L11n318

L11n317

L11n319

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n318 at Knotilus!

Link Presentations

[edit Notes on L11n318's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X20,11,21,12 X22,17,11,18 X16,21,17,22 X10,13,5,14 X19,8,20,9 X7,18,8,19 X2,9,3,10 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, -9, 8, 10, -7}, {4, -2, 7, -3, 11, -6, 5, 9, -8, -4, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{3}-2uv^{2}w^{2}+uv^{2}w+uvw^{4}-3uvw^{3}+5uvw^{2}-3uvw+uw^{3}-2uw^{2}+2uw-2v^{2}w^{3}+2v^{2}w^{2}-v^{2}w+3vw^{3}-5vw^{2}+3vw-v-w^{3}+2w^{2}-w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-11}+4q^{-10}-8q^{-9}+11q^{-8}-14q^{-7}+15q^{-6}-12q^{-5}+11q^{-4}-5q^{-3}+3q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-a^{10}z^{-2}-a^{10}+3z^{4}a^{8}+8z^{2}a^{8}+4a^{8}z^{-2}+8a^{8}-2z^{6}a^{6}-9z^{4}a^{6}-17z^{2}a^{6}-5a^{6}z^{-2}-15a^{6}+3z^{4}a^{4}+9z^{2}a^{4}+2a^{4}z^{-2}+8a^{4}$ (db)
Kauffman polynomial	$a^{13}z^{5}-a^{13}z^{3}+4a^{12}z^{6}-6a^{12}z^{4}+a^{12}z^{2}+7a^{11}z^{7}-13a^{11}z^{5}+6a^{11}z^{3}-2a^{11}z+a^{11}z^{-1}+6a^{10}z^{8}-8a^{10}z^{6}+2a^{10}z^{4}-3a^{10}z^{2}-a^{10}z^{-2}+2a^{10}+2a^{9}z^{9}+9a^{9}z^{7}-28a^{9}z^{5}+27a^{9}z^{3}-13a^{9}z+5a^{9}z^{-1}+11a^{8}z^{8}-27a^{8}z^{6}+35a^{8}z^{4}-27a^{8}z^{2}-4a^{8}z^{-2}+13a^{8}+2a^{7}z^{9}+5a^{7}z^{7}-20a^{7}z^{5}+33a^{7}z^{3}-27a^{7}z+9a^{7}z^{-1}+5a^{6}z^{8}-15a^{6}z^{6}+33a^{6}z^{4}-37a^{6}z^{2}-5a^{6}z^{-2}+20a^{6}+3a^{5}z^{7}-6a^{5}z^{5}+13a^{5}z^{3}-16a^{5}z+5a^{5}z^{-1}+6a^{4}z^{4}-14a^{4}z^{2}-2a^{4}z^{-2}+10a^{4}$ (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 -3                   3 3 -5                 4 2 -2 -7               7 1   6 -9             6 5     -1 -11           9 6       3 -13         5 6         1 -15       6 9           -3 -17     3 6             3 -19   1 5               -4 -21   3                 3 -23 1                   -1

Integral Khovanov Homology (db, data source)

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

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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