L11n224

L11n223

L11n225

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n224 at Knotilus!

Link Presentations

[edit Notes on L11n224's Link Presentations]

Planar diagram presentation	X10,1,11,2 X3,12,4,13 X16,9,17,10 X20,12,21,11 X22,15,9,16 X5,14,6,15 X7,18,8,19 X13,4,14,5 X17,6,18,7 X19,8,20,1 X2,21,3,22
Gauss code	{1, -11, -2, 8, -6, 9, -7, 10}, {3, -1, 4, 2, -8, 6, 5, -3, -9, 7, -10, -4, 11, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{3}t(2)^{3}+t(1)^{2}t(2)^{3}-t(1)t(2)^{3}-t(1)^{2}t(2)^{2}+2t(1)t(2)^{2}+2t(1)^{2}t(2)-t(1)t(2)-t(1)^{2}+t(1)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-{\frac {2}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {3}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {2}{q^{23/2}}}-{\frac {1}{q^{25/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{11}+za^{11}+2z^{3}a^{9}+4za^{9}+a^{9}z^{-1}-z^{7}a^{7}-7z^{5}a^{7}-14z^{3}a^{7}-9za^{7}-a^{7}z^{-1}$ (db)
Kauffman polynomial	$a^{15}z^{5}-3a^{15}z^{3}+2a^{15}z+2a^{14}z^{6}-6a^{14}z^{4}+3a^{14}z^{2}+a^{13}z^{7}-a^{13}z^{5}-3a^{13}z^{3}+a^{13}z+2a^{12}z^{6}-3a^{12}z^{4}-a^{12}z^{2}+3a^{11}z^{5}-6a^{11}z^{3}+3a^{11}z+a^{10}z^{4}+a^{10}z^{2}-2a^{9}z^{5}+8a^{9}z^{3}-5a^{9}z+a^{9}z^{-1}-2a^{8}z^{4}+5a^{8}z^{2}-a^{8}+a^{7}z^{7}-7a^{7}z^{5}+14a^{7}z^{3}-9a^{7}z+a^{7}z^{-1}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                     1 1 -8                     1 1 -10               1 1     0 -12             2         2 -14           2 2 1       -1 -16         2 2           0 -18       1 2 1           0 -20     2 2               0 -22   1 2                 1 -24   1                   -1 -26 1                     1

Integral Khovanov Homology (db, data source)

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

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.

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