L11n277

L11n276

L11n278

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n277 at Knotilus!

Link Presentations

[edit Notes on L11n277's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,18,10,19 X22,17,11,18 X20,11,21,12 X16,21,17,22 X15,1,16,4 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 $u$, $v$, $w$, ...)	$-{\frac {uvw^{4}-uvw^{3}+uvw^{2}-uvw+uw-v^{2}w^{3}+vw^{3}-vw^{2}+vw-v}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$2q^{-2}-q^{-3}+4q^{-4}-3q^{-5}+4q^{-6}-3q^{-7}+2q^{-8}-q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+z^{4}a^{8}+5z^{2}a^{8}+4a^{8}z^{-2}+8a^{8}-z^{6}a^{6}-6z^{4}a^{6}-14z^{2}a^{6}-5a^{6}z^{-2}-15a^{6}+2z^{4}a^{4}+8z^{2}a^{4}+2a^{4}z^{-2}+8a^{4}$ (db)
Kauffman polynomial	$z^{3}a^{11}-2za^{11}+a^{11}z^{-1}+2z^{4}a^{10}-3z^{2}a^{10}-a^{10}z^{-2}+2a^{10}+z^{7}a^{9}-5z^{5}a^{9}+14z^{3}a^{9}-13za^{9}+5a^{9}z^{-1}+z^{8}a^{8}-6z^{6}a^{8}+18z^{4}a^{8}-21z^{2}a^{8}-4a^{8}z^{-2}+13a^{8}+2z^{7}a^{7}-10z^{5}a^{7}+26z^{3}a^{7}-27za^{7}+9a^{7}z^{-1}+z^{8}a^{6}-6z^{6}a^{6}+19z^{4}a^{6}-29z^{2}a^{6}-5a^{6}z^{-2}+20a^{6}+z^{7}a^{5}-5z^{5}a^{5}+13z^{3}a^{5}-16za^{5}+5a^{5}z^{-1}+3z^{4}a^{4}-11z^{2}a^{4}-2a^{4}z^{-2}+10a^{4}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 χ -3               2 2 -5             1 2 1 -7           3     3 -9         1 2     1 -11       3 2       1 -13       1         1 -15   2 3           -1 -17   1             1 -19 1               -1

Integral Khovanov Homology (db, data source)

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

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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