L11a67

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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_14,9,15,10 X_10,20,11,19 X_8,21,9,22 X_18,14,19,13 X_20,15,21,16 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, 5, -6, 11, -2, 8, -5, 9, -3, 4, -8, 6, -9, 7, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-2t(2)^{5}-5t(1)t(2)^{4}+9t(2)^{4}+12t(1)t(2)^{3}-16t(2)^{3}-16t(1)t(2)^{2}+12t(2)^{2}+9t(1)t(2)-5t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {18}{q^{9/2}}}-q^{7/2}+{\frac {24}{q^{7/2}}}+5q^{5/2}-{\frac {29}{q^{5/2}}}-13q^{3/2}+{\frac {30}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {9}{q^{11/2}}}+20{\sqrt {q}}-{\frac {26}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+3z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}-3z^{5}a^{3}-6z^{3}a^{3}-7za^{3}-4a^{3}z^{-1}+z^{7}a+3z^{5}a+6z^{3}a+5za+3az^{-1}-z^{5}a^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{4}z^{10}-3a^{2}z^{10}-7a^{5}z^{9}-19a^{3}z^{9}-12az^{9}-7a^{6}z^{8}-18a^{4}z^{8}-29a^{2}z^{8}-18z^{8}-4a^{7}z^{7}+2a^{5}z^{7}+20a^{3}z^{7}+az^{7}-13z^{7}a^{-1}-a^{8}z^{6}+12a^{6}z^{6}+50a^{4}z^{6}+70a^{2}z^{6}-5z^{6}a^{-2}+28z^{6}+9a^{7}z^{5}+22a^{5}z^{5}+27a^{3}z^{5}+31az^{5}+16z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-6a^{6}z^{4}-37a^{4}z^{4}-45a^{2}z^{4}+2z^{4}a^{-2}-14z^{4}-8a^{7}z^{3}-31a^{5}z^{3}-45a^{3}z^{3}-29az^{3}-7z^{3}a^{-1}-a^{8}z^{2}+a^{6}z^{2}+9a^{4}z^{2}+11a^{2}z^{2}+4z^{2}+3a^{7}z+16a^{5}z+24a^{3}z+15az+4za^{-1}-a^{6}-2a^{4}-3a^{2}-1-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

4

-4

4

9

1

8

2

11

4

-7

0

15

9

6

-2

16

12

-4

13

14

-1

-6

11

16

5

-8

7

13

-6

-10

3

12

9

-12

1

6

-5

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-2$	${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{16}$	${\mathbb {Z} }^{16}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{15}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$