L11a319

L11a318

L11a320

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a319 at Knotilus!

Link Presentations

[edit Notes on L11a319's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(2u^{2}v+2uv^{2}+uv+2u+2v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\frac {8}{q^{9/2}}}-{\frac {9}{q^{7/2}}}-q^{5/2}+{\frac {8}{q^{5/2}}}+q^{3/2}-{\frac {7}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {2}{q^{15/2}}}+{\frac {4}{q^{13/2}}}-{\frac {7}{q^{11/2}}}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-2a^{7}z+a^{5}z^{5}+2a^{5}z^{3}+2a^{3}z^{5}+6a^{3}z^{3}+3a^{3}z+az^{5}+3az^{3}-z^{3}a^{-1}+2az+az^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+2z^{2}a^{10}-2z^{5}a^{9}+3z^{3}a^{9}-3z^{6}a^{8}+5z^{4}a^{8}-3z^{2}a^{8}-4z^{7}a^{7}+11z^{5}a^{7}-15z^{3}a^{7}+4za^{7}-3z^{8}a^{6}+7z^{6}a^{6}-8z^{4}a^{6}-2z^{9}a^{5}+5z^{7}a^{5}-7z^{5}a^{5}+3z^{3}a^{5}-z^{10}a^{4}+3z^{8}a^{4}-8z^{6}a^{4}+11z^{4}a^{4}-3z^{2}a^{4}-3z^{9}a^{3}+13z^{7}a^{3}-26z^{5}a^{3}+27z^{3}a^{3}-6za^{3}-z^{10}a^{2}+5z^{8}a^{2}-14z^{6}a^{2}+22z^{4}a^{2}-10z^{2}a^{2}-z^{9}a+3z^{7}a-5z^{3}a+4za-az^{-1}-z^{8}+4z^{6}-3z^{4}-2z^{2}+1-z^{7}a^{-1}+6z^{5}a^{-1}-11z^{3}a^{-1}+6za^{-1}-a^{-1}z^{-1}$ (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 6                       1 1 4                         0 2                   3 1   2 0                 2       -2 -2               5 3       2 -4             5 4         -1 -6           4 3           1 -8         4 5             1 -10       3 4               -1 -12     1 4                 3 -14   1 3                   -2 -16   1                     1 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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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