L11a136

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_2,9,3,10 X_14,3,15,4 X_10,5,11,6 X_22,11,5,12 X_20,13,21,14 X_12,21,13,22 X_4,19,1,20 X_18,16,19,15 X_16,8,17,7 X_8,18,9,17
Gauss code	{1, -2, 3, -8}, {4, -1, 10, -11, 2, -4, 5, -7, 6, -3, 9, -10, 11, -9, 8, -6, 7, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {3t(1)t(2)^{3}-4t(2)^{3}-8t(1)t(2)^{2}+10t(2)^{2}+10t(1)t(2)-8t(2)-4t(1)+3}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+7{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {14}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {15}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {9}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+za^{7}+a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-4za^{5}-2a^{5}z^{-1}-z^{5}a^{3}+z^{3}a^{3}+3za^{3}+2a^{3}z^{-1}-z^{5}a-z^{3}a-2za-az^{-1}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+4a^{9}z^{3}-a^{9}z+3a^{8}z^{8}-12a^{8}z^{6}+13a^{8}z^{4}-5a^{8}z^{2}+4a^{7}z^{9}-15a^{7}z^{7}+16a^{7}z^{5}-8a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+2a^{6}z^{10}-20a^{6}z^{6}+27a^{6}z^{4}-9a^{6}z^{2}+10a^{5}z^{9}-36a^{5}z^{7}+43a^{5}z^{5}-24a^{5}z^{3}+9a^{5}z-2a^{5}z^{-1}+2a^{4}z^{10}+5a^{4}z^{8}-28a^{4}z^{6}+29a^{4}z^{4}-6a^{4}z^{2}-a^{4}+6a^{3}z^{9}-12a^{3}z^{7}+11a^{3}z^{5}-11a^{3}z^{3}+8a^{3}z-2a^{3}z^{-1}+8a^{2}z^{8}-13a^{2}z^{6}+7a^{2}z^{4}+z^{4}a^{-2}-2a^{2}z^{2}+8az^{7}-8az^{5}+4z^{5}a^{-1}-2az^{3}-3z^{3}a^{-1}+3az-az^{-1}+7z^{6}-7z^{4}$ (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

3

2

4

1

-3

0

7

3

4

-2

8

5

-3

-4

8

6

2

-6

7

8

1

-8

6

8

-2

-10

4

8

4

-12

2

5

-3

-14

1

4

3

-16

2

-2

-18

1

Integral Khovanov Homology

(db, data source)

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