L10a26

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}+2t(1)t(2)^{4}-4t(2)^{4}-6t(1)t(2)^{3}+8t(2)^{3}+8t(1)t(2)^{2}-6t(2)^{2}-4t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {10}{q^{9/2}}}+{\frac {12}{q^{7/2}}}+q^{5/2}-{\frac {15}{q^{5/2}}}-4q^{3/2}+{\frac {13}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+8{\sqrt {q}}-{\frac {11}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-2z^{5}a^{3}-6z^{3}a^{3}-9za^{3}-4a^{3}z^{-1}-z^{5}a+2za+az^{-1}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-8a^{4}z^{8}-5a^{2}z^{8}-3a^{7}z^{7}-10a^{5}z^{7}-16a^{3}z^{7}-9az^{7}-a^{8}z^{6}+3a^{6}z^{6}+8a^{4}z^{6}-4a^{2}z^{6}-8z^{6}+9a^{7}z^{5}+34a^{5}z^{5}+41a^{3}z^{5}+12az^{5}-4z^{5}a^{-1}+3a^{8}z^{4}+9a^{6}z^{4}+16a^{4}z^{4}+20a^{2}z^{4}-z^{4}a^{-2}+9z^{4}-9a^{7}z^{3}-34a^{5}z^{3}-34a^{3}z^{3}-7az^{3}+2z^{3}a^{-1}-3a^{8}z^{2}-13a^{6}z^{2}-21a^{4}z^{2}-14a^{2}z^{2}-3z^{2}+4a^{7}z+16a^{5}z+15a^{3}z+3az+a^{8}+4a^{6}+7a^{4}+4a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-4a^{3}z^{-1}-az^{-1}$ (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

χ

6

1

-1

4

3

2

5

1

-4

0

6

3

-2

8

6

-2

-4

7

5

2

-6

5

8

3

-8

5

7

-2

-10

2

6

4

-12

1

4

-3

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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