L11n276

L11n275

L11n277

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n276 at Knotilus!

Link Presentations

[edit Notes on L11n276's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X22,17,11,18 X20,11,21,12 X16,21,17,22 X4,15,1,16 X19,10,20,5
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11}, {8, 2, -4, -5, 10, -9, 7, 6, -11, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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