L11a33

L11a32

L11a34

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a33 at Knotilus!

Link Presentations

[edit Notes on L11a33's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,11,19,12 X16,7,17,8 X8,17,9,18 X20,13,21,14 X22,15,5,16 X12,19,13,20 X14,21,15,22 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -8, 6, -9, 7, -4, 5, -3, 8, -6, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2uv^{6}+3uv^{5}-3uv^{4}+3uv^{3}-3uv^{2}+2uv-u-v^{7}+2v^{6}-3v^{5}+3v^{4}-3v^{3}+3v^{2}-2v}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{29/2}}}-{\frac {2}{q^{27/2}}}+{\frac {5}{q^{25/2}}}-{\frac {7}{q^{23/2}}}+{\frac {9}{q^{21/2}}}-{\frac {11}{q^{19/2}}}+{\frac {10}{q^{17/2}}}-{\frac {9}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {5}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-4za^{13}-3a^{13}z^{-1}+3z^{5}a^{11}+14z^{3}a^{11}+18za^{11}+7a^{11}z^{-1}-2z^{7}a^{9}-11z^{5}a^{9}-19z^{3}a^{9}-13za^{9}-4a^{9}z^{-1}-z^{7}a^{7}-5z^{5}a^{7}-7z^{3}a^{7}-3za^{7}$ (db)
Kauffman polynomial	$-z^{4}a^{18}+2z^{2}a^{18}-a^{18}-2z^{5}a^{17}+2z^{3}a^{17}-3z^{6}a^{16}+3z^{4}a^{16}-z^{2}a^{16}-3z^{7}a^{15}+2z^{5}a^{15}-3z^{8}a^{14}+4z^{6}a^{14}-3z^{4}a^{14}+z^{2}a^{14}-3z^{9}a^{13}+9z^{7}a^{13}-17z^{5}a^{13}+20z^{3}a^{13}-12za^{13}+3a^{13}z^{-1}-z^{10}a^{12}-3z^{8}a^{12}+22z^{6}a^{12}-39z^{4}a^{12}+29z^{2}a^{12}-7a^{12}-6z^{9}a^{11}+27z^{7}a^{11}-49z^{5}a^{11}+53z^{3}a^{11}-31za^{11}+7a^{11}z^{-1}-z^{10}a^{10}-2z^{8}a^{10}+23z^{6}a^{10}-39z^{4}a^{10}+26z^{2}a^{10}-7a^{10}-3z^{9}a^{9}+14z^{7}a^{9}-23z^{5}a^{9}+24z^{3}a^{9}-16za^{9}+4a^{9}z^{-1}-2z^{8}a^{8}+8z^{6}a^{8}-7z^{4}a^{8}+z^{2}a^{8}-z^{7}a^{7}+5z^{5}a^{7}-7z^{3}a^{7}+3za^{7}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                     2 1 -1 -10                   3     3 -12                 3 2     -1 -14               6 3       3 -16             5 4         -1 -18           6 5           1 -20         3 5             2 -22       4 6               -2 -24     1 3                 2 -26   1 4                   -3 -28   1                     1 -30 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-8$ $i=-6$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-7$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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