L11n81

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L11n82

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n81 at Knotilus!

Link Presentations

[edit Notes on L11n81's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X20,9,21,10 X8,19,9,20 X4,21,1,22 X5,14,6,15 X10,4,11,3 X11,16,12,17 X15,12,16,13 X13,22,14,5 X2,18,3,17
Gauss code	{1, -11, 7, -5}, {-6, -1, 2, -4, 3, -7, -8, 9, -10, 6, -9, 8, 11, -2, 4, -3, 5, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {2}{q^{19/2}}}-{\frac {4}{q^{17/2}}}+{\frac {6}{q^{15/2}}}-{\frac {7}{q^{13/2}}}+{\frac {8}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{9}-a^{9}z^{-1}-z^{5}a^{7}-2z^{3}a^{7}+za^{7}+a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+8z^{3}a^{5}+5za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-5za^{3}-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{2}a^{12}+a^{12}-z^{5}a^{11}-3z^{3}a^{11}+za^{11}-3z^{6}a^{10}+3z^{4}a^{10}-2z^{2}a^{10}-4z^{7}a^{9}+8z^{5}a^{9}-6z^{3}a^{9}+2za^{9}+a^{9}z^{-1}-3z^{8}a^{8}+5z^{6}a^{8}-z^{4}a^{8}+6z^{2}a^{8}-3a^{8}-z^{9}a^{7}-4z^{7}a^{7}+21z^{5}a^{7}-19z^{3}a^{7}+4za^{7}+a^{7}z^{-1}-5z^{8}a^{6}+16z^{6}a^{6}-11z^{4}a^{6}+2z^{2}a^{6}-z^{9}a^{5}-z^{7}a^{5}+17z^{5}a^{5}-25z^{3}a^{5}+10za^{5}-2a^{5}z^{-1}-2z^{8}a^{4}+8z^{6}a^{4}-7z^{4}a^{4}-3z^{2}a^{4}+3a^{4}-z^{7}a^{3}+5z^{5}a^{3}-9z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                   1 1 -2                 1   -1 -4               4 1   3 -6             3 3     0 -8           5 2       3 -10         3 3         0 -12       4 5           -1 -14     2 3             1 -16   2 4               -2 -18   2                 2 -20 2                   -2

Integral Khovanov Homology (db, data source)

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

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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