L11a403

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Link Presentations

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{4}-2uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w-2uvw^{4}+5uvw^{3}-5uvw^{2}+4uvw-uv+uw^{4}-3uw^{3}+4uw^{2}-3uw+u-v^{2}w^{4}+3v^{2}w^{3}-4v^{2}w^{2}+3v^{2}w-v^{2}+vw^{4}-4vw^{3}+5vw^{2}-5vw+2v+w^{3}-2w^{2}+2w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+9q^{5}-15q^{4}+20q^{3}-22q^{2}+23q-18+15q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+3z^{4}a^{-4}+3z^{2}a^{-4}+a^{-4}-z^{8}a^{-2}-5z^{6}a^{-2}-a^{2}z^{4}-10z^{4}a^{-2}-2a^{2}z^{2}-9z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{-2}+2z^{6}+7z^{4}+7z^{2}-2z^{-2}+1$ (db)
Kauffman polynomial	$2z^{10}a^{-2}+2z^{10}+5az^{9}+14z^{9}a^{-1}+9z^{9}a^{-3}+4a^{2}z^{8}+23z^{8}a^{-2}+15z^{8}a^{-4}+12z^{8}+a^{3}z^{7}-11az^{7}-28z^{7}a^{-1}-2z^{7}a^{-3}+14z^{7}a^{-5}-14a^{2}z^{6}-77z^{6}a^{-2}-25z^{6}a^{-4}+9z^{6}a^{-6}-57z^{6}-3a^{3}z^{5}-2az^{5}-6z^{5}a^{-1}-30z^{5}a^{-3}-19z^{5}a^{-5}+4z^{5}a^{-7}+17a^{2}z^{4}+73z^{4}a^{-2}+15z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+67z^{4}+3a^{3}z^{3}+13az^{3}+26z^{3}a^{-1}+27z^{3}a^{-3}+10z^{3}a^{-5}-z^{3}a^{-7}-8a^{2}z^{2}-30z^{2}a^{-2}-7z^{2}a^{-4}+2z^{2}a^{-6}-29z^{2}-a^{3}z-3az-7za^{-1}-7za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

3

-3

11

6

1

5

9

3

-6

7

11

6

5

12

10

-2

3

11

10

1

9

14

5

-1

6

9

-3

3

10

7

-5

1

5

-4

-7

3

-9

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$