L10a57

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_14,6,15,5 X_16,11,17,12 X_20,17,7,18 X_18,14,19,13 X_12,20,13,19 X_4,16,5,15 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -9, 2, -8, 3, -10}, {9, -1, 10, -2, 4, -7, 6, -3, 8, -4, 5, -6, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(2)^{4}+t(2)^{4}-t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-3t(2)^{3}+3t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+3t(2)^{2}-3t(1)^{2}t(2)+5t(1)t(2)-t(2)+t(1)^{2}-t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}-q^{7/2}+{\frac {9}{q^{7/2}}}+3q^{5/2}-{\frac {12}{q^{5/2}}}-6q^{3/2}+{\frac {12}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+9{\sqrt {q}}-{\frac {12}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+za^{5}+a^{5}z^{-1}-z^{5}a^{3}-2z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}-z^{5}a+2za+2az^{-1}+2z^{3}a^{-1}+za^{-1}-a^{-1}z^{-1}-za^{-3}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+3a^{6}z^{6}-6a^{6}z^{4}+2a^{6}z^{2}+5a^{5}z^{7}-12a^{5}z^{5}+11a^{5}z^{3}-5a^{5}z+a^{5}z^{-1}+4a^{4}z^{8}-6a^{4}z^{6}+3a^{4}z^{4}+a^{3}z^{9}+9a^{3}z^{7}-27a^{3}z^{5}+z^{5}a^{-3}+31a^{3}z^{3}-2z^{3}a^{-3}-14a^{3}z+za^{-3}+2a^{3}z^{-1}+7a^{2}z^{8}-12a^{2}z^{6}+3z^{6}a^{-2}+9a^{2}z^{4}-6z^{4}a^{-2}-3a^{2}z^{2}+3z^{2}a^{-2}+a^{2}+az^{9}+8az^{7}+4z^{7}a^{-1}-21az^{5}-6z^{5}a^{-1}+22az^{3}+2z^{3}a^{-1}-12az-2za^{-1}+2az^{-1}+a^{-1}z^{-1}+3z^{8}-6z^{4}+2z^{2}$ (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

7

4

3

-2

6

0

-4

6

0

-6

4

7

3

-8

2

5

-3

-10

1

4

3

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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