L11n187

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_22,12,7,11 X_15,20,16,21 X_18,10,19,9 X_10,20,11,19 X_21,14,22,15 X_16,6,17,5 X₂₇₃₈ X_4,14,5,13 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 3, -2, 10, 7, -4, -8, 11, -5, 6, 4, -7, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-4t(1)t(2)^{3}+2t(2)^{3}-3t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-3t(2)^{2}+2t(1)^{2}t(2)-4t(1)t(2)+2t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$10q^{9/2}-11q^{7/2}+9q^{5/2}-7q^{3/2}+q^{17/2}-3q^{15/2}+6q^{13/2}-9q^{11/2}+4{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-3}-z^{5}a^{-5}+2z^{3}a^{-1}-6z^{3}a^{-3}-z^{3}a^{-5}+z^{3}a^{-7}+4za^{-1}-5za^{-3}+za^{-5}+za^{-7}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-3}-z^{9}a^{-5}-z^{8}a^{-2}-5z^{8}a^{-4}-4z^{8}a^{-6}-6z^{7}a^{-5}-6z^{7}a^{-7}+10z^{6}a^{-4}+5z^{6}a^{-6}-5z^{6}a^{-8}-3z^{5}a^{-1}+17z^{5}a^{-5}+11z^{5}a^{-7}-3z^{5}a^{-9}+4z^{4}a^{-2}-6z^{4}a^{-4}-3z^{4}a^{-6}+6z^{4}a^{-8}-z^{4}a^{-10}+8z^{3}a^{-1}+7z^{3}a^{-3}-15z^{3}a^{-5}-11z^{3}a^{-7}+3z^{3}a^{-9}-z^{2}a^{-2}+2z^{2}a^{-4}-2z^{2}a^{-8}+z^{2}a^{-10}-5za^{-1}-5za^{-3}+4za^{-5}+4za^{-7}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

2

14

4

1

-3

12

5

2

3

10

5

4

-1

8

6

5

1

6

4

6

2

4

3

5

-2

2

5

3

0

2

-2

2

Integral Khovanov Homology

(db, data source)

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