L11a281

L11a280

L11a282

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a281 at Knotilus!

Link Presentations

[edit Notes on L11a281's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X12,4,13,3 X22,16,9,15 X2,17,3,18 X4,22,5,21 X14,5,15,6 X20,13,21,14 X16,12,17,11 X6,19,7,20 X18,7,19,8
Gauss code	{1, -5, 3, -6, 7, -10, 11, -2}, {2, -1, 9, -3, 8, -7, 4, -9, 5, -11, 10, -8, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{5}-t(1)t(2)^{5}+t(1)^{3}t(2)^{4}-5t(1)^{2}t(2)^{4}+5t(1)t(2)^{4}-t(2)^{4}-3t(1)^{3}t(2)^{3}+10t(1)^{2}t(2)^{3}-11t(1)t(2)^{3}+3t(2)^{3}+3t(1)^{3}t(2)^{2}-11t(1)^{2}t(2)^{2}+10t(1)t(2)^{2}-3t(2)^{2}-t(1)^{3}t(2)+5t(1)^{2}t(2)-5t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {18}{q^{9/2}}}-q^{7/2}+{\frac {23}{q^{7/2}}}+4q^{5/2}-{\frac {27}{q^{5/2}}}-9q^{3/2}+{\frac {26}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {5}{q^{13/2}}}+{\frac {11}{q^{11/2}}}+16{\sqrt {q}}-{\frac {23}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-a^{5}z^{3}+a^{5}z+a^{5}z^{-1}+a^{3}z^{7}+2a^{3}z^{5}-a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}+2az-za^{-1}$ (db)
Kauffman polynomial	$-4a^{4}z^{10}-4a^{2}z^{10}-10a^{5}z^{9}-20a^{3}z^{9}-10az^{9}-10a^{6}z^{8}-11a^{4}z^{8}-12a^{2}z^{8}-11z^{8}-5a^{7}z^{7}+17a^{5}z^{7}+43a^{3}z^{7}+13az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+21a^{6}z^{6}+40a^{4}z^{6}+39a^{2}z^{6}-4z^{6}a^{-2}+17z^{6}+9a^{7}z^{5}-4a^{5}z^{5}-30a^{3}z^{5}-5az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}+a^{8}z^{4}-10a^{6}z^{4}-28a^{4}z^{4}-33a^{2}z^{4}+5z^{4}a^{-2}-11z^{4}-3a^{7}z^{3}+a^{5}z^{3}+7a^{3}z^{3}-3az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}+a^{6}z^{2}+5a^{4}z^{2}+7a^{2}z^{2}-z^{2}a^{-2}+2z^{2}-a^{7}z+a^{5}z+2a^{3}z+az+za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 10 3     -7 0               13 6       7 -2             14 11         -3 -4           13 12           1 -6         10 14             4 -8       8 13               -5 -10     4 11                 7 -12   1 7                   -6 -14   4                     4 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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