L11a226

L11a225

L11a227

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a226 at Knotilus!

Link Presentations

[edit Notes on L11a226's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X4758 X16,5,17,6 X6,15,1,16 X22,17,7,18 X18,13,19,14 X14,21,15,22 X2,11,3,12 X12,3,13,4 X10,19,11,20
Gauss code	{1, -9, 10, -3, 4, -5}, {3, -1, 2, -11, 9, -10, 7, -8, 5, -4, 6, -7, 11, -2, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {5u^{2}v^{3}-8u^{2}v^{2}+5u^{2}v-u^{2}+3uv^{4}-11uv^{3}+17uv^{2}-11uv+3u-v^{4}+5v^{3}-8v^{2}+5v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {10}{q^{23/2}}}-{\frac {17}{q^{21/2}}}+{\frac {23}{q^{19/2}}}-{\frac {27}{q^{17/2}}}+{\frac {27}{q^{15/2}}}-{\frac {24}{q^{13/2}}}+{\frac {17}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+4z^{3}a^{11}+6za^{11}+2a^{11}z^{-1}-3z^{5}a^{9}-5z^{3}a^{9}-za^{9}-4z^{5}a^{7}-10z^{3}a^{7}-7za^{7}-a^{7}z^{-1}-z^{5}a^{5}-z^{3}a^{5}$ (db)
Kauffman polynomial	$a^{16}z^{6}-2a^{16}z^{4}+a^{16}z^{2}+4a^{15}z^{7}-8a^{15}z^{5}+5a^{15}z^{3}-a^{15}z+8a^{14}z^{8}-17a^{14}z^{6}+13a^{14}z^{4}-6a^{14}z^{2}+2a^{14}+8a^{13}z^{9}-10a^{13}z^{7}-3a^{13}z^{5}+4a^{13}z^{3}+a^{13}z-a^{13}z^{-1}+3a^{12}z^{10}+16a^{12}z^{8}-51a^{12}z^{6}+49a^{12}z^{4}-25a^{12}z^{2}+5a^{12}+18a^{11}z^{9}-30a^{11}z^{7}+13a^{11}z^{5}-10a^{11}z^{3}+9a^{11}z-2a^{11}z^{-1}+3a^{10}z^{10}+21a^{10}z^{8}-56a^{10}z^{6}+46a^{10}z^{4}-17a^{10}z^{2}+3a^{10}+10a^{9}z^{9}-6a^{9}z^{7}-9a^{9}z^{5}+6a^{9}z^{3}+13a^{8}z^{8}-19a^{8}z^{6}+9a^{8}z^{4}+a^{8}z^{2}-a^{8}+10a^{7}z^{7}-16a^{7}z^{5}+14a^{7}z^{3}-7a^{7}z+a^{7}z^{-1}+4a^{6}z^{6}-3a^{6}z^{4}+a^{5}z^{5}-a^{5}z^{3}$ (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 -4                       1 1 -6                     4 1 -3 -8                   7     7 -10                 10 4     -6 -12               14 7       7 -14             13 10         -3 -16           14 14           0 -18         10 14             4 -20       7 13               -6 -22     3 10                 7 -24   1 7                   -6 -26   3                     3 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-8$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-7$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-6$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=-5$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L11a225 L11a227