L11n326

L11n325

L11n327

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n326 at Knotilus!

Link Presentations

[edit Notes on L11n326's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X13,18,14,19 X9,21,10,20 X19,5,20,12 X11,13,12,22 X21,11,22,10 X4,17,1,18
Gauss code	{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -9, 8}, {-6, 2, -4, -5, 11, 6, -8, 7, -10, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{3}w^{3}+uv^{3}w^{2}+uv^{2}w^{3}-2uv^{2}w^{2}+uv^{2}w+uvw^{2}-v^{2}w-vw^{2}+2vw-v-w+1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-6}+2q^{-5}-2q^{-4}+q^{3}+5q^{-3}-2q^{2}-4q^{-2}+3q+5q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+a^{4}z^{6}-7a^{2}z^{6}+z^{6}+6a^{4}z^{4}-17a^{2}z^{4}+5z^{4}-a^{6}z^{2}+11a^{4}z^{2}-18a^{2}z^{2}+7z^{2}-2a^{6}+7a^{4}-9a^{2}+4+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+2a^{4}z^{8}+4a^{2}z^{8}+2z^{8}+a^{5}z^{7}-3a^{3}z^{7}-2az^{7}+2z^{7}a^{-1}-12a^{4}z^{6}-21a^{2}z^{6}+z^{6}a^{-2}-8z^{6}-5a^{5}z^{5}-3a^{3}z^{5}-6az^{5}-8z^{5}a^{-1}+2a^{6}z^{4}+28a^{4}z^{4}+37a^{2}z^{4}-4z^{4}a^{-2}+7z^{4}+a^{7}z^{3}+11a^{5}z^{3}+17a^{3}z^{3}+13az^{3}+6z^{3}a^{-1}-5a^{6}z^{2}-27a^{4}z^{2}-28a^{2}z^{2}+3z^{2}a^{-2}-3z^{2}-2a^{7}z-7a^{5}z-12a^{3}z-8az-za^{-1}+3a^{6}+11a^{4}+11a^{2}-a^{-2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 1   -1 3               2 1   1 1           1 2 1     0 -1         1 5 2       2 -3         3 4         1 -5       3 2           1 -7   1 1 3             3 -9   2 2               0 -11   1                 1 -13 1                   -1

Integral Khovanov Homology (db, data source)

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

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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