L11a228

L11a227

L11a229

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a228 at Knotilus!

Link Presentations

[edit Notes on L11a228's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X20,10,21,9 X10,22,11,21 X18,16,19,15 X14,20,15,19 X2,11,3,12 X12,3,13,4 X4758 X22,14,7,13 X6,18,1,17
Gauss code	{1, -7, 8, -9, 2, -11}, {9, -1, 3, -4, 7, -8, 10, -6, 5, -2, 11, -5, 6, -3, 4, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+9t(1)t(2)^{3}-3t(2)^{3}+6t(1)^{2}t(2)^{2}-13t(1)t(2)^{2}+6t(2)^{2}-3t(1)^{2}t(2)+9t(1)t(2)-5t(2)-2t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{15/2}-4q^{13/2}+9q^{11/2}-14q^{9/2}+19q^{7/2}-22q^{5/2}+21q^{3/2}-19{\sqrt {q}}+{\frac {13}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+2z^{3}a^{-5}+za^{-5}-z^{7}a^{-3}-3z^{5}a^{-3}-2z^{3}a^{-3}+2za^{-3}+a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+3az^{3}-8z^{3}a^{-1}+4az-8za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-9z^{5}a^{-7}+5z^{3}a^{-7}+7z^{8}a^{-6}-16z^{6}a^{-6}+10z^{4}a^{-6}-2z^{2}a^{-6}+6z^{9}a^{-5}-8z^{7}a^{-5}-z^{5}a^{-5}-z^{3}a^{-5}+za^{-5}+2z^{10}a^{-4}+10z^{8}a^{-4}-26z^{6}a^{-4}+11z^{4}a^{-4}+2z^{2}a^{-4}-a^{-4}+11z^{9}a^{-3}-15z^{7}a^{-3}+a^{3}z^{5}-2z^{5}a^{-3}-2a^{3}z^{3}+9z^{3}a^{-3}+a^{3}z-5za^{-3}+a^{-3}z^{-1}+2z^{10}a^{-2}+10z^{8}a^{-2}+3a^{2}z^{6}-20z^{6}a^{-2}-4a^{2}z^{4}+5z^{4}a^{-2}+a^{2}z^{2}+8z^{2}a^{-2}-3a^{-2}+5z^{9}a^{-1}+6az^{7}+3z^{7}a^{-1}-10az^{5}-21z^{5}a^{-1}+10az^{3}+27z^{3}a^{-1}-7az-14za^{-1}+2az^{-1}+3a^{-1}z^{-1}+7z^{8}-8z^{6}+2z^{4}+4z^{2}-3$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     3   3 12                   6 1   -5 10                 8 3     5 8               11 6       -5 6             11 8         3 4           10 11           1 2         9 11             -2 0       5 11               6 -2     3 8                 -5 -4   1 6                   5 -6   2                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

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

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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