L11a128

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_16,10,17,9 X_18,12,19,11 X_10,18,11,17 X_12,16,13,15 X_22,19,5,20 X_20,7,21,8 X_8,21,9,22 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -9, 3, -5, 4, -6, 11, -2, 6, -3, 5, -4, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(v^{2}-v+1\right)^{2}(uv-2u-2v+1)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-8q^{3/2}+13{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {17}{q^{3/2}}}-{\frac {18}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-2a^{7}z^{-1}+3z^{3}a^{5}+9za^{5}+7a^{5}z^{-1}-3z^{5}a^{3}-11z^{3}a^{3}-15za^{3}-7a^{3}z^{-1}+z^{7}a+4z^{5}a+7z^{3}a+6za+2az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-2a^{5}z^{9}-6a^{3}z^{9}-4az^{9}-2a^{6}z^{8}-4a^{4}z^{8}-9a^{2}z^{8}-7z^{8}-2a^{7}z^{7}-a^{5}z^{7}+6a^{3}z^{7}-2az^{7}-7z^{7}a^{-1}-a^{8}z^{6}+5a^{4}z^{6}+16a^{2}z^{6}-4z^{6}a^{-2}+8z^{6}+6a^{7}z^{5}+11a^{5}z^{5}+5a^{3}z^{5}+12az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}+4a^{8}z^{4}+13a^{6}z^{4}+11a^{4}z^{4}-2a^{2}z^{4}+6z^{4}a^{-2}+2z^{4}-6a^{7}z^{3}-17a^{5}z^{3}-15a^{3}z^{3}-9az^{3}-4z^{3}a^{-1}+z^{3}a^{-3}-5a^{8}z^{2}-18a^{6}z^{2}-24a^{4}z^{2}-13a^{2}z^{2}-2z^{2}a^{-2}-4z^{2}+4a^{7}z+16a^{5}z+15a^{3}z+4az+za^{-1}+2a^{8}+8a^{6}+13a^{4}+8a^{2}+2-2a^{7}z^{-1}-7a^{5}z^{-1}-7a^{3}z^{-1}-2az^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

5

1

4

2

8

3

-5

0

8

5

3

-2

10

9

-1

-4

8

7

1

-6

5

10

5

-8

5

8

-3

-10

1

6

5

-12

1

4

-3

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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