L11a221

L11a220

L11a222

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a221 at Knotilus!

Link Presentations

[edit Notes on L11a221's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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