L11a469

L11a468

L11a470

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a469 at Knotilus!

Link Presentations

[edit Notes on L11a469's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,11,19,12 X16,8,17,7 X8,16,9,15 X22,17,15,18 X12,21,13,22 X20,13,21,14 X14,19,5,20 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, 6, -3, 9, -8, 7, -6}, {10, -1, 4, -5, 11, -2, 3, -7, 8, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)\left(2uv^{2}w-uv^{2}+2uvw^{2}-4uvw+uv-2uw^{2}+uw+v^{2}w^{2}-2v^{2}w+vw^{3}-4vw^{2}+2vw-w^{3}+2w^{2}\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-10}+3q^{-9}-6q^{-8}+11q^{-7}-13q^{-6}+17q^{-5}-16q^{-4}+15q^{-3}-11q^{-2}+q+7q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+3a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-2a^{6}z^{4}-a^{6}z^{2}-2a^{6}z^{-2}-2a^{6}-3a^{4}z^{4}-3a^{4}z^{2}+a^{4}z^{-2}-a^{4}-a^{2}z^{4}+2a^{2}z^{2}+2a^{2}+z^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-4a^{11}z^{5}+5a^{11}z^{3}-2a^{11}z+3a^{10}z^{8}-12a^{10}z^{6}+16a^{10}z^{4}-9a^{10}z^{2}+3a^{10}+3a^{9}z^{9}-6a^{9}z^{7}-7a^{9}z^{5}+18a^{9}z^{3}-7a^{9}z+a^{8}z^{10}+8a^{8}z^{8}-38a^{8}z^{6}+51a^{8}z^{4}-34a^{8}z^{2}-a^{8}z^{-2}+12a^{8}+7a^{7}z^{9}-11a^{7}z^{7}-13a^{7}z^{5}+25a^{7}z^{3}-14a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+12a^{6}z^{8}-38a^{6}z^{6}+39a^{6}z^{4}-28a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+4a^{5}z^{9}+4a^{5}z^{7}-24a^{5}z^{5}+22a^{5}z^{3}-10a^{5}z+2a^{5}z^{-1}+7a^{4}z^{8}-6a^{4}z^{6}-4a^{4}z^{4}+4a^{4}z^{2}-a^{4}z^{-2}+2a^{4}+8a^{3}z^{7}-11a^{3}z^{5}+8a^{3}z^{3}-a^{3}z+6a^{2}z^{6}-7a^{2}z^{4}+6a^{2}z^{2}-2a^{2}+3az^{5}-2az^{3}+z^{4}-z^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                       1 1 1                     2   -2 -1                   5 1   4 -3                 7 3     -4 -5               8 4       4 -7             9 8         -1 -9           8 7           1 -11         6 10             4 -13       5 7               -2 -15     2 7                 5 -17   1 4                   -3 -19   2                     2 -21 1                       -1

Integral Khovanov Homology (db, data source)

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

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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