L11a226

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X₄₇₅₈ X_16,5,17,6 X_6,15,1,16 X_22,17,7,18 X_18,13,19,14 X_14,21,15,22 X_2,11,3,12 X_12,3,13,4 X_10,19,11,20
Gauss code	{1, -9, 10, -3, 4, -5}, {3, -1, 2, -11, 9, -10, 7, -8, 5, -4, 6, -7, 11, -2, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {5u^{2}v^{3}-8u^{2}v^{2}+5u^{2}v-u^{2}+3uv^{4}-11uv^{3}+17uv^{2}-11uv+3u-v^{4}+5v^{3}-8v^{2}+5v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {10}{q^{23/2}}}-{\frac {17}{q^{21/2}}}+{\frac {23}{q^{19/2}}}-{\frac {27}{q^{17/2}}}+{\frac {27}{q^{15/2}}}-{\frac {24}{q^{13/2}}}+{\frac {17}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+4z^{3}a^{11}+6za^{11}+2a^{11}z^{-1}-3z^{5}a^{9}-5z^{3}a^{9}-za^{9}-4z^{5}a^{7}-10z^{3}a^{7}-7za^{7}-a^{7}z^{-1}-z^{5}a^{5}-z^{3}a^{5}$ (db)
Kauffman polynomial	$a^{16}z^{6}-2a^{16}z^{4}+a^{16}z^{2}+4a^{15}z^{7}-8a^{15}z^{5}+5a^{15}z^{3}-a^{15}z+8a^{14}z^{8}-17a^{14}z^{6}+13a^{14}z^{4}-6a^{14}z^{2}+2a^{14}+8a^{13}z^{9}-10a^{13}z^{7}-3a^{13}z^{5}+4a^{13}z^{3}+a^{13}z-a^{13}z^{-1}+3a^{12}z^{10}+16a^{12}z^{8}-51a^{12}z^{6}+49a^{12}z^{4}-25a^{12}z^{2}+5a^{12}+18a^{11}z^{9}-30a^{11}z^{7}+13a^{11}z^{5}-10a^{11}z^{3}+9a^{11}z-2a^{11}z^{-1}+3a^{10}z^{10}+21a^{10}z^{8}-56a^{10}z^{6}+46a^{10}z^{4}-17a^{10}z^{2}+3a^{10}+10a^{9}z^{9}-6a^{9}z^{7}-9a^{9}z^{5}+6a^{9}z^{3}+13a^{8}z^{8}-19a^{8}z^{6}+9a^{8}z^{4}+a^{8}z^{2}-a^{8}+10a^{7}z^{7}-16a^{7}z^{5}+14a^{7}z^{3}-7a^{7}z+a^{7}z^{-1}+4a^{6}z^{6}-3a^{6}z^{4}+a^{5}z^{5}-a^{5}z^{3}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

4

1

-3

-8

7

-10

10

4

-6

-12

14

7

-14

13

10

-3

-16

14

0

-18

10

14

4

-20

7

13

-6

-22

3

10

7

-24

1

7

-6

-26

3

-28

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-11$	${\mathbb {Z} }$
$r=-10$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-8$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-7$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-6$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=-5$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$