L11n78

L11n77

L11n79

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n78 at Knotilus!

Link Presentations

[edit Notes on L11n78's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X19,1,20,4 X5,12,6,13 X3,10,4,11 X13,22,14,5 X21,14,22,15 X11,20,12,21 X2,16,3,15
Gauss code	{1, -11, -7, 5}, {-6, -1, 2, -4, 3, 7, -10, 6, -8, 9, 11, -2, 4, -3, -5, 10, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {9}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}-2za^{9}-2a^{9}z^{-1}+z^{5}a^{7}+3z^{3}a^{7}+5za^{7}+4a^{7}z^{-1}+z^{5}a^{5}+2z^{3}a^{5}+za^{5}-a^{5}z^{-1}-2z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-5z^{2}a^{12}+2a^{12}-2z^{7}a^{11}+6z^{5}a^{11}-3z^{3}a^{11}-za^{11}-2z^{8}a^{10}+4z^{6}a^{10}+2z^{4}a^{10}-3z^{2}a^{10}+a^{10}-z^{9}a^{9}+3z^{5}a^{9}+4z^{3}a^{9}-6za^{9}+2a^{9}z^{-1}-4z^{8}a^{8}+11z^{6}a^{8}-14z^{4}a^{8}+15z^{2}a^{8}-6a^{8}-z^{9}a^{7}+z^{5}a^{7}+3z^{3}a^{7}-9za^{7}+4a^{7}z^{-1}-2z^{8}a^{6}+5z^{6}a^{6}-12z^{4}a^{6}+12z^{2}a^{6}-5a^{6}-2z^{7}a^{5}+4z^{5}a^{5}-7z^{3}a^{5}+za^{5}+a^{5}z^{-1}-z^{6}a^{4}-z^{2}a^{4}+a^{4}-3z^{3}a^{3}+5za^{3}-a^{3}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 3 2 -1 -6               3     3 -8             4 3     -1 -10           5 3       2 -12         2 4         2 -14       4 5           -1 -16     1 2             1 -18   1 4               -3 -20   1                 1 -22 1                   -1

Integral Khovanov Homology (db, data source)

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

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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