L10a61

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-6t(1)t(2)^{3}+3t(2)^{3}-4t(1)^{2}t(2)^{2}+9t(1)t(2)^{2}-4t(2)^{2}+3t(1)^{2}t(2)-6t(1)t(2)+2t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{19/2}+4q^{17/2}-7q^{15/2}+11q^{13/2}-14q^{11/2}+14q^{9/2}-14q^{7/2}+10q^{5/2}-7q^{3/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-3}-2z^{5}a^{-5}+z^{3}a^{-1}-5z^{3}a^{-5}+3z^{3}a^{-7}+za^{-1}+3za^{-3}-6za^{-5}+4za^{-7}-za^{-9}+2a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-z^{3}a^{-11}+4z^{6}a^{-10}-7z^{4}a^{-10}+3z^{2}a^{-10}+6z^{7}a^{-9}-10z^{5}a^{-9}+5z^{3}a^{-9}-2za^{-9}+4z^{8}a^{-8}+3z^{6}a^{-8}-18z^{4}a^{-8}+12z^{2}a^{-8}-a^{-8}+z^{9}a^{-7}+14z^{7}a^{-7}-34z^{5}a^{-7}+28z^{3}a^{-7}-10za^{-7}+a^{-7}z^{-1}+8z^{8}a^{-6}-5z^{6}a^{-6}-13z^{4}a^{-6}+14z^{2}a^{-6}-3a^{-6}+z^{9}a^{-5}+13z^{7}a^{-5}-33z^{5}a^{-5}+32z^{3}a^{-5}-15za^{-5}+3a^{-5}z^{-1}+4z^{8}a^{-4}-z^{6}a^{-4}-7z^{4}a^{-4}+7z^{2}a^{-4}-3a^{-4}+5z^{7}a^{-3}-9z^{5}a^{-3}+8z^{3}a^{-3}-6za^{-3}+2a^{-3}z^{-1}+3z^{6}a^{-2}-5z^{4}a^{-2}+2z^{2}a^{-2}+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

χ

20

1

18

3

-3

16

4

1

3

14

7

3

-4

12

7

4

3

10

7

0

8

7

0

6

4

8

4

3

6

-3

2

1

5

4

0

2

-2

1

Integral Khovanov Homology

(db, data source)

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