L11a129

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(2)^{3}-3t(2)^{3}-6t(1)t(2)^{2}+6t(2)^{2}+6t(1)t(2)-6t(2)-3t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}+3za^{7}+3a^{7}z^{-1}-3z^{3}a^{5}-5za^{5}-2a^{5}z^{-1}+z^{5}a^{3}+z^{3}a^{3}+z^{5}a+2z^{3}a+za-z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+2z^{2}a^{10}-a^{10}-2z^{5}a^{9}+3z^{3}a^{9}-2za^{9}+a^{9}z^{-1}-2z^{6}a^{8}-z^{4}a^{8}+5z^{2}a^{8}-3a^{8}-2z^{7}a^{7}-2z^{5}a^{7}+7z^{3}a^{7}-7za^{7}+3a^{7}z^{-1}-2z^{8}a^{6}+z^{4}a^{6}+3z^{2}a^{6}-3a^{6}-2z^{9}a^{5}+3z^{7}a^{5}-4z^{5}a^{5}+7z^{3}a^{5}-5za^{5}+2a^{5}z^{-1}-z^{10}a^{4}-z^{8}a^{4}+8z^{6}a^{4}-7z^{4}a^{4}+2z^{2}a^{4}-5z^{9}a^{3}+17z^{7}a^{3}-16z^{5}a^{3}+6z^{3}a^{3}-za^{3}-z^{10}a^{2}-2z^{8}a^{2}+19z^{6}a^{2}-23z^{4}a^{2}+7z^{2}a^{2}-3z^{9}a+11z^{7}a-8z^{5}a-z^{3}a-3z^{8}+13z^{6}-15z^{4}+5z^{2}-z^{7}a^{-1}+4z^{5}a^{-1}-4z^{3}a^{-1}+za^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

2

-2

2

3

1

2

0

4

2

-2

5

3

2

-4

6

5

-1

-6

5

4

1

-8

3

6

3

-10

4

5

-1

-12

1

4

3

-14

1

3

-2

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

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