L11a484

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)(w-1)^{3}\left(w^{2}+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-13q^{4}-q^{-4}+18q^{3}+4q^{-3}-20q^{2}-8q^{-2}+21q+14q^{-1}-16$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+3z^{4}a^{-4}+2z^{2}a^{-4}-z^{8}a^{-2}-5z^{6}a^{-2}-a^{2}z^{4}-9z^{4}a^{-2}-2a^{2}z^{2}-6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{6}+7z^{4}+6z^{2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-8}+4z^{5}a^{-7}-2z^{3}a^{-7}+8z^{6}a^{-6}-7z^{4}a^{-6}+z^{2}a^{-6}+11z^{7}a^{-5}-13z^{5}a^{-5}+4z^{3}a^{-5}-za^{-5}+12z^{8}a^{-4}-20z^{6}a^{-4}+12z^{4}a^{-4}-5z^{2}a^{-4}+8z^{9}a^{-3}+a^{3}z^{7}-6z^{7}a^{-3}-3a^{3}z^{5}-18z^{5}a^{-3}+3a^{3}z^{3}+20z^{3}a^{-3}-a^{3}z-4za^{-3}+2z^{10}a^{-2}+4a^{2}z^{8}+18z^{8}a^{-2}-14a^{2}z^{6}-67z^{6}a^{-2}+16a^{2}z^{4}+66z^{4}a^{-2}-8a^{2}z^{2}-21z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+5az^{9}+13z^{9}a^{-1}-12az^{7}-30z^{7}a^{-1}-az^{5}+z^{5}a^{-1}+12az^{3}+23z^{3}a^{-1}-4az-6za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{10}+10z^{8}-53z^{6}+62z^{4}-23z^{2}+2z^{-2}+1$ (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

15

1

13

3

-3

11

5

1

4

9

8

3

-5

7

10

5

10

8

-2

3

11

10

1

9

14

5

-1

5

7

-2

-3

3

9

6

-5

1

5

-4

-7

3

-9

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$