L11n44

L11n43

L11n45

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n44 at Knotilus!

Link Presentations

[edit Notes on L11n44's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,14,6,15 X3849 X9,18,10,19 X11,20,12,21 X13,22,14,5 X19,10,20,11 X21,12,22,13 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, -11, -2, 3, 6, -9, 7, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{7}-1}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-9 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-4za^{13}-3a^{13}z^{-1}+z^{7}a^{11}+8z^{5}a^{11}+21z^{3}a^{11}+21za^{11}+7a^{11}z^{-1}-z^{9}a^{9}-9z^{7}a^{9}-28z^{5}a^{9}-36z^{3}a^{9}-19za^{9}-4a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{18}+a^{13}z^{5}-6a^{13}z^{3}+9a^{13}z-3a^{13}z^{-1}+a^{12}z^{8}-8a^{12}z^{6}+21a^{12}z^{4}-21a^{12}z^{2}+7a^{12}+a^{11}z^{9}-9a^{11}z^{7}+29a^{11}z^{5}-42a^{11}z^{3}+28a^{11}z-7a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+21a^{10}z^{4}-21a^{10}z^{2}+7a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-36a^{9}z^{3}+19a^{9}z-4a^{9}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                   1 1 -10                   1 1 -12               1     1 -14           1         1 -16           2 1       1 -18       1             1 -20       1 1           0 -22   1 1               0 -24   1 1               0 -26 1                   -1 -28 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-10$ $i=-8$ $i=-6$ $r=-9$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L11n43 L11n45