L11a187

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{4}+2t(1)^{2}t(2)^{3}-7t(1)t(2)^{3}+4t(2)^{3}-5t(1)^{2}t(2)^{2}+11t(1)t(2)^{2}-5t(2)^{2}+4t(1)^{2}t(2)-7t(1)t(2)+2t(2)+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {7}{q^{9/2}}}-3q^{7/2}+{\frac {10}{q^{7/2}}}+6q^{5/2}-{\frac {15}{q^{5/2}}}-10q^{3/2}+{\frac {16}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+14{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-a^{3}z^{3}+z^{3}a^{-3}-a^{3}z+za^{-3}-2az^{5}-z^{5}a^{-1}-4az^{3}-z^{3}a^{-1}-4az-2az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-4a^{3}z^{9}-7az^{9}-3z^{9}a^{-1}-6a^{4}z^{8}-9a^{2}z^{8}-4z^{8}a^{-2}-7z^{8}-6a^{5}z^{7}+4a^{3}z^{7}+16az^{7}+3z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+12a^{4}z^{6}+29a^{2}z^{6}+10z^{6}a^{-2}-z^{6}a^{-4}+25z^{6}-a^{7}z^{5}+15a^{5}z^{5}+a^{3}z^{5}-20az^{5}+4z^{5}a^{-1}+9z^{5}a^{-3}+5a^{6}z^{4}-9a^{4}z^{4}-38a^{2}z^{4}-7z^{4}a^{-2}+3z^{4}a^{-4}-34z^{4}+2a^{7}z^{3}-15a^{5}z^{3}-5a^{3}z^{3}+17az^{3}-2z^{3}a^{-1}-7z^{3}a^{-3}+19a^{2}z^{2}+4z^{2}a^{-2}-2z^{2}a^{-4}+25z^{2}+7a^{5}z-9az-za^{-1}+za^{-3}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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

2

6

4

1

-3

4

6

2

4

2

8

4

-4

0

8

6

2

-2

9

0

-4

6

7

-1

-6

4

9

5

-8

3

6

-3

-10

4

-12

1

3

-2

-14

1

Integral Khovanov Homology

(db, data source)

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