L11a2

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_14,10,15,9 X₈₄₉₃ X_10,5,11,6 X_18,11,19,12 X_20,14,21,13 X_22,19,5,20 X_12,22,13,21 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -10, 8, -4, 11, -2, 3, -7, 9, -8, 10, -9}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-5t(2)^{3}+9t(2)^{2}-5t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {11}{q^{9/2}}}-5q^{7/2}+{\frac {17}{q^{7/2}}}+10q^{5/2}-{\frac {24}{q^{5/2}}}-17q^{3/2}+{\frac {27}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+23{\sqrt {q}}-{\frac {27}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-3a^{3}z^{3}+5az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-a^{3}z+az+a^{3}z^{-1}-2az^{-1}+2a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-8a^{3}z^{9}-15az^{9}-7z^{9}a^{-1}-13a^{4}z^{8}-26a^{2}z^{8}-9z^{8}a^{-2}-22z^{8}-11a^{5}z^{7}-8a^{3}z^{7}+9az^{7}+z^{7}a^{-1}-5z^{7}a^{-3}-5a^{6}z^{6}+16a^{4}z^{6}+58a^{2}z^{6}+19z^{6}a^{-2}-z^{6}a^{-4}+57z^{6}-a^{7}z^{5}+15a^{5}z^{5}+35a^{3}z^{5}+32az^{5}+23z^{5}a^{-1}+10z^{5}a^{-3}+4a^{6}z^{4}-5a^{4}z^{4}-35a^{2}z^{4}-12z^{4}a^{-2}+z^{4}a^{-4}-39z^{4}-6a^{5}z^{3}-21a^{3}z^{3}-27az^{3}-17z^{3}a^{-1}-5z^{3}a^{-3}+a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}+7z^{2}+a^{3}z-2za^{-1}-za^{-3}+1+a^{3}z^{-1}+2az^{-1}+2a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

4

6

1

-5

4

11

4

7

2

12

6

-6

0

15

11

4

-2

14

0

-4

10

13

-3

-6

7

14

7

-8

4

10

-6

-10

1

7

6

-12

4

-4

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{15}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$