L11a385

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-4t(2)t(3)^{3}+2t(3)^{3}+6t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+7t(2)t(3)^{2}-4t(3)^{2}-7t(1)t(3)+4t(1)t(2)t(3)-6t(2)t(3)+4t(3)+4t(1)-2t(1)t(2)+2t(2)-2}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-10q^{6}+14q^{5}-18q^{4}+21q^{3}+q^{-3}-19q^{2}-2q^{-2}+17q+7q^{-1}-10$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}+2a^{2}-3a^{-2}+6a^{-4}-2a^{-6}-3+a^{2}z^{-2}-2a^{-2}z^{-2}+3a^{-4}z^{-2}-a^{-6}z^{-2}-z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+7z^{9}a^{-3}+5z^{9}a^{-5}+8z^{8}a^{-2}+15z^{8}a^{-4}+10z^{8}a^{-6}+3z^{8}+2az^{7}+6z^{7}a^{-1}+z^{7}a^{-3}+6z^{7}a^{-5}+9z^{7}a^{-7}+a^{2}z^{6}-15z^{6}a^{-2}-31z^{6}a^{-4}-17z^{6}a^{-6}+4z^{6}a^{-8}-4z^{6}-4az^{5}-23z^{5}a^{-1}-33z^{5}a^{-3}-31z^{5}a^{-5}-16z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+6z^{4}a^{-2}+18z^{4}a^{-4}+10z^{4}a^{-6}-4z^{4}a^{-8}-2z^{4}+23z^{3}a^{-1}+50z^{3}a^{-3}+39z^{3}a^{-5}+11z^{3}a^{-7}-z^{3}a^{-9}+6a^{2}z^{2}-3z^{2}a^{-2}-4z^{2}a^{-4}-2z^{2}a^{-6}+5z^{2}+4az-10za^{-1}-34za^{-3}-27za^{-5}-7za^{-7}-4a^{2}+4a^{-2}+5a^{-4}+a^{-6}-3-2az^{-1}+2a^{-1}z^{-1}+10a^{-3}z^{-1}+8a^{-5}z^{-1}+2a^{-7}z^{-1}+a^{2}z^{-2}-2a^{-2}z^{-2}-3a^{-4}z^{-2}-a^{-6}z^{-2}+z^{-2}$ (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

17

1

-1

15

3

13

7

1

-6

11

7

3

4

9

11

7

-4

7

10

7

3

5

9

11

2

3

8

10

-2

1

5

12

7

-1

2

5

-3

5

-5

1

2

-1

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$