L11n404

L11n403

L11n405

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n404 at Knotilus!

Link Presentations

[edit Notes on L11n404's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X11,22,12,19 X10,4,11,3 X5,21,6,20 X21,5,22,18 X19,12,20,13 X2,14,3,13
Gauss code	{1, -11, 7, -5}, {-10, 8, -9, 6}, {-8, -1, 2, -4, 3, -7, -6, 10, 11, -2, 4, -3, 5, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$0$ (db)
Jones polynomial	$q^{2}-q+2+2q^{-2}+q^{-3}-q^{-4}+q^{-5}-q^{-6}+q^{-7}-q^{-8}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{4}\right)-4a^{6}z^{2}-a^{6}z^{-2}-4a^{6}+a^{4}z^{6}+7a^{4}z^{4}+16a^{4}z^{2}+4a^{4}z^{-2}+13a^{4}-a^{2}z^{6}-7a^{2}z^{4}-16a^{2}z^{2}-5a^{2}z^{-2}-14a^{2}+z^{4}+4z^{2}+2z^{-2}+5$ (db)
Kauffman polynomial	$z^{5}a^{9}-4z^{3}a^{9}+2za^{9}+z^{6}a^{8}-4z^{4}a^{8}+2z^{2}a^{8}+z^{7}a^{7}-5z^{5}a^{7}+6z^{3}a^{7}-4za^{7}+a^{7}z^{-1}+z^{8}a^{6}-6z^{6}a^{6}+10z^{4}a^{6}-10z^{2}a^{6}-a^{6}z^{-2}+7a^{6}+3z^{7}a^{5}-21z^{5}a^{5}+40z^{3}a^{5}-25za^{5}+5a^{5}z^{-1}+3z^{8}a^{4}-22z^{6}a^{4}+50z^{4}a^{4}-49z^{2}a^{4}-4a^{4}z^{-2}+22a^{4}+z^{9}a^{3}-4z^{7}a^{3}-7z^{5}a^{3}+34z^{3}a^{3}-31za^{3}+9a^{3}z^{-1}+3z^{8}a^{2}-22z^{6}a^{2}+52z^{4}a^{2}-53z^{2}a^{2}-5a^{2}z^{-2}+23a^{2}+z^{9}a-6z^{7}a+8z^{5}a+4z^{3}a-12za+5az^{-1}+z^{8}-7z^{6}+16z^{4}-16z^{2}-2z^{-2}+9$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                       1 1 3                         0 1                   2 1   1 -1               3 1       2 -3             1 5 2       2 -5           3 2 2         3 -7         1 2 1           0 -9       2 4 2             0 -11       1 1               0 -13   1 2 1                 0 -15                         0 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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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