L11a53

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{11/2}+4q^{9/2}-7q^{7/2}+11q^{5/2}-16q^{3/2}+17{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {15}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+4az^{5}+3z^{5}a^{-1}-z^{5}a^{-3}-3a^{3}z^{3}+7az^{3}-2z^{3}a^{-3}-3a^{3}z+8az-6za^{-1}+za^{-3}-a^{3}z^{-1}+4az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+z^{7}a^{-5}+3a^{5}z^{5}-3z^{5}a^{-5}-2a^{5}z^{3}+2z^{3}a^{-5}+4z^{8}a^{-4}+6a^{4}z^{6}-15z^{6}a^{-4}-6a^{4}z^{4}+15z^{4}a^{-4}+4a^{4}z^{2}-z^{2}a^{-4}-a^{4}-a^{-4}+5z^{9}a^{-3}+9a^{3}z^{7}-17z^{7}a^{-3}-15a^{3}z^{5}+16z^{5}a^{-3}+15a^{3}z^{3}-3z^{3}a^{-3}-6a^{3}z-2za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+2z^{10}a^{-2}+8a^{2}z^{8}+3z^{8}a^{-2}-8a^{2}z^{6}-23z^{6}a^{-2}-6a^{2}z^{4}+18z^{4}a^{-2}+12a^{2}z^{2}+3z^{2}a^{-2}-4a^{2}-4a^{-2}+5az^{9}+10z^{9}a^{-1}+2az^{7}-25z^{7}a^{-1}-26az^{5}+11z^{5}a^{-1}+29az^{3}+7z^{3}a^{-1}-15az-11za^{-1}+4az^{-1}+4a^{-1}z^{-1}+2z^{10}+7z^{8}-22z^{6}+4z^{4}+11z^{2}-7$ (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

12

1

10

3

-3

8

4

1

3

6

7

3

-4

4

9

4

5

2

8

7

-1

0

10

9

1

-2

7

10

3

-4

5

8

-3

-6

2

7

5

-8

1

5

-4

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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