L11a244

L11a243

L11a245

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a244 at Knotilus!

Link Presentations

[edit Notes on L11a244's Link Presentations]

Planar diagram presentation	X8192 X10,8,11,7 X12,3,13,4 X18,5,19,6 X16,12,17,11 X14,20,15,19 X20,14,21,13 X2,15,3,16 X22,18,7,17 X4,21,5,22 X6,9,1,10
Gauss code	{1, -8, 3, -10, 4, -11}, {2, -1, 11, -2, 5, -3, 7, -6, 8, -5, 9, -4, 6, -7, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+11t(1)t(2)^{3}-6t(2)^{3}+9t(1)^{2}t(2)^{2}-19t(1)t(2)^{2}+9t(2)^{2}-6t(1)^{2}t(2)+11t(1)t(2)-5t(2)-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {11}{q^{9/2}}}-5q^{7/2}+{\frac {17}{q^{7/2}}}+11q^{5/2}-{\frac {25}{q^{5/2}}}-18q^{3/2}+{\frac {28}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+25{\sqrt {q}}-{\frac {28}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+2a^{5}z^{-1}-2a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-6a^{3}z-3a^{3}z^{-1}+az^{7}+3az^{5}-2z^{5}a^{-1}+6az^{3}-3z^{3}a^{-1}+4az-2za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{10}-3z^{10}-10a^{3}z^{9}-19az^{9}-9z^{9}a^{-1}-13a^{4}z^{8}-24a^{2}z^{8}-10z^{8}a^{-2}-21z^{8}-10a^{5}z^{7}+3a^{3}z^{7}+26az^{7}+8z^{7}a^{-1}-5z^{7}a^{-3}-4a^{6}z^{6}+18a^{4}z^{6}+62a^{2}z^{6}+21z^{6}a^{-2}-z^{6}a^{-4}+62z^{6}-a^{7}z^{5}+16a^{5}z^{5}+17a^{3}z^{5}+5az^{5}+14z^{5}a^{-1}+9z^{5}a^{-3}+3a^{6}z^{4}-8a^{4}z^{4}-46a^{2}z^{4}-13z^{4}a^{-2}+z^{4}a^{-4}-49z^{4}+a^{7}z^{3}-15a^{5}z^{3}-16a^{3}z^{3}-9az^{3}-13z^{3}a^{-1}-4z^{3}a^{-3}-3a^{4}z^{2}+9a^{2}z^{2}+2z^{2}a^{-2}+14z^{2}+8a^{5}z+7a^{3}z+za^{-1}+3a^{4}+3a^{2}+1-2a^{5}z^{-1}-3a^{3}z^{-1}-az^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     4   4 6                   7 1   -6 4                 11 4     7 2               14 7       -7 0             14 11         3 -2           15 15           0 -4         10 13             -3 -6       7 15               8 -8     4 10                 -6 -10     7                   7 -12 1 4                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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