L11a394

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-4t(2)t(3)^{3}+2t(3)^{3}+5t(1)t(3)^{2}-3t(1)t(2)t(3)^{2}+6t(2)t(3)^{2}-3t(3)^{2}-6t(1)t(3)+3t(1)t(2)t(3)-5t(2)t(3)+3t(3)+4t(1)-2t(1)t(2)+2t(2)-2}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+11q^{5}-16q^{4}+18q^{3}+q^{-3}-16q^{2}-2q^{-2}+16q+7q^{-1}-10$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}z^{-2}-2a^{-6}+z^{6}a^{-4}+3z^{4}a^{-4}+6z^{2}a^{-4}+3a^{-4}z^{-2}+6a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}-2z^{2}a^{-2}+a^{2}z^{-2}-2a^{-2}z^{-2}+2a^{2}-3a^{-2}-2z^{4}-4z^{2}-z^{-2}-3$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-5z^{4}a^{-8}+6z^{7}a^{-7}-14z^{5}a^{-7}+14z^{3}a^{-7}-7za^{-7}+2a^{-7}z^{-1}+7z^{8}a^{-6}-16z^{6}a^{-6}+17z^{4}a^{-6}-5z^{2}a^{-6}-a^{-6}z^{-2}+a^{-6}+4z^{9}a^{-5}+z^{7}a^{-5}-21z^{5}a^{-5}+40z^{3}a^{-5}-27za^{-5}+8a^{-5}z^{-1}+z^{10}a^{-4}+10z^{8}a^{-4}-25z^{6}a^{-4}+23z^{4}a^{-4}-9z^{2}a^{-4}-3a^{-4}z^{-2}+5a^{-4}+6z^{9}a^{-3}-2z^{7}a^{-3}-23z^{5}a^{-3}+43z^{3}a^{-3}-34za^{-3}+10a^{-3}z^{-1}+z^{10}a^{-2}+6z^{8}a^{-2}+a^{2}z^{6}-11z^{6}a^{-2}-4a^{2}z^{4}+2z^{4}a^{-2}+6a^{2}z^{2}-4z^{2}a^{-2}+a^{2}z^{-2}-2a^{-2}z^{-2}-4a^{2}+4a^{-2}+2z^{9}a^{-1}+2az^{7}+5z^{7}a^{-1}-4az^{5}-21z^{5}a^{-1}+19z^{3}a^{-1}+4az-10za^{-1}-2az^{-1}+2a^{-1}z^{-1}+3z^{8}-4z^{6}-3z^{4}+6z^{2}+z^{-2}-3$ (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

17

1

-1

15

2

13

5

1

-4

11

6

2

4

9

10

5

-5

7

8

6

2

5

8

10

2

3

8

0

1

5

11

6

-1

2

5

-3

5

-5

1

2

-1

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$