L11n252

L11n251

L11n253

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n252 at Knotilus!

Link Presentations

[edit Notes on L11n252's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,3,15,4 X9,18,10,19 X5,16,6,17 X22,7,11,8 X6,21,7,22 X15,21,16,20 X17,8,18,9 X19,4,20,5 X2,11,3,12 X10,13,1,14
Gauss code	{1, -10, 2, 9, -4, -6, 5, 8, -3, -11}, {10, -1, 11, -2, -7, 4, -8, 3, -9, 7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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