L11a79

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {3t(1)t(2)^{3}-7t(2)^{3}-11t(1)t(2)^{2}+14t(2)^{2}+14t(1)t(2)-11t(2)-7t(1)+3}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-15q^{9/2}+20q^{7/2}-{\frac {1}{q^{7/2}}}-23q^{5/2}+{\frac {3}{q^{5/2}}}+22q^{3/2}-{\frac {8}{q^{3/2}}}+q^{15/2}-4q^{13/2}+10q^{11/2}-20{\sqrt {q}}+{\frac {13}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-3z^{3}a^{-5}-za^{-5}+a^{-5}z^{-1}+2z^{5}a^{-3}+2z^{3}a^{-3}+a^{3}z+za^{-3}-a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}-3z^{3}a^{-1}+az-5za^{-1}+2az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-8z^{5}a^{-7}+4z^{3}a^{-7}-za^{-7}+8z^{8}a^{-6}-19z^{6}a^{-6}+15z^{4}a^{-6}-10z^{2}a^{-6}+4a^{-6}+7z^{9}a^{-5}-9z^{7}a^{-5}-7z^{5}a^{-5}+9z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+2z^{10}a^{-4}+16z^{8}a^{-4}-54z^{6}a^{-4}+59z^{4}a^{-4}-34z^{2}a^{-4}+9a^{-4}+13z^{9}a^{-3}-21z^{7}a^{-3}+a^{3}z^{5}+2z^{5}a^{-3}-2a^{3}z^{3}+9z^{3}a^{-3}+a^{3}z-2za^{-3}-a^{-3}z^{-1}+2z^{10}a^{-2}+15z^{8}a^{-2}+3a^{2}z^{6}-44z^{6}a^{-2}-4a^{2}z^{4}+46z^{4}a^{-2}+a^{2}z^{2}-20z^{2}a^{-2}+4a^{-2}+6z^{9}a^{-1}+6az^{7}-2z^{7}a^{-1}-10az^{5}-10z^{5}a^{-1}+10az^{3}+16z^{3}a^{-1}-7az-9za^{-1}+2az^{-1}+2a^{-1}z^{-1}+7z^{8}-7z^{6}+4z^{2}-2$ (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

8

3

5

8

12

7

-5

6

11

8

3

4

11

12

1

2

9

11

-2

0

5

12

7

-2

3

8

-5

-4

5

-6

1

3

-2

-8

1

Integral Khovanov Homology

(db, data source)

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