L11n416

L11n415

L11n417

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n416 at Knotilus!

Link Presentations

[edit Notes on L11n416's Link Presentations]

Planar diagram presentation	X8192 X14,5,15,6 X10,3,11,4 X4,13,5,14 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X15,20,16,21 X19,22,20,13 X21,16,22,17
Gauss code	{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 7, -8}, {4, -2, -9, 11, 8, -7, -10, 9, -11, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}vw^{3}-u^{2}vw^{2}+u^{2}vw-u^{2}v-u^{2}w^{3}+uv^{2}w^{3}-uv^{2}w^{2}+uv^{2}w-uv^{2}-2uvw^{3}+2uvw^{2}-2uvw+2uv+uw^{3}-uw^{2}+uw-u+v^{2}+vw^{3}-vw^{2}+vw-v}{uvw^{3/2}}}$ (db)
Jones polynomial	$-2q^{-9}+4q^{-8}-6q^{-7}+9q^{-6}-8q^{-5}+9q^{-4}-6q^{-3}+5q^{-2}-2q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+a^{8}z^{4}+4a^{8}z^{2}+4a^{8}z^{-2}+6a^{8}-a^{6}z^{6}-4a^{6}z^{4}-7a^{6}z^{2}-5a^{6}z^{-2}-9a^{6}-a^{4}z^{6}-3a^{4}z^{4}-a^{4}z^{2}+2a^{4}z^{-2}+2a^{4}+a^{2}z^{4}+3a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$3a^{11}z^{3}-5a^{11}z+a^{11}z^{-1}+a^{10}z^{6}+2a^{10}z^{4}-5a^{10}z^{2}-a^{10}z^{-2}+4a^{10}+4a^{9}z^{7}-15a^{9}z^{5}+30a^{9}z^{3}-21a^{9}z+5a^{9}z^{-1}+4a^{8}z^{8}-16a^{8}z^{6}+34a^{8}z^{4}-32a^{8}z^{2}-4a^{8}z^{-2}+17a^{8}+a^{7}z^{9}+4a^{7}z^{7}-23a^{7}z^{5}+43a^{7}z^{3}-33a^{7}z+9a^{7}z^{-1}+6a^{6}z^{8}-22a^{6}z^{6}+36a^{6}z^{4}-36a^{6}z^{2}-5a^{6}z^{-2}+20a^{6}+a^{5}z^{9}+2a^{5}z^{7}-14a^{5}z^{5}+19a^{5}z^{3}-16a^{5}z+5a^{5}z^{-1}+2a^{4}z^{8}-4a^{4}z^{6}-4a^{4}z^{2}-2a^{4}z^{-2}+6a^{4}+2a^{3}z^{7}-6a^{3}z^{5}+3a^{3}z^{3}+a^{3}z+a^{2}z^{6}-4a^{2}z^{4}+5a^{2}z^{2}-2a^{2}$ (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 χ 1                   1 1 -1                 1   -1 -3               4 1   3 -5             4 3     -1 -7           5 2       3 -9         4 5         1 -11       5 4           1 -13     1 4             3 -15   3 5               -2 -17   2                 2 -19 2                   -2

Integral Khovanov Homology (db, data source)

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

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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