L11a354

L11a353

L11a355

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a354 at Knotilus!

Link Presentations

[edit Notes on L11a354's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X8394 X16,11,17,12 X14,8,15,7 X6,16,7,15 X22,17,11,18 X4,20,5,19 X18,6,19,5 X20,9,21,10 X10,21,1,22
Gauss code	{1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, {4, -1, 2, -5, 6, -4, 7, -9, 8, -10, 11, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	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 -\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+t(2) t(1)-2 t(1)+t(2)^2-3 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}} (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-8{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {19}{q^{5/2}}}-{\frac {19}{q^{7/2}}}+{\frac {17}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}+2z^{5}a^{5}+5z^{3}a^{5}+3za^{5}-z^{7}a^{3}-3z^{5}a^{3}-3z^{3}a^{3}-2za^{3}+2z^{5}a+5z^{3}a+3za+az^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+2z^{3}a^{9}-6z^{6}a^{8}+5z^{4}a^{8}-2z^{2}a^{8}-10z^{7}a^{7}+17z^{5}a^{7}-16z^{3}a^{7}+4za^{7}-11z^{8}a^{6}+21z^{6}a^{6}-17z^{4}a^{6}+5z^{2}a^{6}-7z^{9}a^{5}+4z^{7}a^{5}+20z^{5}a^{5}-23z^{3}a^{5}+6za^{5}-2z^{10}a^{4}-13z^{8}a^{4}+46z^{6}a^{4}-39z^{4}a^{4}+12z^{2}a^{4}-11z^{9}a^{3}+24z^{7}a^{3}-2z^{5}a^{3}-11z^{3}a^{3}+4za^{3}-2z^{10}a^{2}-5z^{8}a^{2}+29z^{6}a^{2}-26z^{4}a^{2}+6z^{2}a^{2}-4z^{9}a+9z^{7}a+2z^{5}a-12z^{3}a+6za-az^{-1}-3z^{8}+10z^{6}-10z^{4}+2z^{2}+1-z^{7}a^{-1}+4z^{5}a^{-1}-6z^{3}a^{-1}+4za^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed 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 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   6 1   5 0                 6 2     -4 -2               11 6       5 -4             10 8         -2 -6           9 9           0 -8         8 10             2 -10       5 9               -4 -12     2 8                 6 -14   1 5                   -4 -16   2                     2 -18 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ 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 i=-4} $i=-2$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ 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}^{2}\oplus{\mathbb Z}_2} 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}} 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=-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}^{5}\oplus{\mathbb Z}_2^{2}} 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}^{2}} 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=-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}^{8}\oplus{\mathbb Z}_2^{5}} ${\mathbb {Z} }^{5}$ $r=-3$ 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}^{9}\oplus{\mathbb Z}_2^{8}} ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ 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}^{9}} $r=-1$ 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}^{9}\oplus{\mathbb Z}_2^{10}} ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ 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}^{6}\oplus{\mathbb Z}_2^{6}} ${\mathbb {Z} }^{6}$ $r=2$ 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}^{2}\oplus{\mathbb Z}_2^{6}} ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ 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}^{2}} $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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