L11n4

L11n3

L11n5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n4 at Knotilus!

Link Presentations

[edit Notes on L11n4's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{5}-4t(1)t(2)^{4}+6t(1)t(2)^{3}-2t(2)^{3}-2t(1)t(2)^{2}+6t(2)^{2}-4t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {2}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-a^{7}z^{3}-4a^{7}z-3a^{7}z^{-1}+2a^{5}z^{5}+8a^{5}z^{3}+9a^{5}z+4a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-9a^{3}z^{3}-8a^{3}z-2a^{3}z^{-1}+az^{5}+3az^{3}+az$ (db)
Kauffman polynomial	$-3z^{3}a^{9}+4za^{9}-a^{9}z^{-1}-z^{6}a^{8}-2z^{4}a^{8}+4z^{2}a^{8}-a^{8}-3z^{7}a^{7}+9z^{5}a^{7}-19z^{3}a^{7}+15za^{7}-3a^{7}z^{-1}-3z^{8}a^{6}+9z^{6}a^{6}-16z^{4}a^{6}+11z^{2}a^{6}-3a^{6}-z^{9}a^{5}-4z^{7}a^{5}+23z^{5}a^{5}-39z^{3}a^{5}+24za^{5}-4a^{5}z^{-1}-6z^{8}a^{4}+20z^{6}a^{4}-22z^{4}a^{4}+9z^{2}a^{4}-2a^{4}-z^{9}a^{3}-4z^{7}a^{3}+25z^{5}a^{3}-33z^{3}a^{3}+16za^{3}-2a^{3}z^{-1}-3z^{8}a^{2}+9z^{6}a^{2}-5z^{4}a^{2}+z^{2}a^{2}-a^{2}-3z^{7}a+11z^{5}a-10z^{3}a+3za-z^{6}+3z^{4}-z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                   1 -1 2                 2   2 0               2 1   -1 -2             5 2     3 -4           4 3       -1 -6         5 4         1 -8       3 4           1 -10     3 5             -2 -12   2 4               2 -14   2                 -2 -16 2                   2

Integral Khovanov Homology (db, data source)

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

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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