L10a137

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(3)t(2)^{2}+t(3)^{2}t(2)-t(3)t(2)+t(2)+t(3)\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-5q^{6}+10q^{5}-12q^{4}+14q^{3}-12q^{2}+11q-7+4q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+3z^{2}a^{-4}-2z^{2}a^{-6}-z^{2}-a^{-4}+1+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-7z^{4}a^{-8}+4z^{2}a^{-8}+4z^{7}a^{-7}-7z^{5}a^{-7}+3z^{3}a^{-7}+4z^{8}a^{-6}-6z^{6}a^{-6}+3z^{4}a^{-6}+3z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+2z^{9}a^{-5}+2z^{7}a^{-5}-6z^{5}a^{-5}+3z^{3}a^{-5}+4za^{-5}-2a^{-5}z^{-1}+9z^{8}a^{-4}-19z^{6}a^{-4}+18z^{4}a^{-4}-5z^{2}a^{-4}+2a^{-4}z^{-2}-4a^{-4}+2z^{9}a^{-3}+4z^{7}a^{-3}-9z^{5}a^{-3}+z^{3}a^{-3}+4za^{-3}-2a^{-3}z^{-1}+5z^{8}a^{-2}-6z^{6}a^{-2}+z^{4}a^{-2}-2z^{2}a^{-2}+a^{-2}z^{-2}-a^{-2}+6z^{7}a^{-1}+az^{5}-10z^{5}a^{-1}-az^{3}+2z^{3}a^{-1}+4z^{6}-7z^{4}+2z^{2}+1$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

2

13

3

1

-2

11

7

2

5

9

7

5

-2

7

5

2

5

7

2

3

6

7

-1

1

3

7

4

-1

1

4

-3

3

-5

1

-1

Integral Khovanov Homology

(db, data source)

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