L11a471

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(3)-1)^{2}(t(2)+t(3)-1)(t(3)t(2)-t(2)-t(3))}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}-q^{-5}+9q^{4}+4q^{-4}-14q^{3}-9q^{-3}+21q^{2}+15q^{-2}-22q-20q^{-1}+24$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}-a^{4}z^{2}+z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}-z^{6}a^{-2}+2a^{2}z^{4}-2z^{4}a^{-2}+a^{2}z^{2}-4z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}-4a^{-2}-z^{6}+3z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{6}a^{-6}-2z^{4}a^{-6}+4z^{7}a^{-5}+a^{5}z^{5}-9z^{5}a^{-5}-a^{5}z^{3}+4z^{3}a^{-5}+8z^{8}a^{-4}+4a^{4}z^{6}-22z^{6}a^{-4}-5a^{4}z^{4}+21z^{4}a^{-4}+2a^{4}z^{2}-9z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+7z^{9}a^{-3}+8a^{3}z^{7}-12z^{7}a^{-3}-12a^{3}z^{5}+2z^{5}a^{-3}+7a^{3}z^{3}+4z^{3}a^{-3}-a^{3}z-2za^{-3}+2a^{-3}z^{-1}+2z^{10}a^{-2}+9a^{2}z^{8}+15z^{8}a^{-2}-10a^{2}z^{6}-50z^{6}a^{-2}+3a^{2}z^{4}+54z^{4}a^{-2}-3a^{2}z^{2}-27z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}+7a^{-2}+6az^{9}+13z^{9}a^{-1}+4az^{7}-20z^{7}a^{-1}-21az^{5}+3z^{5}a^{-1}+14az^{3}+6z^{3}a^{-1}-3az-4za^{-1}+2a^{-1}z^{-1}+2z^{10}+16z^{8}-41z^{6}+39z^{4}-23z^{2}-z^{-2}+7$ (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

6

1

5

7

8

3

-5

5

13

6

7

3

11

10

-1

1

13

11

2

-1

9

13

4

-3

6

11

-5

3

9

6

-7

1

6

-5

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$