L11n418

L11n417

L11n419

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n418 at Knotilus!

Link Presentations

[edit Notes on L11n418's Link Presentations]

Planar diagram presentation	X8192 X7,16,8,17 X3,10,4,11 X2,18,3,17 X18,9,19,10 X11,20,12,21 X5,14,6,15 X15,13,16,22 X13,6,14,1 X19,5,20,4 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 $u$, $v$, $w$, ...)	${\frac {u^{2}vw^{3}-2u^{2}vw^{2}+u^{2}vw+u^{2}w^{2}-u^{2}w+uv^{2}w-uvw^{3}+2uvw^{2}-2uvw+uv-uw^{2}+v^{2}w^{2}-v^{2}w-vw^{2}+2vw-v}{uvw^{3/2}}}$ (db)
Jones polynomial	$-1+3q^{-1}-4q^{-2}+7q^{-3}-6q^{-4}+7q^{-5}-5q^{-6}+4q^{-7}-2q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-2a^{6}z^{4}-7a^{6}z^{2}-2a^{6}z^{-2}-8a^{6}+a^{4}z^{6}+5a^{4}z^{4}+10a^{4}z^{2}+a^{4}z^{-2}+7a^{4}-a^{2}z^{4}-2a^{2}z^{2}-a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-4a^{10}z^{4}+4a^{10}z^{2}-a^{10}+2a^{9}z^{7}-7a^{9}z^{5}+5a^{9}z^{3}+2a^{8}z^{8}-6a^{8}z^{6}+4a^{8}z^{4}-4a^{8}z^{2}-a^{8}z^{-2}+4a^{8}+a^{7}z^{9}-a^{7}z^{7}-4a^{7}z^{5}+6a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+4a^{6}z^{8}-16a^{6}z^{6}+28a^{6}z^{4}-27a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+a^{5}z^{9}-2a^{5}z^{7}+a^{5}z^{5}+6a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+23a^{4}z^{4}-23a^{4}z^{2}-a^{4}z^{-2}+10a^{4}+a^{3}z^{7}-2a^{3}z^{5}+6a^{3}z^{3}-3a^{3}z+3a^{2}z^{4}-4a^{2}z^{2}+2a^{2}+az^{3}-az$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 1                   1 -1 -1                 2   2 -3               3 2   -1 -5             4 1     3 -7           3 4       1 -9         4 3         1 -11       2 4           2 -13     2 3             -1 -15   1 3               2 -17   1                 -1 -19 1                   1

Integral Khovanov Homology (db, data source)

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

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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