L11n425

L11n424

L11n426

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n425 at Knotilus!

Link Presentations

[edit Notes on L11n425's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X5,14,6,15 X3,10,4,11 X13,4,14,5 X17,2,18,3 X9,19,10,18 X21,7,22,12 X11,13,12,22 X20,16,21,15 X6,19,1,20
Gauss code	{1, 6, -4, 5, -3, -11}, {2, -1, -7, 4, -9, 8}, {-5, 3, 10, -2, -6, 7, 11, -10, -8, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(2)^{2}t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-t(2)^{2}t(3)^{2}-2t(1)t(2)t(3)^{2}+t(2)t(3)^{2}+t(1)^{2}t(3)-t(1)t(3)-t(1)^{2}t(2)t(3)+2t(1)t(2)t(3)+t(1)+t(1)^{2}t(2)-t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{2}-3q+5-5q^{-1}+6q^{-2}-4q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-5a^{2}z^{4}+z^{4}+3a^{4}z^{2}-9a^{2}z^{2}+2z^{2}+3a^{4}-6a^{2}+a^{-2}+2+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+3a^{4}z^{8}+4a^{2}z^{8}+z^{8}+2a^{5}z^{7}-a^{3}z^{7}-3az^{7}+a^{6}z^{6}-14a^{4}z^{6}-19a^{2}z^{6}-4z^{6}-7a^{5}z^{5}-8a^{3}z^{5}+z^{5}a^{-1}-4a^{6}z^{4}+22a^{4}z^{4}+33a^{2}z^{4}+7z^{4}+4a^{5}z^{3}+14a^{3}z^{3}+10az^{3}+3a^{6}z^{2}-19a^{4}z^{2}-27a^{2}z^{2}+2z^{2}a^{-2}-3z^{2}-9a^{3}z-9az+7a^{4}+11a^{2}-2a^{-2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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 5                 2 2 3               2 1 -1 1             3 1   2 -1           4 4     0 -3         2 1       1 -5       2 4         2 -7     2 2           0 -9     2             2 -11 1 2               -1 -13 1                 1

Integral Khovanov Homology (db, data source)

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

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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