L10a160

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,4,15,3 X_20,12,13,11 X_18,10,19,9 X_10,14,11,13 X_12,20,7,19 X_16,6,17,5 X₂₇₃₈ X_4,16,5,15 X_6,18,1,17
Gauss code	{1, -8, 2, -9, 7, -10}, {8, -1, 4, -5, 3, -6}, {5, -2, 9, -7, 10, -4, 6, -3}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{2}w^{2}+u^{2}vw^{3}-2u^{2}vw^{2}+u^{2}vw-u^{2}w^{3}+u^{2}w^{2}-uv^{2}w^{2}+uv^{2}w-uvw^{3}+2uvw^{2}-2uvw+uv-uw^{2}+uw-v^{2}w+v^{2}-vw^{2}+2vw-v-w}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{10}-2q^{9}+4q^{8}-6q^{7}+8q^{6}-7q^{5}+8q^{4}-5q^{3}+4q^{2}-2q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-6z^{2}a^{-6}+3z^{2}a^{-8}+a^{-2}+3a^{-4}-6a^{-6}+2a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}-2z^{2}a^{-12}+2z^{5}a^{-11}-3z^{3}a^{-11}+3z^{6}a^{-10}-6z^{4}a^{-10}+5z^{2}a^{-10}-a^{-10}+3z^{7}a^{-9}-6z^{5}a^{-9}+6z^{3}a^{-9}+3z^{8}a^{-8}-10z^{6}a^{-8}+19z^{4}a^{-8}-14z^{2}a^{-8}-a^{-8}z^{-2}+5a^{-8}+z^{9}a^{-7}+z^{7}a^{-7}-9z^{5}a^{-7}+16z^{3}a^{-7}-9za^{-7}+2a^{-7}z^{-1}+5z^{8}a^{-6}-20z^{6}a^{-6}+35z^{4}a^{-6}-31z^{2}a^{-6}-2a^{-6}z^{-2}+11a^{-6}+z^{9}a^{-5}-8z^{5}a^{-5}+12z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+2z^{8}a^{-4}-6z^{6}a^{-4}+5z^{4}a^{-4}-6z^{2}a^{-4}-a^{-4}z^{-2}+5a^{-4}+2z^{7}a^{-3}-7z^{5}a^{-3}+5z^{3}a^{-3}+z^{6}a^{-2}-4z^{4}a^{-2}+4z^{2}a^{-2}-a^{-2}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

χ

21

1

19

2

1

-1

17

2

15

4

2

-2

13

4

2

11

3

4

1

9

5

4

1

7

2

5

3

5

2

3

-1

3

1

3

2

1

-1

1

Integral Khovanov Homology

(db, data source)

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