L11n285

L11n284

L11n286

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n285 at Knotilus!

Link Presentations

[edit Notes on L11n285's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X4,15,1,16 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11
Gauss code	{1, 4, -3, -6}, {-2, -1, 5, 3, -8, 9}, {-10, 2, -4, -5, 6, 10, -11, 7, -9, 8, -7, 11}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(3)^{2}t(2)^{2}-2t(1)t(3)t(2)^{2}-2t(1)t(3)^{2}t(2)+t(1)t(3)t(2)-t(3)t(2)+2t(2)+2t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$1-q^{-1}+3q^{-2}-3q^{-3}+5q^{-4}-4q^{-5}+5q^{-6}-3q^{-7}+2q^{-8}-q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}+z^{4}a^{8}+4z^{2}a^{8}+2a^{8}-z^{6}a^{6}-4z^{4}a^{6}-3z^{2}a^{6}+a^{6}z^{-2}-z^{6}a^{4}-4z^{4}a^{4}-4z^{2}a^{4}-2a^{4}z^{-2}-4a^{4}+z^{4}a^{2}+4z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{3}-2a^{11}z+2a^{10}z^{4}-4a^{10}z^{2}+2a^{10}+a^{9}z^{7}-4a^{9}z^{5}+8a^{9}z^{3}-4a^{9}z+2a^{8}z^{8}-11a^{8}z^{6}+23a^{8}z^{4}-16a^{8}z^{2}+4a^{8}+a^{7}z^{9}-4a^{7}z^{7}+5a^{7}z^{5}+3a^{6}z^{8}-14a^{6}z^{6}+21a^{6}z^{4}-9a^{6}z^{2}+a^{6}z^{-2}-2a^{6}+a^{5}z^{9}-4a^{5}z^{7}+6a^{5}z^{5}-8a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}+a^{4}z^{8}-2a^{4}z^{6}-5a^{4}z^{4}+10a^{4}z^{2}+2a^{4}z^{-2}-7a^{4}+a^{3}z^{7}-3a^{3}z^{5}-a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+a^{2}z^{6}-5a^{2}z^{4}+7a^{2}z^{2}+a^{2}z^{-2}-4a^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                     0 -3               3 1   2 -5           1 2 1     0 -7           4 2       2 -9       1 3 3         1 -11       3 2           1 -13     1 3             2 -15   1 2               -1 -17   1                 1 -19 1                   -1

Integral Khovanov Homology (db, data source)

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

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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