L11n253

L11n252

L11n254

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n253 at Knotilus!

Link Presentations

[edit Notes on L11n253'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 X20,15,21,16 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 $u$, $v$, $w$, ...)	${\frac {u^{4}\left(-v^{3}\right)-u^{3}v^{4}+u^{3}v^{3}-u^{3}v-u^{2}v^{2}-uv^{3}+uv-u-v}{u^{2}v^{2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{25/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{15/2}}}+{\frac {1}{q^{13/2}}}-{\frac {2}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}+z^{5}a^{11}+6z^{3}a^{11}+7za^{11}+a^{11}z^{-1}-z^{7}a^{9}-6z^{5}a^{9}-10z^{3}a^{9}-6za^{9}-a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-10z^{3}a^{7}-5za^{7}$ (db)
Kauffman polynomial	$a^{15}z^{5}-5a^{15}z^{3}+5a^{15}z-a^{14}z^{2}-2a^{13}z^{3}+2a^{13}z+a^{12}z^{8}-7a^{12}z^{6}+14a^{12}z^{4}-10a^{12}z^{2}+a^{11}z^{9}-7a^{11}z^{7}+16a^{11}z^{5}-18a^{11}z^{3}+9a^{11}z-a^{11}z^{-1}+2a^{10}z^{8}-12a^{10}z^{6}+19a^{10}z^{4}-9a^{10}z^{2}+a^{10}+a^{9}z^{9}-6a^{9}z^{7}+11a^{9}z^{5}-11a^{9}z^{3}+7a^{9}z-a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+5a^{8}z^{4}+a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-5a^{7}z$ (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             1 1 1     1 -14           1 2 1       0 -16           1 1         0 -18       1 2 2           -1 -20                       0 -22     1 1               0 -24 1                     1 -26 1                     1

Integral Khovanov Homology (db, data source)

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

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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