L11n417

L11n416

L11n418

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n417 at Knotilus!

Link Presentations

[edit Notes on L11n417's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X15,21,16,20 X5,15,6,14 X13,5,14,4 X19,7,20,12 X11,19,12,18 X17,13,18,22 X21,17,22,16 X2738 X6,9,1,10
Gauss code	{1, -10, 2, 5, -4, -11}, {10, -1, 11, -2, -7, 6}, {-5, 4, -3, 9, -8, 7, -6, 3, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}vw^{3}-u^{2}w^{3}+u^{2}w^{2}-u^{2}w+u^{2}+uv^{2}w^{3}-uvw^{3}+uv-u-v^{2}w^{3}+v^{2}w^{2}-v^{2}w+v^{2}-v}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+2q^{6}-3q^{5}+4q^{4}-4q^{3}+5q^{2}-3q+4-q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}a^{-4}-5z^{4}a^{-2}-4z^{4}a^{-4}+z^{4}a^{-6}+z^{4}-9z^{2}a^{-2}-2z^{2}a^{-4}+4z^{2}a^{-6}+4z^{2}-10a^{-2}+4a^{-4}+2a^{-6}-a^{-8}+5-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$za^{-9}+2z^{2}a^{-8}-a^{-8}+z^{5}a^{-7}-z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+3z^{6}a^{-6}-10z^{4}a^{-6}+6z^{2}a^{-6}-a^{-6}z^{-2}+5z^{7}a^{-5}-23z^{5}a^{-5}+31z^{3}a^{-5}-19za^{-5}+5a^{-5}z^{-1}+4z^{8}a^{-4}-19z^{6}a^{-4}+27z^{4}a^{-4}-21z^{2}a^{-4}-4a^{-4}z^{-2}+13a^{-4}+z^{9}a^{-3}+z^{7}a^{-3}-25z^{5}a^{-3}+50z^{3}a^{-3}-35za^{-3}+9a^{-3}z^{-1}+5z^{8}a^{-2}-29z^{6}a^{-2}+55z^{4}a^{-2}-46z^{2}a^{-2}-5a^{-2}z^{-2}+22a^{-2}+z^{9}a^{-1}-4z^{7}a^{-1}-z^{5}a^{-1}+18z^{3}a^{-1}-19za^{-1}+5a^{-1}z^{-1}+z^{8}-7z^{6}+18z^{4}-21z^{2}-2z^{-2}+11$ (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 χ 15                   1 -1 13                 1   1 11               3 2   -1 9             2 2 1   1 7           3 3       0 5         2 2 1       1 3       2 4           2 1     2 1             1 -1   1 4               3 -3                     0 -5 1                   1

Integral Khovanov Homology (db, data source)

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

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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