L11a348

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_10,11,1,12 X_14,5,15,6 X_18,9,19,10 X_2,17,3,18 X_16,8,17,7 X₈₄₉₃ X_20,16,21,15 X_22,13,11,14 X_4,20,5,19 X_6,21,7,22
Gauss code	{1, -5, 7, -10, 3, -11, 6, -7, 4, -2}, {2, -1, 9, -3, 8, -6, 5, -4, 10, -8, 11, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(2t(2)t(1)-t(1)-t(2)+2)\left(t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-2t(1)+t(2)^{2}-2t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+17{\sqrt {q}}-{\frac {25}{\sqrt {q}}}+{\frac {28}{q^{3/2}}}-{\frac {30}{q^{5/2}}}+{\frac {26}{q^{7/2}}}-{\frac {20}{q^{9/2}}}+{\frac {13}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-5a^{3}z^{3}+5az^{3}-2z^{3}a^{-1}+3a^{5}z-8a^{3}z+4az-za^{-1}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-4a^{4}z^{10}-4a^{2}z^{10}-12a^{5}z^{9}-22a^{3}z^{9}-10az^{9}-13a^{6}z^{8}-20a^{4}z^{8}-18a^{2}z^{8}-11z^{8}-6a^{7}z^{7}+16a^{5}z^{7}+36a^{3}z^{7}+6az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+26a^{6}z^{6}+59a^{4}z^{6}+49a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}+8a^{7}z^{5}+6a^{5}z^{5}-2a^{3}z^{5}+11az^{5}+10z^{5}a^{-1}-z^{5}a^{-3}-10a^{6}z^{4}-33a^{4}z^{4}-33a^{2}z^{4}+5z^{4}a^{-2}-5z^{4}-6a^{5}z^{3}-17a^{3}z^{3}-18az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-2a^{6}z^{2}+a^{4}z^{2}+3a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}-2a^{7}z+3a^{5}z+10a^{3}z+7az+2za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

11

3

-8

0

14

6

8

-2

15

12

-3

-4

15

13

2

-6

11

15

4

-8

9

15

-6

-10

5

12

7

-12

1

8

-7

-14

5

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=-3$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-2$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{15}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{15}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$