L11n427

L11n426

L11n428

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n427 at Knotilus!

Link Presentations

[edit Notes on L11n427's Link Presentations]

Planar diagram presentation	X8192 X11,18,12,19 X3,10,4,11 X19,2,20,3 X7,16,8,17 X20,9,21,10 X17,12,18,7 X22,16,13,15 X14,6,15,5 X4,14,5,13 X6,21,1,22
Gauss code	{1, 4, -3, -10, 9, -11}, {-5, -1, 6, 3, -2, 7}, {10, -9, 8, 5, -7, 2, -4, -6, 11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)t(3)^{3}-t(1)t(2)t(3)^{3}+t(1)^{2}t(2)^{2}t(3)^{2}-t(1)t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-2t(1)^{2}t(2)t(3)^{2}+4t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(3)^{2}-t(1)^{2}t(2)^{2}t(3)+2t(1)t(2)^{2}t(3)+t(1)t(3)+t(1)^{2}t(2)t(3)-4t(1)t(2)t(3)+2t(2)t(3)-t(3)+t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{-7}-3q^{-6}+7q^{-5}-8q^{-4}+10q^{-3}-q^{2}-9q^{-2}+3q+8q^{-1}-5$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+a^{8}-a^{6}z^{4}-4a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}+a^{4}z^{6}+4a^{4}z^{4}+6a^{4}z^{2}+a^{4}z^{-2}+4a^{4}+a^{2}z^{6}+3a^{2}z^{4}+2a^{2}z^{2}-z^{4}-2z^{2}$ (db)
Kauffman polynomial	$3a^{8}z^{4}-8a^{8}z^{2}-a^{8}z^{-2}+6a^{8}+a^{7}z^{7}-a^{7}z^{5}+3a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}-7a^{6}z^{6}+20a^{6}z^{4}-25a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+a^{5}z^{9}-a^{5}z^{5}+7a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+5a^{4}z^{8}-15a^{4}z^{6}+27a^{4}z^{4}-22a^{4}z^{2}-a^{4}z^{-2}+8a^{4}+a^{3}z^{9}+3a^{3}z^{7}-10a^{3}z^{5}+11a^{3}z^{3}-3a^{3}z+3a^{2}z^{8}-5a^{2}z^{6}+3a^{2}z^{4}-2a^{2}z^{2}+a^{2}+4az^{7}-9az^{5}+z^{5}a^{-1}+5az^{3}-2z^{3}a^{-1}-az+3z^{6}-7z^{4}+3z^{2}$ (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 3 χ 5                   1 -1 3                 2   2 1               3 1   -2 -1             5 2     3 -3           5 4       -1 -5         5 4         1 -7       4 6           2 -9     3 4             -1 -11   1 5               4 -13 1 2                 -1 -15 2                   2

Integral Khovanov Homology (db, data source)

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

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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