L11n217

L11n216

L11n218

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n217 at Knotilus!

Link Presentations

[edit Notes on L11n217's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X3,12,4,13 X15,22,16,9 X17,3,18,2 X21,4,22,5 X5,15,6,14 X13,21,14,20 X11,16,12,17 X19,7,20,6 X7,19,8,18
Gauss code	{1, 5, -3, 6, -7, 10, -11, -2}, {2, -1, -9, 3, -8, 7, -4, 9, -5, 11, -10, 8, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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