L11n259

L11n258

L11n260

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n259 at Knotilus!

Link Presentations

[edit Notes on L11n259's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X22,20,9,19 X20,16,21,15 X16,22,17,21 X17,12,18,13 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 9, -4, 3, 7, -8, -9, 5, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{3}-2uvw^{2}+uvw-uv-uw^{3}+2uw^{2}-2uw+u-vw^{3}+2vw^{2}-2vw+v+w^{3}-w^{2}+2w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+4q^{5}-6q^{4}+8q^{3}-7q^{2}-q^{-2}+8q+4q^{-1}-5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+3z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+z^{2}a^{-2}-z^{2}a^{-4}-z^{2}-3a^{-2}+a^{-4}+2-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-7}+4z^{4}a^{-6}-z^{2}a^{-6}+z^{7}a^{-5}+2z^{5}a^{-5}+z^{8}a^{-4}+2z^{6}a^{-4}-z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+5z^{7}a^{-3}-6z^{5}a^{-3}+z^{3}a^{-3}-3za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}+6z^{6}a^{-2}-14z^{4}a^{-2}+z^{2}a^{-2}-2a^{-2}z^{-2}+5a^{-2}+4z^{7}a^{-1}+az^{5}-7z^{5}a^{-1}-az^{3}+z^{3}a^{-1}-3za^{-1}+2a^{-1}z^{-1}+4z^{6}-9z^{4}+3z^{2}-z^{-2}+3$ (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   \ -3 -2 -1 0 1 2 3 4 5 χ 13                 1 -1 11               3   3 9             4 2   -2 7           4 2     2 5         3 4       1 3       5 4         1 1     3 6           3 -1   1 2             -1 -3   3               3 -5 1                 -1

Integral Khovanov Homology (db, data source)

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

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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