L11a448

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}-t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}-t(3)t(2)^{3}-2t(1)t(3)^{3}t(2)^{2}+t(3)^{3}t(2)^{2}+2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-2t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)+2t(1)t(3)t(2)-2t(3)t(2)+2t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)+t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-5q^{7}+7q^{6}-9q^{5}+11q^{4}-9q^{3}+9q^{2}+q^{-2}-6q-2q^{-1}+5$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-6}-4z^{4}a^{-6}-4z^{2}a^{-6}-2a^{-6}+z^{8}a^{-4}+6z^{6}a^{-4}+13z^{4}a^{-4}+14z^{2}a^{-4}+a^{-4}z^{-2}+7a^{-4}-2z^{6}a^{-2}-10z^{4}a^{-2}-15z^{2}a^{-2}-2a^{-2}z^{-2}-9a^{-2}+z^{4}+4z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+z^{8}a^{-2}+6z^{8}a^{-4}+6z^{8}a^{-6}+z^{8}-10z^{7}a^{-1}-25z^{7}a^{-3}-9z^{7}a^{-5}+6z^{7}a^{-7}-25z^{6}a^{-2}-39z^{6}a^{-4}-14z^{6}a^{-6}+6z^{6}a^{-8}-6z^{6}+14z^{5}a^{-1}+24z^{5}a^{-3}-3z^{5}a^{-5}-8z^{5}a^{-7}+5z^{5}a^{-9}+54z^{4}a^{-2}+56z^{4}a^{-4}+5z^{4}a^{-6}-7z^{4}a^{-8}+3z^{4}a^{-10}+13z^{4}-3z^{3}a^{-1}+4z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-7}-4z^{3}a^{-9}+z^{3}a^{-11}-41z^{2}a^{-2}-31z^{2}a^{-4}+2z^{2}a^{-8}-z^{2}a^{-10}-13z^{2}-5za^{-1}-8za^{-3}-3za^{-5}+za^{-7}+za^{-9}+13a^{-2}+9a^{-4}-a^{-8}+6+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

19

1

-1

17

2

15

3

1

-2

13

4

2

11

6

4

-2

9

5

3

2

7

4

6

2

5

0

3

4

7

3

1

2

-1

1

4

3

-3

1

-1

-5

1

Integral Khovanov Homology

(db, data source)

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