L11a476

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(3)-1)^{2}(t(3)t(2)-t(2)+1)(t(2)t(3)-t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-10q^{3}+16q^{2}-20q+25-22q^{-1}+20q^{-2}-14q^{-3}+8q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-8a^{2}z^{4}-3z^{4}a^{-2}+11z^{4}+3a^{4}z^{2}-13a^{2}z^{2}-4z^{2}a^{-2}+14z^{2}+3a^{4}-10a^{2}-3a^{-2}+10+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-7a^{5}z^{5}+z^{5}a^{-5}+5a^{5}z^{3}-z^{3}a^{-5}-a^{5}z+5a^{4}z^{8}-9a^{4}z^{6}+4z^{6}a^{-4}+4a^{4}z^{4}-4z^{4}a^{-4}-2a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+5a^{3}z^{9}-4a^{3}z^{7}+9z^{7}a^{-3}-7a^{3}z^{5}-14z^{5}a^{-3}+9a^{3}z^{3}+8z^{3}a^{-3}-7a^{3}z-3za^{-3}+2a^{3}z^{-1}+2a^{2}z^{10}+12a^{2}z^{8}+12z^{8}a^{-2}-38a^{2}z^{6}-23z^{6}a^{-2}+43a^{2}z^{4}+21z^{4}a^{-2}-31a^{2}z^{2}-12z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+5a^{-2}+13az^{9}+8z^{9}a^{-1}-20az^{7}-4z^{7}a^{-1}+az^{5}-14z^{5}a^{-1}+15az^{3}+20z^{3}a^{-1}-13az-10za^{-1}+2az^{-1}+2z^{10}+19z^{8}-55z^{6}+61z^{4}-38z^{2}-z^{-2}+15$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

1

-6

5

9

3

6

3

11

7

-4

1

14

9

5

-1

10

13

3

-3

10

12

-2

-5

6

12

6

-7

2

8

-6

-9

1

6

5

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$