L11a306

L11a305

L11a307

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a306 at Knotilus!

Link Presentations

[edit Notes on L11a306's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(2)^{5}-3t(1)^{2}t(2)^{4}+4t(1)t(2)^{4}-t(2)^{4}-3t(1)^{3}t(2)^{3}+7t(1)^{2}t(2)^{3}-8t(1)t(2)^{3}+3t(2)^{3}+3t(1)^{3}t(2)^{2}-8t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-3t(2)^{2}-t(1)^{3}t(2)+4t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {8}{q^{23/2}}}-{\frac {13}{q^{21/2}}}+{\frac {17}{q^{19/2}}}-{\frac {19}{q^{17/2}}}+{\frac {19}{q^{15/2}}}-{\frac {17}{q^{13/2}}}+{\frac {11}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-a^{13}z^{-1}+4a^{11}z^{3}+8a^{11}z+3a^{11}z^{-1}-3a^{9}z^{5}-8a^{9}z^{3}-6a^{9}z-2a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-5a^{7}z-a^{5}z^{5}-2a^{5}z^{3}$ (db)
Kauffman polynomial	$a^{16}z^{6}-2a^{16}z^{4}+a^{16}z^{2}+4a^{15}z^{7}-10a^{15}z^{5}+6a^{15}z^{3}+6a^{14}z^{8}-13a^{14}z^{6}+5a^{14}z^{4}-a^{14}z^{2}+a^{14}+5a^{13}z^{9}-6a^{13}z^{7}-5a^{13}z^{5}+2a^{13}z^{3}+a^{13}z-a^{13}z^{-1}+2a^{12}z^{10}+7a^{12}z^{8}-27a^{12}z^{6}+30a^{12}z^{4}-19a^{12}z^{2}+3a^{12}+11a^{11}z^{9}-28a^{11}z^{7}+34a^{11}z^{5}-27a^{11}z^{3}+14a^{11}z-3a^{11}z^{-1}+2a^{10}z^{10}+8a^{10}z^{8}-31a^{10}z^{6}+42a^{10}z^{4}-20a^{10}z^{2}+3a^{10}+6a^{9}z^{9}-12a^{9}z^{7}+14a^{9}z^{5}-8a^{9}z^{3}+8a^{9}z-2a^{9}z^{-1}+7a^{8}z^{8}-15a^{8}z^{6}+14a^{8}z^{4}-3a^{8}z^{2}+6a^{7}z^{7}-14a^{7}z^{5}+13a^{7}z^{3}-5a^{7}z+3a^{6}z^{6}-5a^{6}z^{4}+a^{5}z^{5}-2a^{5}z^{3}$ (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 -4                       1 1 -6                     3 1 -2 -8                   4     4 -10                 7 3     -4 -12               10 4       6 -14             9 7         -2 -16           10 10           0 -18         7 9             2 -20       6 10               -4 -22     3 8                 5 -24   1 5                   -4 -26   3                     3 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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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