L11n174

L11n173

L11n175

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n174 at Knotilus!

Link Presentations

[edit Notes on L11n174's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X14,5,15,6 X11,18,12,19 X3,10,4,11 X7,13,8,12 X16,13,17,14 X17,7,18,22 X6,15,1,16 X4,21,5,22 X19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {-6, -1, 2, 5, -4, 6, 7, -3, 9, -7, -8, 4, -11, -2, 10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(t(2)^{2}-3t(2)+1\right)\left(t(1)^{2}t(2)^{2}+t(1)t(2)+1\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {2}{q^{17/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-a^{7}z^{5}-4a^{7}z^{3}-4a^{7}z-2a^{7}z^{-1}+a^{5}z^{7}+5a^{5}z^{5}+7a^{5}z^{3}+4a^{5}z+2a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$2a^{11}z+a^{10}z^{4}+a^{9}z^{7}-3a^{9}z^{5}+4a^{9}z^{3}-3a^{9}z+a^{9}z^{-1}+2a^{8}z^{8}-8a^{8}z^{6}+10a^{8}z^{4}-6a^{8}z^{2}+a^{7}z^{9}-a^{7}z^{7}-9a^{7}z^{5}+15a^{7}z^{3}-9a^{7}z+2a^{7}z^{-1}+4a^{6}z^{8}-17a^{6}z^{6}+19a^{6}z^{4}-8a^{6}z^{2}+a^{6}+a^{5}z^{9}-a^{5}z^{7}-11a^{5}z^{5}+19a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+10a^{4}z^{4}-2a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 0                 1 1 -2               1   -1 -4             3 1   2 -6           2 2     0 -8         3 2       1 -10       2 2         0 -12     2 3           -1 -14   1 2             1 -16 1 2               -1 -18 2                 2

Integral Khovanov Homology (db, data source)

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

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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