L11n312

L11n311

L11n313

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n312 at Knotilus!

Link Presentations

[edit Notes on L11n312's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X9,20,10,21 X7,16,8,17 X13,18,14,19 X19,14,20,15 X15,22,16,11 X17,10,18,5 X21,8,22,9 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, -3, 8}, {-11, 2, -5, 6, -7, 4, -8, 5, -6, 3, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{3}-2uv^{2}w^{2}+2uv^{2}w-uv^{2}+uvw^{4}-2uvw^{3}+3uvw^{2}-3uvw+uv+uw-v^{2}w^{3}-vw^{4}+3vw^{3}-3vw^{2}+2vw-v+w^{4}-2w^{3}+2w^{2}-w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$1-3q^{-1}+7q^{-2}-8q^{-3}+12q^{-4}-11q^{-5}+11q^{-6}-8q^{-7}+5q^{-8}-2q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+z^{4}a^{8}+4z^{2}a^{8}+4a^{8}z^{-2}+7a^{8}-z^{6}a^{6}-4z^{4}a^{6}-9z^{2}a^{6}-5a^{6}z^{-2}-12a^{6}-z^{6}a^{4}-2z^{4}a^{4}+2z^{2}a^{4}+2a^{4}z^{-2}+5a^{4}+z^{4}a^{2}+2z^{2}a^{2}+a^{2}$ (db)
Kauffman polynomial	$3z^{3}a^{11}-4za^{11}+a^{11}z^{-1}+z^{6}a^{10}+4z^{4}a^{10}-6z^{2}a^{10}-a^{10}z^{-2}+3a^{10}+5z^{7}a^{9}-15z^{5}a^{9}+28z^{3}a^{9}-19za^{9}+5a^{9}z^{-1}+6z^{8}a^{8}-21z^{6}a^{8}+39z^{4}a^{8}-32z^{2}a^{8}-4a^{8}z^{-2}+15a^{8}+2z^{9}a^{7}+4z^{7}a^{7}-27z^{5}a^{7}+46z^{3}a^{7}-33za^{7}+9a^{7}z^{-1}+10z^{8}a^{6}-33z^{6}a^{6}+44z^{4}a^{6}-37z^{2}a^{6}-5a^{6}z^{-2}+20a^{6}+2z^{9}a^{5}+2z^{7}a^{5}-20z^{5}a^{5}+25z^{3}a^{5}-18za^{5}+5a^{5}z^{-1}+4z^{8}a^{4}-10z^{6}a^{4}+6z^{4}a^{4}-8z^{2}a^{4}-2a^{4}z^{-2}+8a^{4}+3z^{7}a^{3}-8z^{5}a^{3}+4z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$ (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 1 2 χ 1                   1 1 -1                 2   -2 -3               5 1   4 -5             5 4     -1 -7           7 3       4 -9         5 6         1 -11       6 6           0 -13     2 5             3 -15   3 6               -3 -17   3                 3 -19 2                   -2

Integral Khovanov Homology (db, data source)

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

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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