L11n149

L11n148

L11n150

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n149 at Knotilus!

Link Presentations

[edit Notes on L11n149's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X22,19,7,20 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X20,13,21,14 X14,21,15,22 X17,2,18,3
Gauss code	{1, 11, -6, -7, 5, -3}, {3, -1, -2, 6, 8, -5, 9, -10, 7, -8, -11, 2, 4, -9, 10, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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