L11n435

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,4,15,3 X_12,14,7,13 X₂₇₃₈ X_22,10,13,9 X_6,22,1,21 X_20,16,21,15 X_16,5,17,6 X_11,19,12,18 X_17,11,18,10 X_4,19,5,20
Gauss code	{1, -4, 2, -11, 8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, 7, -8, -10, 9, 11, -7, 6, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)^{2}t(3)^{3}+t(1)t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+2t(1)^{2}t(3)^{2}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-2t(1)^{2}t(2)t(3)^{2}+5t(1)t(2)t(3)^{2}-2t(2)t(3)^{2}-t(1)^{2}t(3)+2t(1)t(2)^{2}t(3)-2t(2)^{2}t(3)+2t(1)t(3)+2t(1)^{2}t(2)t(3)-5t(1)t(2)t(3)+2t(2)t(3)-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+2t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+12q^{5}-14q^{4}+16q^{3}-13q^{2}+11q+3q^{-1}-6$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}-4z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-10z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}+3z^{2}-8a^{-2}+5a^{-4}-a^{-6}+4-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-z^{3}a^{-9}+4z^{6}a^{-8}-6z^{4}a^{-8}+z^{2}a^{-8}+7z^{7}a^{-7}-12z^{5}a^{-7}+4z^{3}a^{-7}-za^{-7}+7z^{8}a^{-6}-12z^{6}a^{-6}+7z^{4}a^{-6}-4z^{2}a^{-6}+a^{-6}+3z^{9}a^{-5}+4z^{7}a^{-5}-17z^{5}a^{-5}+14z^{3}a^{-5}-3za^{-5}+13z^{8}a^{-4}-34z^{6}a^{-4}+41z^{4}a^{-4}-24z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}+3z^{9}a^{-3}-7z^{5}a^{-3}+12z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+6z^{8}a^{-2}-18z^{6}a^{-2}+34z^{4}a^{-2}-31z^{2}a^{-2}-2a^{-2}z^{-2}+12a^{-2}+3z^{7}a^{-1}-3z^{5}a^{-1}+3z^{3}a^{-1}-6za^{-1}+2a^{-1}z^{-1}+6z^{4}-12z^{2}-z^{-2}+6$ (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

χ

17

1

-1

15

3

13

5

1

-4

11

7

3

4

9

8

6

-2

7

8

6

2

5

6

9

3

5

7

-2

1

2

7

5

-1

1

4

-3

3

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-2$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$