L11n155

L11n154

L11n156

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n155 at Knotilus!

Link Presentations

[edit Notes on L11n155's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X7,16,8,17 X13,20,14,21 X15,22,16,7 X6,19,1,20 X18,11,19,12 X12,6,13,5 X21,14,22,15 X4,18,5,17
Gauss code	{1, -2, 3, -11, 9, -7}, {-4, -1, 2, -3, 8, -9, -5, 10, -6, 4, 11, -8, 7, 5, -10, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-2t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-t(2)^{3}+2t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-2t(2)-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {10}{q^{13/2}}}+{\frac {10}{q^{15/2}}}-{\frac {8}{q^{17/2}}}+{\frac {6}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-3z^{3}a^{9}-3za^{9}-2a^{9}z^{-1}+z^{7}a^{7}+5z^{5}a^{7}+10z^{3}a^{7}+11za^{7}+5a^{7}z^{-1}-2z^{5}a^{5}-8z^{3}a^{5}-9za^{5}-3a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{4}-a^{14}z^{2}+3a^{13}z^{5}-3a^{13}z^{3}+5a^{12}z^{6}-7a^{12}z^{4}+4a^{12}z^{2}-a^{12}+5a^{11}z^{7}-7a^{11}z^{5}+5a^{11}z^{3}-a^{11}z+3a^{10}z^{8}-a^{10}z^{6}-3a^{10}z^{4}+3a^{10}z^{2}+a^{9}z^{9}+3a^{9}z^{7}-5a^{9}z^{5}-a^{9}z^{3}+4a^{9}z-2a^{9}z^{-1}+4a^{8}z^{8}-8a^{8}z^{6}+9a^{8}z^{4}-11a^{8}z^{2}+5a^{8}+a^{7}z^{9}-2a^{7}z^{7}+8a^{7}z^{5}-19a^{7}z^{3}+15a^{7}z-5a^{7}z^{-1}+a^{6}z^{8}-2a^{6}z^{6}+4a^{6}z^{4}-9a^{6}z^{2}+5a^{6}+3a^{5}z^{5}-10a^{5}z^{3}+10a^{5}z-3a^{5}z^{-1}$ (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 -4                   2 2 -6                 2 1 -1 -8               5 1   4 -10             4 3     -1 -12           6 4       2 -14         4 4         0 -16       4 6           -2 -18     2 4             2 -20   1 4               -3 -22   2                 2 -24 1                   -1

Integral Khovanov Homology (db, data source)

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

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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