L11a223

L11a222

L11a224

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a223 at Knotilus!

Link Presentations

[edit Notes on L11a223's Link Presentations]

Planar diagram presentation	X8192 X14,9,15,10 X4758 X16,6,17,5 X18,16,19,15 X6,18,1,17 X22,19,7,20 X20,12,21,11 X10,22,11,21 X2,14,3,13 X12,4,13,3
Gauss code	{1, -10, 11, -3, 4, -6}, {3, -1, 2, -9, 8, -11, 10, -2, 5, -4, 6, -5, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-4t(1)^{2}t(2)^{3}+10t(1)t(2)^{3}-5t(2)^{3}+7t(1)^{2}t(2)^{2}-17t(1)t(2)^{2}+7t(2)^{2}-5t(1)^{2}t(2)+10t(1)t(2)-4t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-16q^{9/2}+21q^{7/2}-{\frac {1}{q^{7/2}}}-25q^{5/2}+{\frac {4}{q^{5/2}}}+25q^{3/2}-{\frac {10}{q^{3/2}}}+q^{15/2}-4q^{13/2}+9q^{11/2}-22{\sqrt {q}}+{\frac {16}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+3z^{5}a^{-3}+2az^{3}-9z^{3}a^{-1}+7z^{3}a^{-3}-3z^{3}a^{-5}+3az-7za^{-1}+7za^{-3}-3za^{-5}+za^{-7}+az^{-1}-2a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-7z^{9}a^{-1}-13z^{9}a^{-3}-6z^{9}a^{-5}-20z^{8}a^{-2}-16z^{8}a^{-4}-7z^{8}a^{-6}-11z^{8}-9az^{7}-6z^{7}a^{-1}+9z^{7}a^{-3}+2z^{7}a^{-5}-4z^{7}a^{-7}-4a^{2}z^{6}+45z^{6}a^{-2}+41z^{6}a^{-4}+14z^{6}a^{-6}-z^{6}a^{-8}+15z^{6}-a^{3}z^{5}+14az^{5}+33z^{5}a^{-1}+27z^{5}a^{-3}+18z^{5}a^{-5}+9z^{5}a^{-7}+4a^{2}z^{4}-28z^{4}a^{-2}-26z^{4}a^{-4}-8z^{4}a^{-6}+2z^{4}a^{-8}-8z^{4}+a^{3}z^{3}-11az^{3}-32z^{3}a^{-1}-32z^{3}a^{-3}-19z^{3}a^{-5}-7z^{3}a^{-7}-a^{2}z^{2}+6z^{2}a^{-2}+5z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}+2z^{2}+5az+13za^{-1}+13za^{-3}+7za^{-5}+2za^{-7}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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 16                       1 -1 14                     3   3 12                   6 1   -5 10                 10 3     7 8               11 6       -5 6             14 10         4 4           12 12           0 2         10 13             -3 0       7 13               6 -2     3 9                 -6 -4   1 7                   6 -6   3                     -3 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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