L11a332

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a332 at Knotilus!
	[edit L11a332 Quick Notes]

[edit L11a332 Further Notes and Views]

Link Presentations

[edit Notes on L11a332's Link Presentations]

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_20,5,21,6 X_22,13,9,14 X_18,21,19,22 X_16,7,17,8 X_14,17,15,18 X_8,9,1,10 X_6,15,7,16 X_4,19,5,20
Gauss code	{1, -2, 3, -11, 4, -10, 7, -9}, {9, -1, 2, -3, 5, -8, 10, -7, 8, -6, 11, -4, 6, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} , $v$ , $w$ , ...)	$-{\frac {3t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+t(2)t(1)^{3}-3t(2)^{3}t(1)^{2}+8t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+t(1)^{2}+t(2)^{3}t(1)-5t(2)^{2}t(1)+8t(2)t(1)-3t(1)+t(2)^{2}-3t(2)+3}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{29/2}}}-{\frac {4}{q^{27/2}}}+{\frac {7}{q^{25/2}}}-{\frac {11}{q^{23/2}}}+{\frac {14}{q^{21/2}}}-{\frac {15}{q^{19/2}}}+{\frac {15}{q^{17/2}}}-{\frac {12}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {6}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-2a^{13}z+3a^{11}z^{5}+11a^{11}z^{3}+9a^{11}z-2a^{9}z^{7}-10a^{9}z^{5}-15a^{9}z^{3}-6a^{9}z+a^{9}z^{-1}-a^{7}z^{7}-5a^{7}z^{5}-8a^{7}z^{3}-5a^{7}z-a^{7}z^{-1}$ (db)
Kauffman polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{18} z^4+4 a^{17} z^5-3 a^{17} z^3+7 a^{16} z^6-7 a^{16} z^4+a^{16} z^2+8 a^{15} z^7-8 a^{15} z^5-a^{15} z^3+2 a^{15} z+7 a^{14} z^8-8 a^{14} z^6+a^{14} z^4+4 a^{13} z^9-12 a^{13} z^5+7 a^{13} z^3+a^{13} z+a^{12} z^{10}+10 a^{12} z^8-37 a^{12} z^6+42 a^{12} z^4-18 a^{12} z^2+7 a^{11} z^9-20 a^{11} z^7+18 a^{11} z^5-12 a^{11} z^3+7 a^{11} z+a^{10} z^{10}+5 a^{10} z^8-29 a^{10} z^6+38 a^{10} z^4-16 a^{10} z^2+3 a^9 z^9-11 a^9 z^7+13 a^9 z^5-9 a^9 z^3+3 a^9 z+a^9 z^{-1} +2 a^8 z^8-7 a^8 z^6+5 a^8 z^4+a^8 z^2-a^8+a^7 z^7-5 a^7 z^5+8 a^7 z^3-5 a^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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

7

4

-3

-18

8

0

-20

6

7

1

-22

5

8

-3

-24

3

7

4

-26

1

4

-3

-28

3

-30

1

-1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	$i=-8$	$i=-6$
$r=-11$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
$r=-10$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-9}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}}	${\mathbb {Z} }^{3}$
$r=-8$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}}
$r=-7$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-6$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{8}}
$r=-5$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}}
$r=-4$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}}	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$