L11n316

L11n315

L11n317

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n316 at Knotilus!

Link Presentations

[edit Notes on L11n316's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,17,8,16 X20,10,21,9 X18,12,19,11 X22,20,11,19 X15,9,16,8 X10,22,5,21 X17,14,18,15 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, 4, -8}, {5, -2, 11, 9, -7, 3, -9, -5, 6, -4, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{2}w-uv^{2}+uvw^{2}-3uvw+2uv-uw^{2}+uw-v^{2}w+v^{2}-2vw^{2}+3vw-v+w^{2}-2w}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{9}-3q^{8}+5q^{7}-6q^{6}+8q^{5}-7q^{4}+7q^{3}-4q^{2}+3q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}-z^{4}a^{-6}+a^{-6}z^{-2}+2a^{-6}-2z^{4}a^{-4}-5z^{2}a^{-4}-2a^{-4}z^{-2}-6a^{-4}+3z^{2}a^{-2}+a^{-2}z^{-2}+4a^{-2}$ (db)
Kauffman polynomial	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}a^{-8}-2z^{7}a^{-5}+z^{7}a^{-7}+3z^{7}a^{-9}-z^{6}a^{-4}-11z^{6}a^{-6}-9z^{6}a^{-8}+z^{6}a^{-10}+3z^{5}a^{-3}+5z^{5}a^{-5}-8z^{5}a^{-7}-10z^{5}a^{-9}-z^{4}a^{-4}+8z^{4}a^{-6}+6z^{4}a^{-8}-3z^{4}a^{-10}-4z^{3}a^{-3}-8z^{3}a^{-5}+3z^{3}a^{-7}+7z^{3}a^{-9}+5z^{2}a^{-2}+10z^{2}a^{-4}-3z^{2}a^{-8}+2z^{2}a^{-10}+6za^{-3}+6za^{-5}-5a^{-2}-8a^{-4}-3a^{-6}+a^{-8}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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   \ 0 1 2 3 4 5 6 7 8 χ 19                 1 1 17               2   -2 15             3 1   2 13           4 3     -1 11         4 2       2 9       3 4         1 7     4 4           0 5   1 4             3 3 2 3               -1 1 3                 3

Integral Khovanov Homology (db, data source)

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

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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