L11n214

L11n213

L11n215

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n214 at Knotilus!

Link Presentations

[edit Notes on L11n214's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,13,21,14 X5,14,6,15 X7,16,8,17 X15,6,16,7 X4,20,5,19
Gauss code	{1, -2, 3, -11, -8, 10, -9, 6}, {-4, -1, 2, -3, 7, 8, -10, 9, -5, 4, 11, -7, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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