L11a417

L11a416

L11a418

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a417 at Knotilus!

Link Presentations

[edit Notes on L11a417's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X22,14,11,13 X20,16,21,15 X8,18,9,17 X16,8,17,7 X18,10,19,9 X10,20,5,19 X14,22,15,21 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 6, -5, 7, -8}, {11, -2, 3, -9, 4, -6, 5, -7, 8, -4, 9, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{2}w^{2}-2uv^{2}w-2uvw^{2}+4uvw-2uv-2uw+3u-3v^{2}w^{2}+2v^{2}w+2vw^{2}-4vw+2v+2w-2}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-5q^{6}+8q^{5}-10q^{4}+11q^{3}+q^{-3}-10q^{2}-q^{-2}+9q+4q^{-1}-5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+3z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}+3z^{2}a^{-2}+2z^{2}a^{-4}-2z^{2}a^{-6}-7z^{2}+3a^{2}+3a^{-2}-6+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+z^{9}a^{-1}+4z^{9}a^{-3}+3z^{9}a^{-5}-3z^{8}a^{-2}+4z^{8}a^{-6}+z^{8}+az^{7}-z^{7}a^{-1}-15z^{7}a^{-3}-9z^{7}a^{-5}+4z^{7}a^{-7}+a^{2}z^{6}+8z^{6}a^{-2}-4z^{6}a^{-4}-10z^{6}a^{-6}+3z^{6}a^{-8}-2az^{5}+26z^{5}a^{-3}+14z^{5}a^{-5}-9z^{5}a^{-7}+z^{5}a^{-9}-5a^{2}z^{4}-10z^{4}a^{-2}+10z^{4}a^{-4}+9z^{4}a^{-6}-7z^{4}a^{-8}-9z^{4}-3az^{3}-5z^{3}a^{-1}-14z^{3}a^{-3}-6z^{3}a^{-5}+4z^{3}a^{-7}-2z^{3}a^{-9}+8a^{2}z^{2}+5z^{2}a^{-2}-4z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}a^{-8}+13z^{2}+6az+6za^{-1}-5a^{2}-3a^{-2}+a^{-4}-8-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     2   2 13                   3 1   -2 11                 5 2     3 9               5 3       -2 7             6 5         1 5           6 7           1 3         3 4             -1 1       3 7               4 -1     1 2                 -1 -3     3                   3 -5 1 1                     0 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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