L11a163

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a163 at Knotilus!
	[edit L11a163 Quick Notes]

[edit L11a163 Further Notes and Views]

Link Presentations

[edit Notes on L11a163's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_18,11,19,12 X_22,19,7,20 X_20,15,21,16 X_16,21,17,22 X_12,17,13,18 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -9, 11, -4, 7, -8, 9, -5, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {3t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-t(2)^{3}+4t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+4t(2)^{2}-t(1)^{2}t(2)+5t(1)t(2)-5t(2)-t(1)+3}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{29/2}}}-{\frac {3}{q^{27/2}}}+{\frac {6}{q^{25/2}}}-{\frac {10}{q^{23/2}}}+{\frac {13}{q^{21/2}}}-{\frac {14}{q^{19/2}}}+{\frac {14}{q^{17/2}}}-{\frac {12}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {6}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-3a^{13}z-a^{13}z^{-1}+3a^{11}z^{5}+12a^{11}z^{3}+12a^{11}z+2a^{11}z^{-1}-2a^{9}z^{7}-10a^{9}z^{5}-15a^{9}z^{3}-7a^{9}z-a^{7}z^{7}-5a^{7}z^{5}-8a^{7}z^{3}-5a^{7}z-a^{7}z^{-1}$ (db)
Kauffman polynomial	$a^{18}z^{4}-a^{18}z^{2}+3a^{17}z^{5}-3a^{17}z^{3}+a^{17}z+5a^{16}z^{6}-5a^{16}z^{4}+2a^{16}z^{2}+6a^{15}z^{7}-6a^{15}z^{5}+a^{15}z^{3}+a^{15}z+6a^{14}z^{8}-9a^{14}z^{6}+7a^{14}z^{4}-5a^{14}z^{2}+2a^{14}+4a^{13}z^{9}-4a^{13}z^{7}-2a^{13}z^{5}+3a^{13}z-a^{13}z^{-1}+a^{12}z^{10}+9a^{12}z^{8}-37a^{12}z^{6}+49a^{12}z^{4}-30a^{12}z^{2}+5a^{12}+7a^{11}z^{9}-22a^{11}z^{7}+25a^{11}z^{5}-22a^{11}z^{3}+13a^{11}z-2a^{11}z^{-1}+a^{10}z^{10}+5a^{10}z^{8}-30a^{10}z^{6}+41a^{10}z^{4}-21a^{10}z^{2}+3a^{10}+3a^{9}z^{9}-11a^{9}z^{7}+13a^{9}z^{5}-10a^{9}z^{3}+5a^{9}z+2a^{8}z^{8}-7a^{8}z^{6}+5a^{8}z^{4}+a^{8}z^{2}-a^{8}+a^{7}z^{7}-5a^{7}z^{5}+8a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}$ (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

-6

1

-8

2

1

-1

-10

4

-12

4

2

-2

-14

8

4

-16

6

4

-2

-18

8

0

-20

6

7

1

-22

4

7

-3

-24

2

6

4

-26

1

4

-3

-28

2

-30

1

-1

Integral Khovanov Homology

(db, data source)

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