L11a530

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_20,14,21,13 X_18,10,19,9 X_10,22,11,21 X_14,20,7,19 X_16,5,17,6 X_22,18,15,17 X₂₇₃₈ X_4,11,5,12 X_6,15,1,16
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 4, -5, 10, -2, 3, -6}, {11, -7, 8, -4, 6, -3, 5, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{3}-t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+2t(1)^{2}t(2)^{2}+t(1)^{2}t(3)^{2}t(2)^{2}-3t(1)t(3)^{2}t(2)^{2}+3t(3)^{2}t(2)^{2}-4t(1)t(2)^{2}-3t(1)^{2}t(3)t(2)^{2}+8t(1)t(3)t(2)^{2}-5t(3)t(2)^{2}+2t(2)^{2}-3t(1)^{2}t(2)-2t(1)^{2}t(3)^{2}t(2)+4t(1)t(3)^{2}t(2)-2t(3)^{2}t(2)+3t(1)t(2)+5t(1)^{2}t(3)t(2)-8t(1)t(3)t(2)+3t(3)t(2)-t(2)+t(1)^{2}t(3)^{2}-t(1)t(3)^{2}-2t(1)^{2}t(3)+t(1)t(3)}{t(1)t(2)^{3/2}t(3)}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+16q^{2}-20q+24-23q^{-1}+20q^{-2}-14q^{-3}+9q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3z^{2}a^{4}+a^{4}z^{-2}+3z^{4}a^{2}+z^{2}a^{2}-2a^{2}z^{-2}-3a^{2}-z^{6}-z^{4}-2z^{2}+z^{-2}+1+2z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}-z^{2}a^{-4}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-6a^{5}z^{5}+z^{5}a^{-5}+3a^{5}z^{3}-z^{3}a^{-5}+6a^{4}z^{8}-14a^{4}z^{6}+4z^{6}a^{-4}+15a^{4}z^{4}-5z^{4}a^{-4}-12a^{4}z^{2}+2z^{2}a^{-4}-a^{4}z^{-2}+5a^{4}+5a^{3}z^{9}-a^{3}z^{7}+8z^{7}a^{-3}-18a^{3}z^{5}-11z^{5}a^{-3}+21a^{3}z^{3}+5z^{3}a^{-3}-9a^{3}z+2a^{3}z^{-1}+2a^{2}z^{10}+12a^{2}z^{8}+10z^{8}a^{-2}-34a^{2}z^{6}-14z^{6}a^{-2}+32a^{2}z^{4}+8z^{4}a^{-2}-23a^{2}z^{2}-2z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}-a^{-2}+12az^{9}+7z^{9}a^{-1}-13az^{7}-z^{7}a^{-1}-11az^{5}-11z^{5}a^{-1}+18az^{3}+6z^{3}a^{-1}-9az+2az^{-1}+2z^{10}+16z^{8}-37z^{6}+27z^{4}-12z^{2}-z^{-2}+5$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

6

1

-5

5

10

3

7

3

11

7

-4

1

13

9

4

-1

12

13

1

-3

8

11

-3

-5

6

12

6

-7

3

8

-5

-9

6

-11

1

3

-2

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$