L11a103

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {3(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-6q^{5/2}+7q^{3/2}-10{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{5}-za^{5}+a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}-2a^{3}z^{-1}+z^{5}a+z^{3}a+az^{-1}+z^{5}a^{-1}+2z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}-z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-3a^{5}z^{9}-8a^{3}z^{9}-5az^{9}-a^{6}z^{8}+7a^{4}z^{8}+a^{2}z^{8}-7z^{8}+17a^{5}z^{7}+38a^{3}z^{7}+14az^{7}-7z^{7}a^{-1}+5a^{6}z^{6}+20a^{2}z^{6}-7z^{6}a^{-2}+18z^{6}-30a^{5}z^{5}-51a^{3}z^{5}-5az^{5}+10z^{5}a^{-1}-6z^{5}a^{-3}-7a^{6}z^{4}-13a^{4}z^{4}-29a^{2}z^{4}+8z^{4}a^{-2}-3z^{4}a^{-4}-12z^{4}+17a^{5}z^{3}+15a^{3}z^{3}-6az^{3}+3z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+3a^{6}z^{2}+6a^{4}z^{2}+10a^{2}z^{2}+7z^{2}+a^{5}z+7a^{3}z+6az-3za^{-1}-3za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-2a^{3}z^{-1}-az^{-1}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

10

1

8

2

-2

6

4

1

3

4

3

2

-1

2

7

4

3

0

6

5

-1

-2

4

5

-1

-4

5

6

1

-6

2

4

-2

-8

2

5

3

-10

1

2

-1

-12

2

-14

1

-1

Integral Khovanov Homology

(db, data source)

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