L11a119

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(v-2)(2v-1)(uv-2u-2v+1)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+7q^{5/2}-12q^{3/2}+16{\sqrt {q}}-{\frac {17}{\sqrt {q}}}+{\frac {17}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3za^{5}-a^{5}z^{-1}+3z^{3}a^{3}-a^{3}z^{-1}-z^{5}a+2z^{3}a+4za+2az^{-1}-z^{5}a^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}+z^{3}a^{-3}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-7az^{9}-4z^{9}a^{-1}-4a^{4}z^{8}-9a^{2}z^{8}-6z^{8}a^{-2}-11z^{8}-3a^{5}z^{7}-a^{3}z^{7}+8az^{7}+2z^{7}a^{-1}-4z^{7}a^{-3}-2a^{6}z^{6}+5a^{4}z^{6}+25a^{2}z^{6}+16z^{6}a^{-2}-z^{6}a^{-4}+35z^{6}-a^{7}z^{5}+2a^{5}z^{5}+8a^{3}z^{5}+10az^{5}+16z^{5}a^{-1}+11z^{5}a^{-3}+3a^{6}z^{4}-6a^{4}z^{4}-30a^{2}z^{4}-12z^{4}a^{-2}+2z^{4}a^{-4}-35z^{4}+3a^{7}z^{3}+3a^{5}z^{3}-12a^{3}z^{3}-23az^{3}-18z^{3}a^{-1}-7z^{3}a^{-3}+5a^{4}z^{2}+16a^{2}z^{2}+4z^{2}a^{-2}+15z^{2}-3a^{7}z-3a^{5}z+7a^{3}z+13az+6za^{-1}-a^{6}-2a^{4}-3a^{2}-1+a^{7}z^{-1}+a^{5}z^{-1}-a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

4

1

-3

4

8

3

5

2

8

4

-4

0

9

8

1

-2

9

0

-4

6

8

-2

-6

4

9

5

-8

2

6

-4

-10

1

5

4

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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