L11n236

L11n235

L11n237

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n236 at Knotilus!

Link Presentations

[edit Notes on L11n236's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X14,3,15,4 X20,13,21,14 X12,21,13,22 X22,5,9,6 X7,16,8,17 X15,18,16,19 X17,8,18,1 X6,9,7,10 X4,19,5,20
Gauss code	{1, -2, 3, -11, 6, -10, -7, 9}, {10, -1, 2, -5, 4, -3, -8, 7, -9, 8, 11, -4, 5, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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