L11n89

L11n88

L11n90

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n89 at Knotilus!

Link Presentations

[edit Notes on L11n89's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X9,15,10,14 X19,10,20,11 X21,9,22,8 X13,18,14,19 X15,21,16,20 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, -5, 6, 11, -2, -8, 5, -9, 3, -4, 8, -6, 9, -7, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{5}-3t(1)t(2)^{4}-t(2)^{4}+2t(1)t(2)^{3}+2t(2)^{3}+2t(1)t(2)^{2}+2t(2)^{2}-t(1)t(2)-3t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {2}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {2}{q^{13/2}}}+{\frac {2}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-4a^{7}z-3a^{7}z^{-1}+a^{5}z^{5}+6a^{5}z^{3}+8a^{5}z+4a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-7a^{3}z^{3}-6a^{3}z-2a^{3}z^{-1}+az^{5}+3az^{3}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+5z^{5}a^{9}-6z^{3}a^{9}+4za^{9}-a^{9}z^{-1}-2z^{8}a^{8}+11z^{6}a^{8}-14z^{4}a^{8}+6z^{2}a^{8}-a^{8}-z^{9}a^{7}+4z^{7}a^{7}+6z^{5}a^{7}-24z^{3}a^{7}+16za^{7}-3a^{7}z^{-1}-3z^{8}a^{6}+21z^{6}a^{6}-36z^{4}a^{6}+18z^{2}a^{6}-3a^{6}-z^{9}a^{5}+3z^{7}a^{5}+14z^{5}a^{5}-40z^{3}a^{5}+25za^{5}-4a^{5}z^{-1}-3z^{8}a^{4}+18z^{6}a^{4}-28z^{4}a^{4}+15z^{2}a^{4}-2a^{4}-5z^{7}a^{3}+25z^{5}a^{3}-32z^{3}a^{3}+15za^{3}-2a^{3}z^{-1}-2z^{8}a^{2}+7z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}-a^{2}-3z^{7}a+12z^{5}a-10z^{3}a+2za-z^{6}+3z^{4}-z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     2   2 0                   1 1   0 -2               1 4 2     1 -4             1 2 2       1 -6           1 3 3         -1 -8         2 3 2           1 -10       1 2 2             -1 -12     1 2 3               2 -14   1 1                   0 -16   1                     -1 -18 1                       1

Integral Khovanov Homology (db, data source)

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

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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