L11a45

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_18,9,19,10 X_20,15,21,16 X_16,19,17,20 X_8,17,9,18 X_4,21,1,22 X_12,6,13,5 X_10,4,11,3 X_22,12,5,11 X_2,14,3,13
Gauss code	{1, -11, 9, -7}, {8, -1, 2, -6, 3, -9, 10, -8, 11, -2, 4, -5, 6, -3, 5, -4, 7, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)^{2}}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+15{\sqrt {q}}-{\frac {21}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {24}{q^{5/2}}}+{\frac {20}{q^{7/2}}}-{\frac {14}{q^{9/2}}}+{\frac {9}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+3a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-2a^{5}z^{3}+3a^{3}z^{3}+3az^{3}-2z^{3}a^{-1}-a^{5}z+2a^{3}z-za^{-1}-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-6a^{5}z^{9}-12a^{3}z^{9}-6az^{9}-7a^{6}z^{8}-12a^{4}z^{8}-14a^{2}z^{8}-9z^{8}-4a^{7}z^{7}+7a^{5}z^{7}+15a^{3}z^{7}-4az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+16a^{6}z^{6}+31a^{4}z^{6}+27a^{2}z^{6}-4z^{6}a^{-2}+9z^{6}+9a^{7}z^{5}+2a^{5}z^{5}+20az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-11a^{6}z^{4}-19a^{4}z^{4}-10a^{2}z^{4}+5z^{4}a^{-2}+z^{4}-5a^{7}z^{3}+a^{3}z^{3}-13az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}-a^{8}z^{2}+2a^{6}z^{2}+5a^{4}z^{2}+a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}-2a^{5}z-3a^{3}z+az+2za^{-1}-a^{6}-3a^{4}-3a^{2}+a^{5}z^{-1}+3a^{3}z^{-1}+2az^{-1}$ (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

8

1

6

3

-3

4

6

1

5

2

9

3

-6

0

12

6

-2

12

11

-1

-4

12

10

2

-6

8

12

4

-8

6

12

-6

-10

3

8

5

-12

1

6

-5

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

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