L10a171

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw^{2}x-uvw^{2}+uvwx^{2}-3uvwx+2uvw+uvx-uv-uw^{2}x-uwx^{2}+3uwx-uw+ux^{2}-2ux+u+vw^{2}x^{2}-2vw^{2}x+vw^{2}-vwx^{2}+3vwx-vw-vx-w^{2}x^{2}+w^{2}x+2wx^{2}-3wx+w-x^{2}+x}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {14}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {14}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}+a^{9}z^{-3}-3z^{3}a^{7}-6za^{7}-6a^{7}z^{-1}-3a^{7}z^{-3}+2z^{5}a^{5}+6z^{3}a^{5}+10za^{5}+9a^{5}z^{-1}+3a^{5}z^{-3}+z^{5}a^{3}-4za^{3}-4a^{3}z^{-1}-a^{3}z^{-3}-z^{3}a-za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+5z^{4}a^{10}-z^{2}a^{10}-5z^{7}a^{9}+10z^{5}a^{9}-10z^{3}a^{9}+11za^{9}-5a^{9}z^{-1}+a^{9}z^{-3}-4z^{8}a^{8}+2z^{6}a^{8}+9z^{4}a^{8}-14z^{2}a^{8}-3a^{8}z^{-2}+10a^{8}-z^{9}a^{7}-13z^{7}a^{7}+42z^{5}a^{7}-54z^{3}a^{7}+33za^{7}-12a^{7}z^{-1}+3a^{7}z^{-3}-8z^{8}a^{6}+10z^{6}a^{6}+8z^{4}a^{6}-26z^{2}a^{6}-6a^{6}z^{-2}+19a^{6}-z^{9}a^{5}-13z^{7}a^{5}+42z^{5}a^{5}-54z^{3}a^{5}+33za^{5}-12a^{5}z^{-1}+3a^{5}z^{-3}-4z^{8}a^{4}+2z^{6}a^{4}+9z^{4}a^{4}-14z^{2}a^{4}-3a^{4}z^{-2}+10a^{4}-5z^{7}a^{3}+10z^{5}a^{3}-10z^{3}a^{3}+11za^{3}-5a^{3}z^{-1}+a^{3}z^{-3}-3z^{6}a^{2}+5z^{4}a^{2}-z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

2

-2

5

1

4

-4

6

4

-2

-6

8

3

5

-8

6

8

2

-10

8

6

2

-12

3

8

5

-14

4

6

-2

-16

1

5

4

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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