L11a23

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,11,17,12 X_14,7,15,8 X_8,15,9,16 X_20,13,21,14 X_22,17,5,18 X_18,21,19,22 X_12,19,13,20 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9, 6, -4, 5, -3, 7, -8, 9, -6, 8, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)^{5}+4t(1)t(2)^{4}-6t(2)^{4}-9t(1)t(2)^{3}+10t(2)^{3}+10t(1)t(2)^{2}-9t(2)^{2}-6t(1)t(2)+4t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {3}{q^{25/2}}}+{\frac {8}{q^{23/2}}}-{\frac {13}{q^{21/2}}}+{\frac {17}{q^{19/2}}}-{\frac {20}{q^{17/2}}}+{\frac {20}{q^{15/2}}}-{\frac {18}{q^{13/2}}}+{\frac {12}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-a^{13}z^{-1}+3a^{11}z^{3}+4a^{11}z+a^{11}z^{-1}-2a^{9}z^{5}-2a^{9}z^{3}+3a^{9}z+2a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-7a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$-z^{6}a^{16}+3z^{4}a^{16}-3z^{2}a^{16}+a^{16}-3z^{7}a^{15}+7z^{5}a^{15}-5z^{3}a^{15}+za^{15}-5z^{8}a^{14}+10z^{6}a^{14}-5z^{4}a^{14}+z^{2}a^{14}-4z^{9}a^{13}+z^{7}a^{13}+11z^{5}a^{13}-6z^{3}a^{13}-za^{13}+a^{13}z^{-1}-z^{10}a^{12}-13z^{8}a^{12}+34z^{6}a^{12}-29z^{4}a^{12}+14z^{2}a^{12}-3a^{12}-8z^{9}a^{11}+5z^{7}a^{11}+11z^{5}a^{11}-5z^{3}a^{11}-3za^{11}+a^{11}z^{-1}-z^{10}a^{10}-14z^{8}a^{10}+30z^{6}a^{10}-20z^{4}a^{10}+3z^{2}a^{10}-4z^{9}a^{9}-5z^{7}a^{9}+19z^{5}a^{9}-18z^{3}a^{9}+8za^{9}-2a^{9}z^{-1}-6z^{8}a^{8}+4z^{6}a^{8}+5z^{4}a^{8}-8z^{2}a^{8}+3a^{8}-6z^{7}a^{7}+11z^{5}a^{7}-12z^{3}a^{7}+8za^{7}-2a^{7}z^{-1}-3z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}$ (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

3

1

-2

-8

5

-10

7

3

-4

-12

11

5

6

-14

10

8

-2

-16

10

0

-18

7

10

3

-20

6

10

-4

-22

2

7

5

-24

1

6

-5

-26

2

-28

1

-1

Integral Khovanov Homology

(db, data source)

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