L11n16

L11n15

L11n17

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n16 at Knotilus!

Link Presentations

[edit Notes on L11n16's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X13,2,14,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 10, -7, 4, -11, -2, 3, 6, -8, 9, -10, 7, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{5}-2uv^{4}+uv^{3}+v^{2}-2v+2}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {2}{q^{21/2}}}-{\frac {1}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+z^{5}a^{11}+5z^{3}a^{11}+5za^{11}+a^{11}z^{-1}-z^{7}a^{9}-5z^{5}a^{9}-5z^{3}a^{9}+2za^{9}+2a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-11z^{3}a^{7}-8za^{7}-2a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{16}+a^{16}-z^{3}a^{15}+za^{15}-z^{4}a^{14}+z^{2}a^{14}-z^{7}a^{13}+4z^{5}a^{13}-4z^{3}a^{13}+a^{13}z^{-1}-2z^{8}a^{12}+11z^{6}a^{12}-19z^{4}a^{12}+13z^{2}a^{12}-3a^{12}-z^{9}a^{11}+4z^{7}a^{11}-3z^{5}a^{11}+2z^{3}a^{11}-3za^{11}+a^{11}z^{-1}-3z^{8}a^{10}+15z^{6}a^{10}-19z^{4}a^{10}+6z^{2}a^{10}-z^{9}a^{9}+4z^{7}a^{9}-z^{5}a^{9}-6z^{3}a^{9}+6za^{9}-2a^{9}z^{-1}-z^{8}a^{8}+4z^{6}a^{8}-z^{4}a^{8}-5z^{2}a^{8}+3a^{8}-z^{7}a^{7}+6z^{5}a^{7}-11z^{3}a^{7}+8za^{7}-2a^{7}z^{-1}$ (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 -6                   1 1 -8                 1 1 0 -10               2     2 -12           1 1 1     1 -14         1 4 2       1 -16       1 2 1         0 -18       3 3           0 -20   1 2 2             1 -22   1 2               -1 -24 1 1                 0 -26 1                   -1

Integral Khovanov Homology (db, data source)

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

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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