L11a359

L11a358

L11a360

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a359 at Knotilus!

Link Presentations

[edit Notes on L11a359's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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