L10a37

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

1

3

2

5

2

-3

0

6

4

2

-2

6

0

-4

5

0

-6

3

6

3

-8

2

5

-3

-10

1

4

3

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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