L11a344

L11a343

L11a345

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a344 at Knotilus!

Link Presentations

[edit Notes on L11a344's Link Presentations]

Planar diagram presentation	X12,1,13,2 X18,7,19,8 X10,5,1,6 X6374 X4,9,5,10 X20,14,21,13 X22,16,11,15 X14,22,15,21 X16,20,17,19 X2,11,3,12 X8,17,9,18
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 6, -8, 7, -9, 11, -2, 9, -6, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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