L11a397

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_22,12,9,11 X_20,14,21,13 X_18,16,19,15 X_8,18,5,17 X_16,8,17,7 X_14,20,15,19 X_12,22,13,21 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6}, {11, -2, 3, -9, 4, -8, 5, -7, 6, -5, 8, -4, 9, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-3t(2)t(1)+3t(2)t(3)t(1)-3t(3)t(1)+4t(1)+3t(2)-4t(2)t(3)+3t(3)-3}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$-q^{7}+3q^{6}-4q^{5}+5q^{4}-7q^{3}+8q^{2}-7q+7-4q^{-1}+4q^{-2}-q^{-3}+q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-6}+z^{4}a^{-4}+a^{4}z^{-2}+z^{2}a^{-4}+a^{4}+z^{4}a^{-2}-2a^{2}z^{2}-2a^{2}z^{-2}-3a^{2}+z^{4}+z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$z^{7}a^{-7}-4z^{5}a^{-7}+3z^{3}a^{-7}+3z^{8}a^{-6}-14z^{6}a^{-6}+16z^{4}a^{-6}-3z^{2}a^{-6}+3z^{9}a^{-5}-14z^{7}a^{-5}+18z^{5}a^{-5}-7z^{3}a^{-5}+z^{10}a^{-4}-z^{8}a^{-4}-8z^{6}a^{-4}+a^{4}z^{4}+10z^{4}a^{-4}-3a^{4}z^{2}-2z^{2}a^{-4}-a^{4}z^{-2}+3a^{4}+4z^{9}a^{-3}-17z^{7}a^{-3}+a^{3}z^{5}+22z^{5}a^{-3}-10z^{3}a^{-3}-3a^{3}z+2a^{3}z^{-1}+z^{10}a^{-2}-3z^{8}a^{-2}+a^{2}z^{6}+5z^{6}a^{-2}+2a^{2}z^{4}-6z^{4}a^{-2}-6a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+5a^{2}+z^{9}a^{-1}+az^{7}-z^{7}a^{-1}+az^{5}-3az+2az^{-1}+z^{8}+z^{4}-3z^{2}-z^{-2}+3$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

15

1

-1

13

2

11

2

1

-1

9

3

2

1

7

4

2

-2

5

4

3

1

3

4

1

4

0

-1

3

6

3

-3

1

0

-5

3

-7

1

0

-9

1

Integral Khovanov Homology

(db, data source)

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