L11n438

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_11,19,12,18 X_3,11,4,10 X_9,1,10,4 X_7,15,8,14 X_13,5,14,8 X_19,13,20,22 X_15,21,16,20 X_21,17,22,16 X_17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 11}, {-7, 6, -9, 10, -11, 3, -8, 9, -10, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(4)^{3}-t(1)t(2)t(4)^{3}+t(3)t(4)^{3}-t(4)^{3}-2t(1)t(4)^{2}+3t(1)t(2)t(4)^{2}-t(2)t(4)^{2}+t(1)t(3)t(4)^{2}-2t(1)t(2)t(3)t(4)^{2}+2t(2)t(3)t(4)^{2}-3t(3)t(4)^{2}+2t(4)^{2}+2t(1)t(4)-3t(1)t(2)t(4)+t(2)t(4)-t(1)t(3)t(4)+2t(1)t(2)t(3)t(4)-2t(2)t(3)t(4)+3t(3)t(4)-2t(4)+t(1)t(2)-t(1)t(2)t(3)+t(2)t(3)-t(3)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$11q^{9/2}-13q^{7/2}+6q^{5/2}-4q^{3/2}+q^{21/2}-2q^{19/2}+6q^{17/2}-10q^{15/2}+12q^{13/2}-15q^{11/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-3z^{5}a^{-5}+4z^{3}a^{-3}-12z^{3}a^{-5}+6z^{3}a^{-7}+9za^{-3}-22za^{-5}+17za^{-7}-4za^{-9}+7a^{-3}z^{-1}-20a^{-5}z^{-1}+20a^{-7}z^{-1}-8a^{-9}z^{-1}+a^{-11}z^{-1}+2a^{-3}z^{-3}-7a^{-5}z^{-3}+9a^{-7}z^{-3}-5a^{-9}z^{-3}+a^{-11}z^{-3}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-4z^{4}a^{-12}+6z^{2}a^{-12}+a^{-12}z^{-2}-4a^{-12}+2z^{7}a^{-11}-5z^{5}a^{-11}+4z^{3}a^{-11}-a^{-11}z^{-3}-2za^{-11}+2a^{-11}z^{-1}+2z^{8}a^{-10}+2z^{6}a^{-10}-21z^{4}a^{-10}+34z^{2}a^{-10}+7a^{-10}z^{-2}-24a^{-10}+z^{9}a^{-9}+8z^{7}a^{-9}-25z^{5}a^{-9}+22z^{3}a^{-9}-5a^{-9}z^{-3}-16za^{-9}+12a^{-9}z^{-1}+7z^{8}a^{-8}-46z^{4}a^{-8}+75z^{2}a^{-8}+18a^{-8}z^{-2}-58a^{-8}+z^{9}a^{-7}+16z^{7}a^{-7}-46z^{5}a^{-7}+50z^{3}a^{-7}-9a^{-7}z^{-3}-37za^{-7}+24a^{-7}z^{-1}+5z^{8}a^{-6}+5z^{6}a^{-6}-41z^{4}a^{-6}+73z^{2}a^{-6}+19a^{-6}z^{-2}-60a^{-6}+10z^{7}a^{-5}-26z^{5}a^{-5}+42z^{3}a^{-5}-7a^{-5}z^{-3}-39za^{-5}+23a^{-5}z^{-1}+6z^{6}a^{-4}-12z^{4}a^{-4}+26z^{2}a^{-4}+7a^{-4}z^{-2}-23a^{-4}+10z^{3}a^{-3}-2a^{-3}z^{-3}-16za^{-3}+9a^{-3}z^{-1}$ (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		\

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

1

18

5

1

-4

16

5

1

4

14

7

5

-2

12

8

5

3

10

7

11

4

8

6

4

2

6

7

4

6

-2

2

4

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=0$	${\mathbb {Z} }^{4}$	${\mathbb {Z} }^{4}$
$r=1$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{8}$
$r=5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=9$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$