L11n25

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_20,9,21,10 X₈₄₉₃ X_18,22,19,21 X_11,14,12,15 X_5,13,6,12 X_13,5,14,22 X_10,19,11,20 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, -1, 2, -5, 4, -10, -7, 8, -9, 7, 11, -2, 3, -6, 10, -4, 6, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-2)(v-1)(2v-1)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+6q^{5/2}-10q^{3/2}+12{\sqrt {q}}-{\frac {12}{\sqrt {q}}}+{\frac {11}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {2}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}+a^{3}z^{3}+z^{3}a^{-3}-2a^{3}z-2a^{3}z^{-1}-a^{-3}z^{-1}-az^{5}-z^{5}a^{-1}-z^{3}a^{-1}+2az+az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}+a^{4}z^{6}+z^{6}a^{-4}+4a^{4}z^{4}-2z^{4}a^{-4}-3a^{4}z^{2}+a^{4}+4a^{3}z^{7}+4z^{7}a^{-3}-7a^{3}z^{5}-12z^{5}a^{-3}+14a^{3}z^{3}+8z^{3}a^{-3}-11a^{3}z+za^{-3}+2a^{3}z^{-1}-a^{-3}z^{-1}+5a^{2}z^{8}+5z^{8}a^{-2}-12a^{2}z^{6}-14z^{6}a^{-2}+19a^{2}z^{4}+10z^{4}a^{-2}-13a^{2}z^{2}-4z^{2}a^{-2}+3a^{2}+a^{-2}+2az^{9}+2z^{9}a^{-1}+4az^{7}+4z^{7}a^{-1}-19az^{5}-24z^{5}a^{-1}+22az^{3}+19z^{3}a^{-1}-10az-za^{-1}+az^{-1}-a^{-1}z^{-1}+10z^{8}-28z^{6}+27z^{4}-14z^{2}+2$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

3

1

-2

4

7

3

4

2

5

3

-2

0

7

0

-2

6

7

1

-4

3

5

-2

-6

2

6

4

-8

3

-3

-10

2

Integral Khovanov Homology

(db, data source)

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