L11n255

L11n254

L11n256

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n255 at Knotilus!

Link Presentations

[edit Notes on L11n255's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X11,16,12,17 X21,18,22,19 X13,20,14,21 X19,12,20,13 X17,22,18,9 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -9, 8}, {-11, 2, -3, 6, -5, 9, -8, 3, -7, 4, -6, 5, -4, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+2t(3)^{3}+t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-4t(3)^{2}-2t(1)t(3)+4t(1)t(2)t(3)-t(2)t(3)+4t(3)+2t(1)-2t(1)t(2)-2}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-3q^{-9}+6q^{-8}-9q^{-7}+11q^{-6}-11q^{-5}+12q^{-4}-7q^{-3}+6q^{-2}-2q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+z^{4}a^{8}+3z^{2}a^{8}+3a^{8}z^{-2}+5a^{8}-z^{6}a^{6}-3z^{4}a^{6}-4z^{2}a^{6}-2a^{6}z^{-2}-5a^{6}-z^{6}a^{4}-3z^{4}a^{4}-3z^{2}a^{4}-a^{4}z^{-2}-2a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$6z^{3}a^{11}-7za^{11}+2a^{11}z^{-1}+3z^{6}a^{10}-2z^{2}a^{10}-a^{10}z^{-2}+a^{10}+7z^{7}a^{9}-20z^{5}a^{9}+35z^{3}a^{9}-27za^{9}+8a^{9}z^{-1}+5z^{8}a^{8}-10z^{6}a^{8}+13z^{4}a^{8}-8z^{2}a^{8}-3a^{8}z^{-2}+5a^{8}+z^{9}a^{7}+10z^{7}a^{7}-34z^{5}a^{7}+46z^{3}a^{7}-34za^{7}+10a^{7}z^{-1}+7z^{8}a^{6}-15z^{6}a^{6}+8z^{4}a^{6}-3z^{2}a^{6}-2a^{6}z^{-2}+4a^{6}+z^{9}a^{5}+5z^{7}a^{5}-19z^{5}a^{5}+18z^{3}a^{5}-10za^{5}+2a^{5}z^{-1}+2z^{8}a^{4}-z^{6}a^{4}-9z^{4}a^{4}+9z^{2}a^{4}+a^{4}z^{-2}-3a^{4}+2z^{7}a^{3}-5z^{5}a^{3}+z^{3}a^{3}+4za^{3}-2a^{3}z^{-1}+z^{6}a^{2}-4z^{4}a^{2}+6z^{2}a^{2}+a^{2}z^{-2}-4a^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                 1   -1 -3               5 1   4 -5             5 4     -1 -7           7 2       5 -9         4 5         1 -11       7 7           0 -13     3 5             2 -15   3 6               -3 -17   3                 3 -19 3                   -3

Integral Khovanov Homology (db, data source)

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

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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