L11n27

L11n26

L11n28

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n27 at Knotilus!

Link Presentations

[edit Notes on L11n27's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X5,12,6,13 X3849 X13,22,14,5 X21,14,22,15 X11,18,12,19 X9,20,10,21 X19,10,20,11 X2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -9, 10, -8, 4, -6, 7, 11, -2, -3, 8, -10, 9, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {1}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{9}-a^{9}z^{-1}+z^{3}a^{7}+za^{7}+a^{7}z^{-1}+2z^{3}a^{5}+4za^{5}+2a^{5}z^{-1}-3za^{3}-2a^{3}z^{-1}-za$ (db)
Kauffman polynomial	$-z^{8}a^{10}+7z^{6}a^{10}-17z^{4}a^{10}+16z^{2}a^{10}-4a^{10}-z^{9}a^{9}+5z^{7}a^{9}-6z^{5}a^{9}+a^{9}z^{-1}-4z^{8}a^{8}+23z^{6}a^{8}-42z^{4}a^{8}+31z^{2}a^{8}-9a^{8}-z^{9}a^{7}+z^{7}a^{7}+13z^{5}a^{7}-22z^{3}a^{7}+5za^{7}+a^{7}z^{-1}-3z^{8}a^{6}+14z^{6}a^{6}-18z^{4}a^{6}+11z^{2}a^{6}-4a^{6}-4z^{7}a^{5}+19z^{5}a^{5}-25z^{3}a^{5}+13za^{5}-2a^{5}z^{-1}-2z^{6}a^{4}+7z^{4}a^{4}-5z^{2}a^{4}+2a^{4}-3z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}-z^{2}a^{2}-za$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ 0                   1 1 -2                 2 2 0 -4               1   1 2 -6             3 3     0 -8           2 1       1 -10         1 3 1       1 -12       3 2           1 -14       1             1 -16   1 3               -2 -18                     0 -20 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $i=0$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$

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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