L11n254

L11n253

L11n255

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n254 at Knotilus!

Link Presentations

[edit Notes on L11n254's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X22,7,11,8 X9,18,10,19 X17,20,18,21 X19,10,20,1 X8,11,9,12 X4,15,5,16 X6,21,7,22
Gauss code	{1, -2, 3, -10, 4, -11, 5, -9, -6, 8}, {9, -1, 2, -3, 10, -4, -7, 6, -8, 7, 11, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{4}v^{4}+u^{3}v^{2}+u^{2}v^{3}-u^{2}v^{2}+u^{2}v+uv^{2}+1}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}+{\frac {2}{q^{21/2}}}-{\frac {2}{q^{23/2}}}+{\frac {1}{q^{25/2}}}-{\frac {1}{q^{27/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-2za^{13}+z^{7}a^{11}+8z^{5}a^{11}+19z^{3}a^{11}+14za^{11}+a^{11}z^{-1}-z^{9}a^{9}-9z^{7}a^{9}-28z^{5}a^{9}-36z^{3}a^{9}-17za^{9}-a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{17}z^{3}-2a^{17}z+a^{16}z^{4}-a^{16}z^{2}+a^{15}z^{5}-2a^{15}z^{3}+2a^{15}z+a^{14}z^{4}+2a^{13}z+a^{12}z^{8}-8a^{12}z^{6}+19a^{12}z^{4}-13a^{12}z^{2}+a^{11}z^{9}-9a^{11}z^{7}+27a^{11}z^{5}-33a^{11}z^{3}+15a^{11}z-a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+19a^{10}z^{4}-14a^{10}z^{2}+a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-36a^{9}z^{3}+17a^{9}z-a^{9}z^{-1}$ (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 -8                     1 1 -10                     1 1 -12                 1     1 -14             1         1 -16           1 1 1       -1 -18         1 1           0 -20       1 1 1           -1 -22     1 1               0 -24     1                 1 -26 1 1                   0 -28 1                     1

Integral Khovanov Homology (db, data source)

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

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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