L11n420

L11n419

L11n421

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n420 at Knotilus!

Link Presentations

[edit Notes on L11n420's Link Presentations]

Planar diagram presentation	X8192 X7,16,8,17 X3,10,4,11 X17,2,18,3 X18,9,19,10 X11,20,12,21 X14,6,15,5 X22,15,13,16 X6,14,1,13 X4,19,5,20 X21,12,22,7
Gauss code	{1, 4, -3, -10, 7, -9}, {-2, -1, 5, 3, -6, 11}, {9, -7, 8, 2, -4, -5, 10, 6, -11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{2}w^{3}-u^{2}v^{2}w^{2}-uv^{2}w^{3}+uvw^{2}-uvw+u+w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-7}+2q^{-5}-q^{-4}+2q^{-3}-q^{-2}+q^{-1}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-a^{10}+a^{8}z^{6}+6a^{8}z^{4}+9a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-a^{6}z^{8}-7a^{6}z^{6}-16a^{6}z^{4}-16a^{6}z^{2}-2a^{6}z^{-2}-8a^{6}+a^{4}z^{6}+6a^{4}z^{4}+10a^{4}z^{2}+a^{4}z^{-2}+5a^{4}$ (db)
Kauffman polynomial	$a^{11}(-z)-2a^{10}z^{2}+a^{10}+a^{9}z^{7}-6a^{9}z^{5}+8a^{9}z^{3}-3a^{9}z+2a^{8}z^{8}-13a^{8}z^{6}+25a^{8}z^{4}-20a^{8}z^{2}-a^{8}z^{-2}+8a^{8}+a^{7}z^{9}-5a^{7}z^{7}+3a^{7}z^{5}+8a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+3a^{6}z^{8}-20a^{6}z^{6}+41a^{6}z^{4}-33a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+a^{5}z^{9}-6a^{5}z^{7}+9a^{5}z^{5}-6a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}-7a^{4}z^{6}+16a^{4}z^{4}-15a^{4}z^{2}-a^{4}z^{-2}+6a^{4}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ -1                 1 1 -3                   0 -5             2 1   1 -7         1 1 1     1 -9         2 1       1 -11     2 1 1         2 -13     2 1           1 -15 1 1 1             1 -17 1 1               0

Integral Khovanov Homology (db, data source)

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

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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