L11a183

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_18,7,19,8 X_20,10,21,9 X_22,17,7,18 X_14,6,15,5 X_16,12,17,11 X_12,16,13,15 X_4,14,5,13 X_6,20,1,19 X_2,21,3,22
Gauss code	{1, -11, 2, -9, 6, -10}, {3, -1, 4, -2, 7, -8, 9, -6, 8, -7, 5, -3, 10, -4, 11, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+t(2)^{4}-5t(1)^{2}t(2)^{3}+9t(1)t(2)^{3}-4t(2)^{3}+6t(1)^{2}t(2)^{2}-13t(1)t(2)^{2}+6t(2)^{2}-4t(1)^{2}t(2)+9t(1)t(2)-5t(2)+t(1)^{2}-3t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-15q^{9/2}+20q^{7/2}-{\frac {1}{q^{7/2}}}-24q^{5/2}+{\frac {4}{q^{5/2}}}+23q^{3/2}-{\frac {9}{q^{3/2}}}+q^{15/2}-4q^{13/2}+9q^{11/2}-21{\sqrt {q}}+{\frac {15}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+2z^{3}a^{-5}+za^{-5}-z^{7}a^{-3}-3z^{5}a^{-3}-3z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-3z^{5}a^{-1}+2az^{3}-3z^{3}a^{-1}+az+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-9z^{5}a^{-7}+6z^{3}a^{-7}-za^{-7}+7z^{8}a^{-6}-15z^{6}a^{-6}+9z^{4}a^{-6}-z^{2}a^{-6}+6z^{9}a^{-5}-5z^{7}a^{-5}-9z^{5}a^{-5}+8z^{3}a^{-5}-za^{-5}+2z^{10}a^{-4}+13z^{8}a^{-4}-33z^{6}a^{-4}+19z^{4}a^{-4}-3z^{2}a^{-4}+12z^{9}a^{-3}-12z^{7}a^{-3}+a^{3}z^{5}-10z^{5}a^{-3}-a^{3}z^{3}+9z^{3}a^{-3}-a^{-3}z^{-1}+2z^{10}a^{-2}+15z^{8}a^{-2}+4a^{2}z^{6}-30z^{6}a^{-2}-5a^{2}z^{4}+13z^{4}a^{-2}+2a^{2}z^{2}-z^{2}a^{-2}+a^{-2}+6z^{9}a^{-1}+8az^{7}+5z^{7}a^{-1}-12az^{5}-23z^{5}a^{-1}+8az^{3}+16z^{3}a^{-1}-2az-2za^{-1}-a^{-1}z^{-1}+9z^{8}-9z^{6}+2z^{2}$ (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

16

1

-1

14

3

12

6

1

-5

10

9

3

6

8

11

6

-5

6

13

9

4

11

12

1

2

10

12

-2

0

6

12

6

-2

3

9

-6

-4

1

6

5

-6

3

-3

-8

1

Integral Khovanov Homology

(db, data source)

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