L11n209

L11n208

L11n210

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n209 at Knotilus!

Link Presentations

[edit Notes on L11n209's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X22,18,9,17 X19,5,20,4 X6,22,7,21 X16,7,17,8 X8,9,1,10 X13,18,14,19 X20,15,21,16
Gauss code	{1, -2, 3, 6, -4, -7, 8, -9}, {9, -1, 2, -3, -10, 4, 11, -8, 5, 10, -6, -11, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)t(2)+1)(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {1}{\sqrt {q}}}-{\frac {3}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+4a^{7}z^{3}+4a^{7}z-a^{5}z^{7}-6a^{5}z^{5}-12a^{5}z^{3}-7a^{5}z+a^{5}z^{-1}+a^{3}z^{5}+3a^{3}z^{3}+a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{11}z^{5}-3a^{11}z^{3}+a^{11}z+2a^{10}z^{6}-6a^{10}z^{4}+3a^{10}z^{2}+2a^{9}z^{7}-5a^{9}z^{5}+2a^{9}z^{3}+2a^{8}z^{8}-7a^{8}z^{6}+10a^{8}z^{4}-5a^{8}z^{2}+a^{7}z^{9}-3a^{7}z^{7}+5a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z+3a^{6}z^{8}-13a^{6}z^{6}+24a^{6}z^{4}-12a^{6}z^{2}+a^{5}z^{9}-5a^{5}z^{7}+14a^{5}z^{5}-15a^{5}z^{3}+6a^{5}z+a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+9a^{4}z^{4}-6a^{4}z^{2}-a^{4}+3a^{3}z^{5}-7a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}+a^{2}z^{4}-2a^{2}z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 0                   1 -1 -2                 2   2 -4               2 2   0 -6             4 1     3 -8           2 2       0 -10         4 4         0 -12       2 3           1 -14     1 3             -2 -16   1 2               1 -18   1                 -1 -20 1                   1

Integral Khovanov Homology (db, data source)

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

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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