L11a232

L11a231

L11a233

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a232 at Knotilus!

Link Presentations

[edit Notes on L11a232's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X18,6,19,5 X22,16,7,15 X16,11,17,12 X20,17,21,18 X10,21,11,22 X14,10,15,9 X4,20,5,19 X2738 X6,14,1,13
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 8, -7, 5, -2, 11, -8, 4, -5, 6, -3, 9, -6, 7, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+2t(2)^{4}-4t(1)^{2}t(2)^{3}+11t(1)t(2)^{3}-6t(2)^{3}+7t(1)^{2}t(2)^{2}-17t(1)t(2)^{2}+7t(2)^{2}-6t(1)^{2}t(2)+11t(1)t(2)-4t(2)+2t(1)^{2}-3t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-10q^{9/2}+{\frac {1}{q^{9/2}}}+17q^{7/2}-{\frac {5}{q^{7/2}}}-24q^{5/2}+{\frac {11}{q^{5/2}}}+27q^{3/2}-{\frac {18}{q^{3/2}}}-q^{13/2}+4q^{11/2}-28{\sqrt {q}}+{\frac {24}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+4z^{3}a^{-3}+5za^{-3}+2a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+3az^{3}-6z^{3}a^{-1}+2az-5za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-4z^{4}a^{-6}+z^{2}a^{-6}+9z^{7}a^{-5}-13z^{5}a^{-5}+10z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+12z^{8}a^{-4}+a^{4}z^{6}-17z^{6}a^{-4}-a^{4}z^{4}+11z^{4}a^{-4}-3z^{2}a^{-4}+9z^{9}a^{-3}+5a^{3}z^{7}-9a^{3}z^{5}-24z^{5}a^{-3}+4a^{3}z^{3}+27z^{3}a^{-3}-12za^{-3}+2a^{-3}z^{-1}+3z^{10}a^{-2}+10a^{2}z^{8}+21z^{8}a^{-2}-21a^{2}z^{6}-53z^{6}a^{-2}+12a^{2}z^{4}+38z^{4}a^{-2}-a^{2}z^{2}-10z^{2}a^{-2}+a^{-2}+9az^{9}+18z^{9}a^{-1}-9az^{7}-23z^{7}a^{-1}-12az^{5}-13z^{5}a^{-1}+13az^{3}+25z^{3}a^{-1}-4az-12za^{-1}+az^{-1}+2a^{-1}z^{-1}+3z^{10}+19z^{8}-54z^{6}+36z^{4}-7z^{2}$ (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                     3   -3 10                   7 1   6 8                 10 3     -7 6               14 7       7 4             14 11         -3 2           14 13           1 0         11 15             4 -2       7 13               -6 -4     4 11                 7 -6   1 7                   -6 -8   4                     4 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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