L11a479

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}(t(2)+t(3))}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+9q^{2}-14q+19-20q^{-1}+21q^{-2}-16q^{-3}+13q^{-4}-7q^{-5}+3q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}+2z^{4}a^{4}+3z^{2}a^{4}+a^{4}z^{-2}+2a^{4}-z^{6}a^{2}-z^{4}a^{2}-2a^{2}z^{-2}-a^{2}-z^{6}-2z^{4}-3z^{2}+z^{-2}-1+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-4a^{7}z^{5}+5a^{7}z^{3}-2a^{7}z+3a^{6}z^{8}-11a^{6}z^{6}+14a^{6}z^{4}-9a^{6}z^{2}+3a^{6}+4a^{5}z^{9}-10a^{5}z^{7}+a^{5}z^{5}+11a^{5}z^{3}-7a^{5}z+2a^{4}z^{10}+6a^{4}z^{8}-37a^{4}z^{6}+48a^{4}z^{4}+z^{4}a^{-4}-27a^{4}z^{2}-a^{4}z^{-2}+9a^{4}+12a^{3}z^{9}-28a^{3}z^{7}+7a^{3}z^{5}+4z^{5}a^{-3}+15a^{3}z^{3}-z^{3}a^{-3}-10a^{3}z+2a^{3}z^{-1}+2a^{2}z^{10}+16a^{2}z^{8}-54a^{2}z^{6}+9z^{6}a^{-2}+48a^{2}z^{4}-8z^{4}a^{-2}-20a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}-a^{-2}+8az^{9}-4az^{7}+13z^{7}a^{-1}-19az^{5}-17z^{5}a^{-1}+17az^{3}+7z^{3}a^{-1}-6az-za^{-1}+2az^{-1}+13z^{8}-19z^{6}+5z^{4}+z^{2}-z^{-2}+1$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

3

-3

5

6

1

5

3

8

3

-5

1

11

6

5

-1

11

10

-1

-3

10

9

1

-5

8

13

5

-7

5

8

-3

-9

2

8

6

-11

1

5

-4

-13

2

-15

1

-1

Integral Khovanov Homology

(db, data source)

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