L11a177

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_16,7,17,8 X_18,10,19,9 X_20,11,21,12 X_22,15,7,16 X_12,21,13,22 X_14,6,15,5 X_4,14,5,13 X_6,18,1,17 X_2,19,3,20
Gauss code	{1, -11, 2, -9, 8, -10}, {3, -1, 4, -2, 5, -7, 9, -8, 6, -3, 10, -4, 11, -5, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(v^{2}-v+1\right)\left(uv^{2}-uv+u+v-1\right)\left(uv^{2}-uv-v^{2}+v-1\right)}{uv^{3}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {9}{q^{9/2}}}-4q^{7/2}+{\frac {16}{q^{7/2}}}+9q^{5/2}-{\frac {21}{q^{5/2}}}-16q^{3/2}+{\frac {24}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+20{\sqrt {q}}-{\frac {25}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+4a^{3}z^{5}+5a^{3}z^{3}+a^{3}z-a^{3}z^{-1}-az^{9}-6az^{7}+z^{7}a^{-1}-13az^{5}+4z^{5}a^{-1}-11az^{3}+5z^{3}a^{-1}-az+za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{10}-3z^{10}-8a^{3}z^{9}-15az^{9}-7z^{9}a^{-1}-10a^{4}z^{8}-11a^{2}z^{8}-7z^{8}a^{-2}-8z^{8}-8a^{5}z^{7}+6a^{3}z^{7}+28az^{7}+10z^{7}a^{-1}-4z^{7}a^{-3}-4a^{6}z^{6}+13a^{4}z^{6}+26a^{2}z^{6}+14z^{6}a^{-2}-z^{6}a^{-4}+24z^{6}-a^{7}z^{5}+11a^{5}z^{5}+a^{3}z^{5}-20az^{5}+9z^{5}a^{-3}+5a^{6}z^{4}-5a^{4}z^{4}-14a^{2}z^{4}-6z^{4}a^{-2}+2z^{4}a^{-4}-12z^{4}+a^{7}z^{3}-5a^{5}z^{3}+2a^{3}z^{3}+12az^{3}-2z^{3}a^{-1}-6z^{3}a^{-3}-2a^{6}z^{2}-a^{4}z^{2}-z^{2}a^{-2}-z^{2}a^{-4}-z^{2}-2a^{3}z-az+2za^{-1}+za^{-3}+a^{4}+3a^{2}+3-a^{3}z^{-1}-3az^{-1}-2a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

1

-5

4

10

3

7

2

11

7

-4

0

14

9

5

-2

11

12

1

-4

10

13

-3

-6

6

11

5

-8

3

10

-7

-10

1

6

5

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

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