L11a318

L11a317

L11a319

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a318 at Knotilus!

Link Presentations

[edit Notes on L11a318's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X20,13,21,14 X16,8,17,7 X18,6,19,5 X6,18,7,17 X4,20,5,19 X22,15,9,16 X14,21,15,22
Gauss code	{1, -2, 3, -9, 7, -8, 6, -4}, {4, -1, 2, -3, 5, -11, 10, -6, 8, -7, 9, -5, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}+t(1)t(2)^{3}+t(2)^{3}+t(1)t(2)^{2}+t(1)^{2}t(2)+t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{5/2}}}-q^{3/2}+{\frac {4}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {2}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{7}-4z^{3}a^{7}-3za^{7}+z^{7}a^{5}+5z^{5}a^{5}+6z^{3}a^{5}-a^{5}z^{-1}+z^{7}a^{3}+6z^{5}a^{3}+12z^{3}a^{3}+10za^{3}+3a^{3}z^{-1}-z^{5}a-5z^{3}a-7za-2az^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{2}+2a^{11}z^{3}+3a^{10}z^{4}-2a^{10}z^{2}+4a^{9}z^{5}-6a^{9}z^{3}+5a^{8}z^{6}-12a^{8}z^{4}+3a^{8}z^{2}+6a^{7}z^{7}-22a^{7}z^{5}+20a^{7}z^{3}-6a^{7}z+5a^{6}z^{8}-21a^{6}z^{6}+23a^{6}z^{4}-8a^{6}z^{2}+a^{6}+3a^{5}z^{9}-12a^{5}z^{7}+8a^{5}z^{5}+5a^{5}z^{3}-a^{5}z^{-1}+a^{4}z^{10}-a^{4}z^{8}-16a^{4}z^{6}+37a^{4}z^{4}-21a^{4}z^{2}+3a^{4}+4a^{3}z^{9}-26a^{3}z^{7}+57a^{3}z^{5}-51a^{3}z^{3}+19a^{3}z-3a^{3}z^{-1}+a^{2}z^{10}-6a^{2}z^{8}+10a^{2}z^{6}-a^{2}z^{4}-7a^{2}z^{2}+3a^{2}+az^{9}-8az^{7}+23az^{5}-28az^{3}+13az-2az^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 4                       1 1 2                         0 0                   3 1   2 -2                 1       -1 -4               4 3       1 -6             3 3         0 -8           3 2           1 -10         2 3             1 -12       2 3               -1 -14     1 2                 1 -16   1 2                   -1 -18   1                     1 -20 1                       -1

Integral Khovanov Homology (db, data source)

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

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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