L11a220

L11a219

L11a221

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a220 at Knotilus!

Link Presentations

[edit Notes on L11a220's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X16,6,17,5 X20,15,21,16 X18,21,19,22 X14,19,15,20 X2738 X4,11,5,12 X6,18,1,17
Gauss code	{1, -9, 2, -10, 5, -11}, {9, -1, 3, -4, 10, -2, 4, -8, 6, -5, 11, -7, 8, -6, 7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-4t(1)^{2}t(2)^{3}+8t(1)t(2)^{3}-5t(2)^{3}+7t(1)^{2}t(2)^{2}-13t(1)t(2)^{2}+7t(2)^{2}-5t(1)^{2}t(2)+8t(1)t(2)-4t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {18}{q^{7/2}}}+4q^{5/2}-{\frac {23}{q^{5/2}}}-9q^{3/2}+{\frac {22}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+5za^{5}+3a^{5}z^{-1}-3z^{5}a^{3}-8z^{3}a^{3}-8za^{3}-2a^{3}z^{-1}+z^{7}a+4z^{5}a+8z^{3}a+5za-z^{5}a^{-1}-2z^{3}a^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-4a^{5}z^{9}-9a^{3}z^{9}-5az^{9}-5a^{6}z^{8}-16a^{4}z^{8}-20a^{2}z^{8}-9z^{8}-3a^{7}z^{7}-2a^{5}z^{7}-a^{3}z^{7}-10az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+9a^{6}z^{6}+39a^{4}z^{6}+43a^{2}z^{6}-4z^{6}a^{-2}+10z^{6}+7a^{7}z^{5}+20a^{5}z^{5}+35a^{3}z^{5}+35az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}-3a^{6}z^{4}-27a^{4}z^{4}-28a^{2}z^{4}+5z^{4}a^{-2}-2z^{4}-6a^{7}z^{3}-20a^{5}z^{3}-33a^{3}z^{3}-28az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}-3a^{8}z^{2}-3a^{6}z^{2}+4a^{4}z^{2}+6a^{2}z^{2}-2z^{2}a^{-2}+3a^{7}z+9a^{5}z+10a^{3}z+7az+3za^{-1}+a^{8}+3a^{6}+3a^{4}-a^{7}z^{-1}-3a^{5}z^{-1}-2a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 9 3     -6 0               11 6       5 -2             12 10         -2 -4           11 10           1 -6         8 13             5 -8       6 10               -4 -10     2 8                 6 -12   1 6                   -5 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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