L11a190

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_14,17,15,18 X_16,5,17,6 X_4,15,5,16 X_18,13,19,14 X_22,20,7,19 X_20,12,21,11 X_12,22,13,21 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 8, -9, 6, -3, 5, -4, 3, -6, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-6t(1)t(2)^{3}+4t(2)^{3}-4t(1)^{2}t(2)^{2}+9t(1)t(2)^{2}-4t(2)^{2}+4t(1)^{2}t(2)-6t(1)t(2)+2t(2)-t(1)^{2}+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{5/2}-3q^{3/2}+6{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {13}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {14}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {9}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+za^{7}-z^{5}a^{5}-z^{3}a^{5}+za^{5}+2a^{5}z^{-1}-2z^{5}a^{3}-5z^{3}a^{3}-6za^{3}-3a^{3}z^{-1}-z^{5}a-z^{3}a+az^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+5a^{9}z^{3}-2a^{9}z+3a^{8}z^{8}-12a^{8}z^{6}+14a^{8}z^{4}-4a^{8}z^{2}+3a^{7}z^{9}-8a^{7}z^{7}+a^{7}z^{5}+6a^{7}z^{3}-a^{7}z+a^{6}z^{10}+5a^{6}z^{8}-24a^{6}z^{6}+23a^{6}z^{4}-6a^{6}z^{2}+6a^{5}z^{9}-11a^{5}z^{7}-6a^{5}z^{5}+15a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+a^{4}z^{10}+7a^{4}z^{8}-18a^{4}z^{6}+4a^{4}z^{4}+6a^{4}z^{2}-3a^{4}+3a^{3}z^{9}+4a^{3}z^{7}-20a^{3}z^{5}+20a^{3}z^{3}-12a^{3}z+3a^{3}z^{-1}+5a^{2}z^{8}-a^{2}z^{6}-11a^{2}z^{4}+z^{4}a^{-2}+12a^{2}z^{2}-z^{2}a^{-2}-3a^{2}+6az^{7}-6az^{5}+3z^{5}a^{-1}+3az^{3}-3z^{3}a^{-1}-az+za^{-1}+az^{-1}+5z^{6}-5z^{4}+3z^{2}-1$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

2

4

1

-3

0

6

2

4

-2

8

5

-3

-4

7

5

2

-6

7

8

1

-8

6

7

-1

-10

3

7

4

-12

3

6

-3

-14

1

4

3

-16

2

-2

-18

1

Integral Khovanov Homology

(db, data source)

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