L11a134

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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_10,5,11,6 X_14,3,15,4 X_18,13,19,14 X_22,20,5,19 X_20,12,21,11 X_12,22,13,21 X_2,9,3,10 X_8,15,9,16
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 10, -4, 8, -9, 6, -5, 11, -2, 3, -6, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}+4t(1)t(2)^{4}-6t(2)^{4}-10t(1)t(2)^{3}+12t(2)^{3}+12t(1)t(2)^{2}-10t(2)^{2}-6t(1)t(2)+4t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+9{\sqrt {q}}-{\frac {14}{\sqrt {q}}}+{\frac {18}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {20}{q^{7/2}}}-{\frac {18}{q^{9/2}}}+{\frac {13}{q^{11/2}}}-{\frac {8}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}-a^{7}z^{-1}-z^{5}a^{5}+z^{3}a^{5}+5za^{5}+4a^{5}z^{-1}-3z^{5}a^{3}-8z^{3}a^{3}-10za^{3}-4a^{3}z^{-1}-z^{5}a+z^{3}a+3za+az^{-1}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-3a^{9}z^{5}+3a^{9}z^{3}-a^{9}z+4a^{8}z^{8}-14a^{8}z^{6}+15a^{8}z^{4}-4a^{8}z^{2}-a^{8}+5a^{7}z^{9}-13a^{7}z^{7}+4a^{7}z^{5}+8a^{7}z^{3}-4a^{7}z+a^{7}z^{-1}+2a^{6}z^{10}+8a^{6}z^{8}-39a^{6}z^{6}+35a^{6}z^{4}-2a^{6}z^{2}-4a^{6}+12a^{5}z^{9}-23a^{5}z^{7}-8a^{5}z^{5}+27a^{5}z^{3}-14a^{5}z+4a^{5}z^{-1}+2a^{4}z^{10}+16a^{4}z^{8}-47a^{4}z^{6}+24a^{4}z^{4}+8a^{4}z^{2}-7a^{4}+7a^{3}z^{9}+4a^{3}z^{7}-37a^{3}z^{5}+34a^{3}z^{3}-15a^{3}z+4a^{3}z^{-1}+12a^{2}z^{8}-13a^{2}z^{6}-5a^{2}z^{4}+z^{4}a^{-2}+9a^{2}z^{2}-4a^{2}+13az^{7}-18az^{5}+4z^{5}a^{-1}+11az^{3}-z^{3}a^{-1}-4az+az^{-1}+9z^{6}-8z^{4}+3z^{2}-1$ (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

3

2

6

1

-5

0

8

3

5

-2

11

7

-4

11

7

4

-6

9

11

2

-8

9

11

-2

-10

5

10

5

-12

3

8

-5

-14

1

5

4

-16

3

-3

-18

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$