L11n441

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_18,11,19,12 X_10,3,11,4 X_4,9,1,10 X_14,7,15,8 X_8,13,5,14 X_19,13,20,22 X_15,21,16,20 X_21,17,22,16 X_12,17,9,18
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, -9, 10, 11, -3, -8, 9, -10, 8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(4)^{3}-t(1)t(2)t(4)^{3}+t(2)t(4)^{3}-t(1)t(3)t(4)^{3}-t(2)t(3)t(4)^{3}+t(3)t(4)^{3}-t(4)^{3}-2t(1)t(4)^{2}+t(1)t(2)t(4)^{2}-t(2)t(4)^{2}+t(1)t(3)t(4)^{2}+2t(2)t(3)t(4)^{2}-t(3)t(4)^{2}+2t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)-t(1)t(3)t(4)-2t(2)t(3)t(4)+t(3)t(4)-t(1)+t(1)t(2)-t(2)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-t(3)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}+{\frac {9}{q^{7/2}}}-{\frac {11}{q^{5/2}}}+{\frac {9}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{15/2}}}-{\frac {6}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+3{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}+2a^{9}z^{-3}-4za^{7}-11a^{7}z^{-1}-7a^{7}z^{-3}+6z^{3}a^{5}+20za^{5}+23a^{5}z^{-1}+9a^{5}z^{-3}-3z^{5}a^{3}-13z^{3}a^{3}-22za^{3}-17a^{3}z^{-1}-5a^{3}z^{-3}+3z^{3}a+6za+4az^{-1}+az^{-3}$ (db)
Kauffman polynomial	$a^{9}z^{7}-6a^{9}z^{5}+14a^{9}z^{3}-2a^{9}z^{-3}-16a^{9}z+9a^{9}z^{-1}+a^{8}z^{8}-a^{8}z^{6}-10a^{8}z^{4}+26a^{8}z^{2}+7a^{8}z^{-2}-23a^{8}+a^{7}z^{9}+2a^{7}z^{7}-21a^{7}z^{5}+41a^{7}z^{3}-7a^{7}z^{-3}-39a^{7}z+23a^{7}z^{-1}+6a^{6}z^{8}-10a^{6}z^{6}-29a^{6}z^{4}+73a^{6}z^{2}+19a^{6}z^{-2}-60a^{6}+a^{5}z^{9}+12a^{5}z^{7}-47a^{5}z^{5}+51a^{5}z^{3}-9a^{5}z^{-3}-37a^{5}z+24a^{5}z^{-1}+5a^{4}z^{8}+a^{4}z^{6}-44a^{4}z^{4}+75a^{4}z^{2}+18a^{4}z^{-2}-58a^{4}+11a^{3}z^{7}-29a^{3}z^{5}+27a^{3}z^{3}-5a^{3}z^{-3}-16a^{3}z+12a^{3}z^{-1}+10a^{2}z^{6}-25a^{2}z^{4}+34a^{2}z^{2}+7a^{2}z^{-2}-24a^{2}+3az^{5}+3az^{3}-az^{-3}-2az+2az^{-1}+6z^{2}+z^{-2}-4$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

3

-3

0

3

-2

7

4

-3

-4

4

3

1

2

-6

5

7

2

-8

7

4

3

-10

5

11

6

-12

1

0

-14

5

-16

1

0

-18

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$	$i=0$
$r=-8$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$		${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$		${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{7}$
$r=-3$		${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$		${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=1$		${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$