L11a8

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_20,9,21,10 X₈₄₉₃ X_18,22,19,21 X_14,12,15,11 X_12,5,13,6 X_22,13,5,14 X_10,19,11,20 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -1, 2, -5, 4, -10, 7, -8, 9, -7, 11, -2, 3, -6, 10, -4, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-5t(2)^{3}+9t(2)^{2}-5t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+5q^{5/2}-11q^{3/2}+17{\sqrt {q}}-{\frac {25}{\sqrt {q}}}+{\frac {27}{q^{3/2}}}-{\frac {27}{q^{5/2}}}+{\frac {23}{q^{7/2}}}-{\frac {17}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+3z^{3}a^{5}+2za^{5}-a^{5}z^{-1}-3z^{5}a^{3}-5z^{3}a^{3}+3a^{3}z^{-1}+z^{7}a+3z^{5}a+4z^{3}a-za-2az^{-1}-z^{5}a^{-1}-z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-8a^{7}z^{5}+6a^{7}z^{3}-2a^{7}z+8a^{6}z^{8}-17a^{6}z^{6}+14a^{6}z^{4}-6a^{6}z^{2}+a^{6}+7a^{5}z^{9}-3a^{5}z^{7}-18a^{5}z^{5}+18a^{5}z^{3}-4a^{5}z-a^{5}z^{-1}+2a^{4}z^{10}+22a^{4}z^{8}-59a^{4}z^{6}+47a^{4}z^{4}-16a^{4}z^{2}+3a^{4}+15a^{3}z^{9}-8a^{3}z^{7}-35a^{3}z^{5}+z^{5}a^{-3}+30a^{3}z^{3}-a^{3}z-3a^{3}z^{-1}+2a^{2}z^{10}+27a^{2}z^{8}-61a^{2}z^{6}+5z^{6}a^{-2}+39a^{2}z^{4}-4z^{4}a^{-2}-11a^{2}z^{2}+3a^{2}+8az^{9}+10az^{7}+11z^{7}a^{-1}-41az^{5}-15z^{5}a^{-1}+25az^{3}+7z^{3}a^{-1}+az-2az^{-1}+13z^{8}-15z^{6}+4z^{4}-2z^{2}$ (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

4

-4

4

7

1

6

2

10

4

-6

0

15

7

8

-2

14

12

-2

-4

13

0

-6

10

14

4

-8

7

13

-6

-10

3

10

7

-12

1

7

-6

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{15}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$