L11a463

L11a462

L11a464

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a463 at Knotilus!

Link Presentations

[edit Notes on L11a463's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X8,14,9,13 X22,8,13,7 X20,15,21,16 X16,6,17,5 X18,12,19,11 X12,18,5,17 X10,20,11,19 X2,9,3,10 X4,22,1,21
Gauss code	{1, -10, 2, -11}, {6, -1, 4, -3, 10, -9, 7, -8}, {3, -2, 5, -6, 8, -7, 9, -5, 11, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}w^{2}-2uv^{3}w+uv^{2}w^{3}-3uv^{2}w^{2}+4uv^{2}w-2uv^{2}-2uvw^{3}+4uvw^{2}-3uvw+uv+uw^{3}-2uw^{2}+uw-v^{3}w^{2}+2v^{3}w-v^{3}-v^{2}w^{3}+3v^{2}w^{2}-4v^{2}w+2v^{2}+2vw^{3}-4vw^{2}+3vw-v+2w^{2}-w}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{9}-3q^{8}+7q^{7}-9q^{6}+15q^{5}-17q^{4}+17q^{3}-15q^{2}+12q-7+4q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}a^{-6}-z^{4}+z^{2}a^{-2}+2z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+a^{-4}-3a^{-6}+a^{-8}+1+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-3z^{4}a^{-10}+z^{2}a^{-10}+3z^{7}a^{-9}-8z^{5}a^{-9}+2z^{3}a^{-9}+6z^{8}a^{-8}-22z^{6}a^{-8}+28z^{4}a^{-8}-20z^{2}a^{-8}-a^{-8}z^{-2}+8a^{-8}+5z^{9}a^{-7}-13z^{7}a^{-7}+5z^{5}a^{-7}+11z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}+4z^{8}a^{-6}-30z^{6}a^{-6}+49z^{4}a^{-6}-31z^{2}a^{-6}-2a^{-6}z^{-2}+13a^{-6}+10z^{9}a^{-5}-30z^{7}a^{-5}+32z^{5}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+4z^{8}a^{-4}-18z^{6}a^{-4}+24z^{4}a^{-4}-12z^{2}a^{-4}-a^{-4}z^{-2}+5a^{-4}+5z^{9}a^{-3}-8z^{7}a^{-3}+9z^{5}a^{-3}-7z^{3}a^{-3}+6z^{8}a^{-2}-7z^{6}a^{-2}-z^{4}a^{-2}+6z^{7}a^{-1}+az^{5}-9z^{5}a^{-1}-az^{3}+z^{3}a^{-1}+4z^{6}-7z^{4}+2z^{2}+1$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3 1 -2 15                   4     4 13                 5 3     -2 11               10 4       6 9             8 6         -2 7           9 9           0 5         6 8             2 3       6 9               -3 1     3 8                 5 -1   1 4                   -3 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

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

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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