L11a105

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Link Presentations

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

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

Polynomial invariants

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

12

1

-1

10

3

8

4

1

-3

6

7

3

4

8

4

-4

2

8

7

1

0

8

9

1

-2

5

7

-2

-4

4

8

4

-6

2

5

-3

-8

1

5

4

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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