L11a239

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{3}-6u^{2}v^{2}+7u^{2}v-2u^{2}+uv^{4}-7uv^{3}+13uv^{2}-7uv+u-2v^{4}+7v^{3}-6v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {17}{q^{9/2}}}+{\frac {20}{q^{7/2}}}+q^{5/2}-{\frac {20}{q^{5/2}}}-4q^{3/2}+{\frac {18}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {8}{q^{13/2}}}+{\frac {12}{q^{11/2}}}+8{\sqrt {q}}-{\frac {14}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-4a^{7}z-2a^{7}z^{-1}+5a^{5}z^{3}+6a^{5}z+2a^{5}z^{-1}-2a^{3}z^{5}-3a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}-az^{5}+z^{3}a^{-1}-az$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+6a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-10a^{8}z^{6}+10a^{8}z^{4}-3a^{8}z^{2}+4a^{7}z^{9}-9a^{7}z^{7}-2a^{7}z^{5}+12a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+2a^{6}z^{10}+6a^{6}z^{8}-35a^{6}z^{6}+39a^{6}z^{4}-15a^{6}z^{2}+a^{6}+12a^{5}z^{9}-28a^{5}z^{7}+6a^{5}z^{5}+15a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+2a^{4}z^{10}+16a^{4}z^{8}-57a^{4}z^{6}+53a^{4}z^{4}-17a^{4}z^{2}+8a^{3}z^{9}-6a^{3}z^{7}-16a^{3}z^{5}+17a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}+13a^{2}z^{8}-24a^{2}z^{6}+17a^{2}z^{4}+z^{4}a^{-2}-5a^{2}z^{2}+12az^{7}-16az^{5}+4z^{5}a^{-1}+6az^{3}-2z^{3}a^{-1}+az+8z^{6}-6z^{4}$ (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

5

1

-4

0

9

3

6

-2

10

6

-4

10

8

2

-6

10

0

-8

7

10

-3

-10

5

10

5

-12

3

7

-4

-14

1

6

5

-16

2

-2

-18

1

Integral Khovanov Homology

(db, data source)

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