L10a23

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_18,9,19,10 X_8,17,9,18 X_4,19,1,20 X_14,12,15,11 X_10,4,11,3 X_12,5,13,6 X_20,13,5,14 X_2,16,3,15
Gauss code	{1, -10, 7, -5}, {8, -1, 2, -4, 3, -7, 6, -8, 9, -6, 10, -2, 4, -3, 5, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(v^{2}-3v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {14}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-2a^{7}z-a^{7}z^{-1}+2a^{5}z^{5}+6a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-5a^{3}z^{3}-2a^{3}z+az^{5}+2az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-2a^{9}z^{3}+6a^{8}z^{6}-8a^{8}z^{4}+6a^{8}z^{2}-2a^{8}+7a^{7}z^{7}-10a^{7}z^{5}+8a^{7}z^{3}-4a^{7}z+a^{7}z^{-1}+5a^{6}z^{8}-2a^{6}z^{6}-9a^{6}z^{4}+12a^{6}z^{2}-5a^{6}+2a^{5}z^{9}+7a^{5}z^{7}-23a^{5}z^{5}+20a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+10a^{4}z^{8}-23a^{4}z^{6}+12a^{4}z^{4}+3a^{4}z^{2}-3a^{4}+2a^{3}z^{9}+4a^{3}z^{7}-22a^{3}z^{5}+18a^{3}z^{3}-4a^{3}z+5a^{2}z^{8}-14a^{2}z^{6}+10a^{2}z^{4}-2a^{2}z^{2}+a^{2}+4az^{7}-12az^{5}+8az^{3}+az-az^{-1}+z^{6}-2z^{4}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

3

0

3

1

-2

7

3

4

-4

7

5

-2

-6

7

5

2

-8

5

7

2

-10

5

7

-2

-12

2

5

3

-14

1

5

-4

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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