L11n387

L11n386

L11n388

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n387 at Knotilus!

Link Presentations

[edit Notes on L11n387's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X8493 X11,19,12,22 X21,18,22,5 X9,20,10,21 X17,11,18,10 X19,17,20,16 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {-10, 8, -7, 6}, {-4, -1, 2, -5, -8, 9, -6, 4, 11, -2, 3, 10, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{3}+4q^{2}-6q+11-10q^{-1}+11q^{-2}-9q^{-3}+7q^{-4}-4q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-3a^{2}z^{4}+2z^{4}+a^{4}z^{2}-3a^{2}z^{2}-z^{2}a^{-2}+3z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2a^{3}z^{9}+2az^{9}+5a^{4}z^{8}+9a^{2}z^{8}+4z^{8}+4a^{5}z^{7}+3a^{3}z^{7}+az^{7}+2z^{7}a^{-1}+a^{6}z^{6}-13a^{4}z^{6}-27a^{2}z^{6}-13z^{6}-11a^{5}z^{5}-23a^{3}z^{5}-16az^{5}-4z^{5}a^{-1}-2a^{6}z^{4}+5a^{4}z^{4}+25a^{2}z^{4}+4z^{4}a^{-2}+22z^{4}+6a^{5}z^{3}+20a^{3}z^{3}+23az^{3}+10z^{3}a^{-1}+z^{3}a^{-3}+a^{6}z^{2}-a^{4}z^{2}-9a^{2}z^{2}-4z^{2}a^{-2}-11z^{2}-a^{5}z-4a^{3}z-6az-4za^{-1}-za^{-3}+1-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 7                   1 -1 5                 3   3 3               3 1   -2 1             8 3     5 -1           6 7       1 -3         5 5 1       1 -5       4 6           2 -7     3 5             -2 -9   1 4               3 -11   3                 -3 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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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