L10a35

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-2t(2)^{3}+t(2)^{2}-2t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {10}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-3za^{7}-2a^{7}z^{-1}+2z^{5}a^{5}+8z^{3}a^{5}+10za^{5}+5a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-9z^{3}a^{3}-8za^{3}-3a^{3}z^{-1}+z^{5}a+3z^{3}a+za$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+a^{10}+2a^{9}z^{5}-2a^{9}z^{3}+3a^{8}z^{6}-3a^{8}z^{4}+a^{8}z^{2}+3a^{7}z^{7}-3a^{7}z^{5}+4a^{7}z^{3}-5a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}+a^{6}z^{6}-7a^{6}z^{4}+10a^{6}z^{2}-5a^{6}+a^{5}z^{9}+3a^{5}z^{7}-12a^{5}z^{5}+18a^{5}z^{3}-15a^{5}z+5a^{5}z^{-1}+5a^{4}z^{8}-12a^{4}z^{6}+5a^{4}z^{4}+7a^{4}z^{2}-5a^{4}+a^{3}z^{9}+3a^{3}z^{7}-18a^{3}z^{5}+22a^{3}z^{3}-12a^{3}z+3a^{3}z^{-1}+3a^{2}z^{8}-9a^{2}z^{6}+5a^{2}z^{4}+a^{2}z^{2}+3az^{7}-11az^{5}+10az^{3}-2az+z^{6}-3z^{4}+z^{2}$ (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

4

1

-1

2

0

2

1

-1

-2

5

2

3

-4

5

4

-1

-6

5

3

2

-8

3

5

2

-10

4

5

-1

-12

1

3

2

-14

1

4

-3

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

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