L11a173

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(t(1)t(2)^{3}-t(1)t(2)^{2}+t(2)^{2}+t(1)t(2)-t(2)-t(1)+1\right)\left(t(1)t(2)^{3}-t(2)^{3}-t(1)t(2)^{2}+t(2)^{2}+t(1)t(2)-t(2)+1\right)}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-10q^{3/2}+13{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {15}{q^{3/2}}}-{\frac {14}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+5a^{3}z^{5}+8a^{3}z^{3}+4a^{3}z+a^{3}z^{-1}-az^{9}-7az^{7}+z^{7}a^{-1}-18az^{5}+5z^{5}a^{-1}-20az^{3}+8z^{3}a^{-1}-9az+4za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-5a^{3}z^{9}-9az^{9}-4z^{9}a^{-1}-6a^{4}z^{8}-a^{2}z^{8}-4z^{8}a^{-2}+z^{8}-5a^{5}z^{7}+13a^{3}z^{7}+32az^{7}+11z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+15a^{4}z^{6}+12a^{2}z^{6}+10z^{6}a^{-2}-z^{6}a^{-4}+5z^{6}-a^{7}z^{5}+11a^{5}z^{5}-17a^{3}z^{5}-53az^{5}-15z^{5}a^{-1}+9z^{5}a^{-3}+6a^{6}z^{4}-15a^{4}z^{4}-21a^{2}z^{4}-5z^{4}a^{-2}+3z^{4}a^{-4}-8z^{4}+2a^{7}z^{3}-7a^{5}z^{3}+7a^{3}z^{3}+39az^{3}+17z^{3}a^{-1}-6z^{3}a^{-3}-a^{6}z^{2}+4a^{4}z^{2}+8a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+6z^{2}+2a^{5}z-3a^{3}z-11az-6za^{-1}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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

2

6

4

1

-3

4

6

2

4

2

7

4

-3

0

9

6

3

-2

7

8

1

-4

7

8

-1

-6

4

8

4

-8

2

6

-4

-10

1

4

3

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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