L11n220

L11n219

L11n221

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n220 at Knotilus!

Link Presentations

[edit Notes on L11n220's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X3,12,4,13 X15,22,16,9 X2,17,3,18 X21,4,22,5 X14,5,15,6 X20,13,21,14 X11,16,12,17 X6,19,7,20 X18,7,19,8
Gauss code	{1, -5, -3, 6, 7, -10, 11, -2}, {2, -1, -9, 3, 8, -7, -4, 9, 5, -11, 10, -8, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{5}+t(1)^{3}t(2)^{4}-t(1)^{2}t(2)^{4}+t(1)^{2}t(2)^{3}+t(1)t(2)^{2}-t(1)t(2)+t(2)+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {2}{q^{21/2}}}-{\frac {1}{q^{23/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-a^{13}z^{-1}+a^{11}z^{5}+6a^{11}z^{3}+9a^{11}z+3a^{11}z^{-1}-a^{9}z^{7}-6a^{9}z^{5}-11a^{9}z^{3}-8a^{9}z-2a^{9}z^{-1}-a^{7}z^{7}-6a^{7}z^{5}-10a^{7}z^{3}-4a^{7}z$ (db)
Kauffman polynomial	$a^{15}z+a^{14}z^{2}+a^{14}+a^{13}z^{5}-3a^{13}z^{3}+3a^{13}z-a^{13}z^{-1}+a^{12}z^{8}-6a^{12}z^{6}+13a^{12}z^{4}-12a^{12}z^{2}+3a^{12}+a^{11}z^{9}-7a^{11}z^{7}+20a^{11}z^{5}-29a^{11}z^{3}+16a^{11}z-3a^{11}z^{-1}+2a^{10}z^{8}-11a^{10}z^{6}+19a^{10}z^{4}-14a^{10}z^{2}+3a^{10}+a^{9}z^{9}-6a^{9}z^{7}+13a^{9}z^{5}-16a^{9}z^{3}+10a^{9}z-2a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+6a^{8}z^{4}-a^{8}z^{2}+a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-4a^{7}z$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                 1 1 -8               1 1 0 -10             1     1 -12           1 1     0 -14         2 1       1 -16         1         1 -18     2 2           0 -20                   0 -22 1 2               -1 -24 1                 1

Integral Khovanov Homology (db, data source)

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

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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