L11a447

L11a446

L11a448

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a447 at Knotilus!

Link Presentations

[edit Notes on L11a447's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X8,18,9,17 X16,8,17,7 X18,10,19,9 X10,14,11,13 X20,11,21,12 X22,19,13,20 X12,21,5,22 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -3, 5, -6, 7, -9}, {6, -2, 11, -4, 3, -5, 8, -7, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{3}w^{3}-uv^{3}w^{2}-2uv^{2}w^{3}+4uv^{2}w^{2}-uv^{2}w+uvw^{3}-4uvw^{2}+4uvw-uv+uw^{2}-2uw+u-v^{3}w^{3}+2v^{3}w^{2}-v^{3}w+v^{2}w^{3}-4v^{2}w^{2}+4v^{2}w-v^{2}+vw^{2}-4vw+2v+w-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-9q^{4}+13q^{3}-14q^{2}+15q-11+10q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+4z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-a^{2}z^{4}-14z^{4}a^{-2}-3a^{2}z^{2}-16z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-9a^{-2}+2z^{6}+9z^{4}+13z^{2}+z^{-2}+8$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-3z^{3}a^{-7}+5z^{6}a^{-6}-6z^{4}a^{-6}+3z^{2}a^{-6}-a^{-6}+6z^{7}a^{-5}-8z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+6z^{8}a^{-4}-10z^{6}a^{-4}+8z^{4}a^{-4}-5z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+4z^{9}a^{-3}+a^{3}z^{7}-3z^{7}a^{-3}-4a^{3}z^{5}-10z^{5}a^{-3}+5a^{3}z^{3}+16z^{3}a^{-3}-2a^{3}z-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+3a^{2}z^{8}+10z^{8}a^{-2}-12a^{2}z^{6}-41z^{6}a^{-2}+15a^{2}z^{4}+52z^{4}a^{-2}-9a^{2}z^{2}-32z^{2}a^{-2}-2a^{-2}z^{-2}+3a^{2}+11a^{-2}+3az^{9}+7z^{9}a^{-1}-7az^{7}-17z^{7}a^{-1}-5az^{5}+15az^{3}+18z^{3}a^{-1}-7az-12za^{-1}+2a^{-1}z^{-1}+z^{10}+7z^{8}-38z^{6}+52z^{4}-32z^{2}-z^{-2}+11$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     2   -2 11                   4 1   3 9                 6 3     -3 7               7 3       4 5             7 6         -1 3           8 7           1 1         6 10             4 -1       4 5               -1 -3     2 6                 4 -5   1 4                   -3 -7   2                     2 -9 1                       -1

Integral Khovanov Homology (db, data source)

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

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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