L11n433

L11n432

L11n434

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n433 at Knotilus!

Link Presentations

[edit Notes on L11n433's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X20,16,21,15 X5,17,6,16 X11,19,12,18 X17,11,18,10 X19,5,20,4
Gauss code	{1, -4, 2, 11, -8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, 7, 8, -10, 9, -11, -7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(uvw+1)\left(uvw^{2}-uw^{2}+u-vw^{2}+v-1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{10}-q^{9}+q^{8}-q^{7}+q^{6}+q^{5}+2q^{3}-q^{2}+q$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-16z^{4}a^{-6}+6z^{4}a^{-8}+10z^{2}a^{-4}-17z^{2}a^{-6}+8z^{2}a^{-8}-z^{2}a^{-10}+6a^{-4}-9a^{-6}+3a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}-3z^{2}a^{-12}+z^{5}a^{-11}-3z^{3}a^{-11}+z^{6}a^{-10}-5z^{4}a^{-10}+5z^{2}a^{-10}-2a^{-10}+z^{7}a^{-9}-6z^{5}a^{-9}+7z^{3}a^{-9}+2z^{8}a^{-8}-13z^{6}a^{-8}+22z^{4}a^{-8}-11z^{2}a^{-8}-a^{-8}z^{-2}+3a^{-8}+z^{9}a^{-7}-5z^{7}a^{-7}+z^{5}a^{-7}+13z^{3}a^{-7}-9za^{-7}+2a^{-7}z^{-1}+3z^{8}a^{-6}-21z^{6}a^{-6}+44z^{4}a^{-6}-35z^{2}a^{-6}-2a^{-6}z^{-2}+11a^{-6}+z^{9}a^{-5}-6z^{7}a^{-5}+8z^{5}a^{-5}+3z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+16z^{4}a^{-4}-16z^{2}a^{-4}-a^{-4}z^{-2}+7a^{-4}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 21                     1 1 19                   1 1 0 17               1 1     0 15             1 2 1     0 13           1 2 1       0 11         1 1 2         2 9       1 3 1           1 7     1 1 2             2 5   1 2                 1 3                       0 1 1                     1

Integral Khovanov Homology (db, data source)

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

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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