L11n19

L11n18

L11n20

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n19 at Knotilus!

Link Presentations

[edit Notes on L11n19's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X13,22,14,5 X21,14,22,15 X9,18,10,19 X11,20,12,21 X19,10,20,11 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -8, 10, -9, 4, -6, 7, -11, -2, 3, 8, -10, 9, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{5}-uv^{4}-v+2}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{25/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{17/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{13/2}}}-{\frac {2}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}-2a^{13}z^{-1}+z^{5}a^{11}+6z^{3}a^{11}+9za^{11}+4a^{11}z^{-1}-z^{7}a^{9}-6z^{5}a^{9}-10z^{3}a^{9}-5za^{9}-a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-10z^{3}a^{7}-5za^{7}-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{16}+2a^{16}-za^{15}-z^{2}a^{14}+a^{14}-z^{5}a^{13}+5z^{3}a^{13}-7za^{13}+2a^{13}z^{-1}-z^{8}a^{12}+7z^{6}a^{12}-17z^{4}a^{12}+18z^{2}a^{12}-6a^{12}-z^{9}a^{11}+7z^{7}a^{11}-18z^{5}a^{11}+24z^{3}a^{11}-15za^{11}+4a^{11}z^{-1}-2z^{8}a^{10}+12z^{6}a^{10}-22z^{4}a^{10}+17z^{2}a^{10}-5a^{10}-z^{9}a^{9}+6z^{7}a^{9}-11z^{5}a^{9}+9z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-z^{8}a^{8}+5z^{6}a^{8}-5z^{4}a^{8}-z^{2}a^{8}+a^{8}-z^{7}a^{7}+6z^{5}a^{7}-10z^{3}a^{7}+5za^{7}-a^{7}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                   1 1 -8                 1 1 0 -10               1     1 -12           1 1 1     1 -14         1 3 1       1 -16       1 1 1         1 -18       3 2           1 -20   1 1 1             1 -22   1 2               -1 -24 1                   -1 -26 1                   -1

Integral Khovanov Homology (db, data source)

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

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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