z^{6}-19z^{6}a^{-4}-7a^{4}z^{4}+15z^{4}a^{-4}+4a^{4}z^{2}-4z^{2}a^{-4}-a^{4}z^{-2}+a^{4}+6z^{9}a^{-3}+6a^{3}z^{7}-11z^{7}a^{-3}-7a^{3}z^{5}+2z^{5}a^{-3}+a^{3}z^{3}+3z^{3}a^{-3}-a^{3}z+2a^{3}z^{-1}+2z^{10}a^{-2}+6a^{2}z^{8}+9z^{8}a^{-2}-a^{2}z^{6}-32z^{6}a^{-2}-12a^{2}z^{4}+30z^{4}a^{-2}+8a^{2}z^{2}-8z^{2}a^{-2}-2a^{2}z^{-2}+a^{2}+5az^{9}+11z^{9}a^{-1}-az^{7}-22z^{7}a^{-1}-3az^{5}+17z^{5}a^{-1}-2az^{3}-6z^{3}a^{-1}-az+2az^{-1}+2z^{10}+8z^{8}-17z^{6}+8z^{4}-z^{-2}+1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

3

-3

9

5

1

4

7

8

3

-5

5

10

5

3

10

8

-2

1

11

10

1

-1

8

12

4

-3

6

9

-3

-5

3

10

7

-7

1

4

-3

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

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