L11a339

L11a338

L11a340

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a339 at Knotilus!

Link Presentations

[edit Notes on L11a339's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+3t(1)t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}+3t(1)t(2)-t(1)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$14q^{9/2}-17q^{7/2}+16q^{5/2}-{\frac {1}{q^{5/2}}}-14q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-3q^{15/2}+7q^{13/2}-11q^{11/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-3}+z^{7}a^{-1}-7z^{7}a^{-3}+z^{7}a^{-5}+5z^{5}a^{-1}-19z^{5}a^{-3}+5z^{5}a^{-5}+9z^{3}a^{-1}-25z^{3}a^{-3}+9z^{3}a^{-5}+8za^{-1}-15za^{-3}+7za^{-5}+2a^{-1}z^{-1}-3a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-4z^{9}a^{-1}-10z^{9}a^{-3}-6z^{9}a^{-5}-z^{8}a^{-2}-6z^{8}a^{-4}-8z^{8}a^{-6}-3z^{8}-az^{7}+13z^{7}a^{-1}+33z^{7}a^{-3}+11z^{7}a^{-5}-8z^{7}a^{-7}+20z^{6}a^{-2}+29z^{6}a^{-4}+14z^{6}a^{-6}-6z^{6}a^{-8}+11z^{6}+4az^{5}-11z^{5}a^{-1}-41z^{5}a^{-3}-11z^{5}a^{-5}+12z^{5}a^{-7}-3z^{5}a^{-9}-24z^{4}a^{-2}-33z^{4}a^{-4}-12z^{4}a^{-6}+7z^{4}a^{-8}-z^{4}a^{-10}-11z^{4}-5az^{3}+6z^{3}a^{-1}+35z^{3}a^{-3}+13z^{3}a^{-5}-9z^{3}a^{-7}+2z^{3}a^{-9}+10z^{2}a^{-2}+18z^{2}a^{-4}+6z^{2}a^{-6}-4z^{2}a^{-8}+z^{2}a^{-10}+3z^{2}+2az-7za^{-1}-17za^{-3}-6za^{-5}+2za^{-7}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 6 2     4 10               8 5       -3 8             9 6         3 6           7 8           1 4         7 9             -2 2       5 9               4 0     2 5                 -3 -2   1 5                   4 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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