L11a86

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_14,10,15,9 X_10,14,11,13 X_20,17,21,18 X_18,7,19,8 X_8,19,9,20 X_22,15,5,16 X_16,21,17,22 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 6, -7, 3, -4, 11, -2, 4, -3, 8, -9, 5, -6, 7, -5, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(2)^{3}-4t(2)^{3}-8t(1)t(2)^{2}+9t(2)^{2}+9t(1)t(2)-8t(2)-4t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {14}{q^{7/2}}}+{\frac {14}{q^{9/2}}}-{\frac {15}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {9}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+3za^{9}+2a^{9}z^{-1}-3z^{3}a^{7}-2za^{7}+z^{5}a^{5}-z^{3}a^{5}-3za^{5}-a^{5}z^{-1}+z^{5}a^{3}+z^{3}a^{3}+za^{3}-z^{3}a-za$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-5z^{2}a^{12}+2a^{12}-2z^{7}a^{11}+6z^{5}a^{11}-5z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-2z^{8}a^{10}+z^{6}a^{10}+10z^{4}a^{10}-13z^{2}a^{10}+5a^{10}-2z^{9}a^{9}+z^{7}a^{9}+5z^{5}a^{9}-6z^{3}a^{9}+5za^{9}-2a^{9}z^{-1}-z^{10}a^{8}-2z^{8}a^{8}+3z^{6}a^{8}+4z^{4}a^{8}-6z^{2}a^{8}+3a^{8}-5z^{9}a^{7}+9z^{7}a^{7}-9z^{5}a^{7}+4z^{3}a^{7}-z^{10}a^{6}-4z^{8}a^{6}+7z^{6}a^{6}-5z^{4}a^{6}+2z^{2}a^{6}-a^{6}-3z^{9}a^{5}+2z^{7}a^{5}-2z^{5}a^{5}+3z^{3}a^{5}-2za^{5}+a^{5}z^{-1}-4z^{8}a^{4}+3z^{6}a^{4}+3z^{4}a^{4}-3z^{2}a^{4}-4z^{7}a^{3}+5z^{5}a^{3}-3z^{6}a^{2}+6z^{4}a^{2}-3z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

2

-2

4

1

3

-4

7

3

-4

-6

7

3

4

-8

7

0

-10

8

7

1

-12

5

8

3

-14

4

7

-3

-16

1

5

4

-18

1

4

-3

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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