L11a332

L11a331

L11a333

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a332 at Knotilus!

Link Presentations

[edit Notes on L11a332's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+t(2)t(1)^{3}-3t(2)^{3}t(1)^{2}+8t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+t(1)^{2}+t(2)^{3}t(1)-5t(2)^{2}t(1)+8t(2)t(1)-3t(1)+t(2)^{2}-3t(2)+3}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{29/2}}}-{\frac {4}{q^{27/2}}}+{\frac {7}{q^{25/2}}}-{\frac {11}{q^{23/2}}}+{\frac {14}{q^{21/2}}}-{\frac {15}{q^{19/2}}}+{\frac {15}{q^{17/2}}}-{\frac {12}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {6}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-2a^{13}z+3a^{11}z^{5}+11a^{11}z^{3}+9a^{11}z-2a^{9}z^{7}-10a^{9}z^{5}-15a^{9}z^{3}-6a^{9}z+a^{9}z^{-1}-a^{7}z^{7}-5a^{7}z^{5}-8a^{7}z^{3}-5a^{7}z-a^{7}z^{-1}$ (db)
Kauffman polynomial	$a^{18}z^{4}+4a^{17}z^{5}-3a^{17}z^{3}+7a^{16}z^{6}-7a^{16}z^{4}+a^{16}z^{2}+8a^{15}z^{7}-8a^{15}z^{5}-a^{15}z^{3}+2a^{15}z+7a^{14}z^{8}-8a^{14}z^{6}+a^{14}z^{4}+4a^{13}z^{9}-12a^{13}z^{5}+7a^{13}z^{3}+a^{13}z+a^{12}z^{10}+10a^{12}z^{8}-37a^{12}z^{6}+42a^{12}z^{4}-18a^{12}z^{2}+7a^{11}z^{9}-20a^{11}z^{7}+18a^{11}z^{5}-12a^{11}z^{3}+7a^{11}z+a^{10}z^{10}+5a^{10}z^{8}-29a^{10}z^{6}+38a^{10}z^{4}-16a^{10}z^{2}+3a^{9}z^{9}-11a^{9}z^{7}+13a^{9}z^{5}-9a^{9}z^{3}+3a^{9}z+a^{9}z^{-1}+2a^{8}z^{8}-7a^{8}z^{6}+5a^{8}z^{4}+a^{8}z^{2}-a^{8}+a^{7}z^{7}-5a^{7}z^{5}+8a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                     2 1 -1 -10                   4     4 -12                 4 2     -2 -14               8 4       4 -16             7 4         -3 -18           8 8           0 -20         6 7             1 -22       5 8               -3 -24     3 7                 4 -26   1 4                   -3 -28   3                     3 -30 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-8$ $i=-6$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-8$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-7$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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