L11n356

L11n355

L11n357

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n356 at Knotilus!

Link Presentations

[edit Notes on L11n356's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,18,12,19 X16,8,17,7 X8,16,9,15 X13,21,14,20 X19,22,20,15 X21,13,22,12 X17,14,18,5 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, -9, 3, -7, 6, -8, 7}, {10, -1, 4, -5, 11, -2, -3, 8, -6, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(-t(1)t(2)^{2}+t(1)t(3)t(2)^{2}-t(3)t(2)^{2}+t(2)^{2}+t(1)t(2)-2t(1)t(3)t(2)+t(3)t(2)-2t(2)-t(1)+t(1)t(3)-t(3)+1\right)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{-6}+4q^{-5}-6q^{-4}+q^{3}+9q^{-3}-3q^{2}-9q^{-2}+6q+10q^{-1}-7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{2}z^{6}-a^{4}z^{4}+3a^{2}z^{4}-2z^{4}-a^{4}z^{2}+2a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+a^{4}-2a^{2}+a^{-2}+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{3}+4a^{6}z^{4}-2a^{6}z^{2}+a^{5}z^{7}+2a^{5}z^{5}+2a^{4}z^{8}-3a^{4}z^{6}+8a^{4}z^{4}-8a^{4}z^{2}-a^{4}z^{-2}+4a^{4}+a^{3}z^{9}+2a^{3}z^{7}-5a^{3}z^{5}+4a^{3}z^{3}-5a^{3}z+2a^{3}z^{-1}+5a^{2}z^{8}-10a^{2}z^{6}+z^{6}a^{-2}+5a^{2}z^{4}-3z^{4}a^{-2}-6a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}+6a^{2}-a^{-2}+az^{9}+4az^{7}+3z^{7}a^{-1}-16az^{5}-9z^{5}a^{-1}+11az^{3}+6z^{3}a^{-1}-5az+2az^{-1}+3z^{8}-6z^{6}-2z^{4}+3z^{2}-z^{-2}+2$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2   -2 3               4 1   3 1             4 3     -1 -1           6 3       3 -3         4 5         1 -5       5 5           0 -7     2 5             3 -9   2 4               -2 -11   3                 3 -13 1                   -1

Integral Khovanov Homology (db, data source)

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

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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