L11a236

L11a235

L11a237

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a236 at Knotilus!

Link Presentations

[edit Notes on L11a236's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,15,21,16 X16,9,17,10 X10,19,11,20 X22,13,7,14 X14,21,15,22 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, 6, -7, 3, -4, 11, -8, 5, -3, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {5u^{2}v^{2}-6u^{2}v+2u^{2}+5uv^{3}-9uv^{2}+5uv+2v^{4}-6v^{3}+5v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {13}{q^{13/2}}}-{\frac {14}{q^{15/2}}}+{\frac {12}{q^{17/2}}}-{\frac {9}{q^{19/2}}}+{\frac {6}{q^{21/2}}}-{\frac {3}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+4a^{11}z+2a^{11}z^{-1}-3a^{9}z^{3}-a^{9}z-5a^{7}z^{3}-5a^{7}z-a^{7}z^{-1}-3a^{5}z^{3}-a^{5}z-a^{3}z^{3}$ (db)
Kauffman polynomial	$a^{14}z^{8}-5a^{14}z^{6}+9a^{14}z^{4}-7a^{14}z^{2}+2a^{14}+3a^{13}z^{9}-15a^{13}z^{7}+24a^{13}z^{5}-13a^{13}z^{3}+2a^{13}z-a^{13}z^{-1}+2a^{12}z^{10}-3a^{12}z^{8}-18a^{12}z^{6}+44a^{12}z^{4}-29a^{12}z^{2}+5a^{12}+10a^{11}z^{9}-42a^{11}z^{7}+51a^{11}z^{5}-21a^{11}z^{3}+7a^{11}z-2a^{11}z^{-1}+2a^{10}z^{10}+7a^{10}z^{8}-50a^{10}z^{6}+64a^{10}z^{4}-25a^{10}z^{2}+3a^{10}+7a^{9}z^{9}-15a^{9}z^{7}-6a^{9}z^{5}+14a^{9}z^{3}-2a^{9}z+11a^{8}z^{8}-28a^{8}z^{6}+15a^{8}z^{4}-2a^{8}z^{2}-a^{8}+12a^{7}z^{7}-27a^{7}z^{5}+17a^{7}z^{3}-6a^{7}z+a^{7}z^{-1}+9a^{6}z^{6}-11a^{6}z^{4}+a^{6}z^{2}+6a^{5}z^{5}-4a^{5}z^{3}+a^{5}z+3a^{4}z^{4}+a^{3}z^{3}$ (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 -2                       1 1 -4                     3 1 -2 -6                   3     3 -8                 6 3     -3 -10               7 3       4 -12             6 6         0 -14           8 7           1 -16         5 7             2 -18       4 7               -3 -20     2 5                 3 -22   1 4                   -3 -24   2                     2 -26 1                       -1

Integral Khovanov Homology (db, data source)

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

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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