L11a180

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-5t(1)^{2}t(2)^{3}+11t(1)t(2)^{3}-6t(2)^{3}+9t(1)^{2}t(2)^{2}-19t(1)t(2)^{2}+9t(2)^{2}-6t(1)^{2}t(2)+11t(1)t(2)-5t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{15/2}-4q^{13/2}+10q^{11/2}-18q^{9/2}+24q^{7/2}-29q^{5/2}+29q^{3/2}-26{\sqrt {q}}+{\frac {19}{\sqrt {q}}}-{\frac {12}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-3z^{5}a^{-1}+3z^{5}a^{-3}+az^{3}-5z^{3}a^{-1}+5z^{3}a^{-3}-3z^{3}a^{-5}+az-za^{-1}+2za^{-3}-2za^{-5}+za^{-7}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-8z^{5}a^{-7}+6z^{3}a^{-7}-2za^{-7}+8z^{8}a^{-6}-16z^{6}a^{-6}+12z^{4}a^{-6}-4z^{2}a^{-6}+8z^{9}a^{-5}-7z^{7}a^{-5}-10z^{5}a^{-5}+12z^{3}a^{-5}-3za^{-5}+3z^{10}a^{-4}+19z^{8}a^{-4}-54z^{6}a^{-4}+41z^{4}a^{-4}-10z^{2}a^{-4}+19z^{9}a^{-3}-23z^{7}a^{-3}+a^{3}z^{5}-12z^{5}a^{-3}+15z^{3}a^{-3}-a^{-3}z^{-1}+3z^{10}a^{-2}+27z^{8}a^{-2}+5a^{2}z^{6}-65z^{6}a^{-2}-3a^{2}z^{4}+41z^{4}a^{-2}-8z^{2}a^{-2}+a^{-2}+11z^{9}a^{-1}+12az^{7}-15az^{5}-26z^{5}a^{-1}+6az^{3}+15z^{3}a^{-1}-az-a^{-1}z^{-1}+16z^{8}-23z^{6}+11z^{4}-3z^{2}$ (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

7

1

-6

10

11

3

8

13

7

-6

6

16

11

5

4

14

0

2

12

15

-3

0

8

15

7

-2

4

11

-7

-4

1

8

7

-6

4

-4

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{16}$
$r=3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$