L10a47

L10a46

L10a48

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a47 at Knotilus!

Link Presentations

[edit Notes on L10a47's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{3}-4uv^{2}+4uv-3u-3v^{3}+4v^{2}-4v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-5q^{9/2}+7q^{7/2}-{\frac {1}{q^{7/2}}}-8q^{5/2}+{\frac {1}{q^{5/2}}}+9q^{3/2}-{\frac {4}{q^{3/2}}}-q^{13/2}+3q^{11/2}-8{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}-2az^{3}+2z^{3}a^{-1}+2z^{3}a^{-3}-z^{3}a^{-5}+a^{3}z-5az+2za^{-1}+za^{-3}-za^{-5}+2a^{3}z^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-4z^{8}a^{-2}-3z^{8}a^{-4}-z^{8}-az^{7}+2z^{7}a^{-1}-z^{7}a^{-3}-4z^{7}a^{-5}-a^{2}z^{6}+11z^{6}a^{-2}+7z^{6}a^{-4}-3z^{6}a^{-6}-a^{3}z^{5}-az^{5}-7z^{5}a^{-1}+4z^{5}a^{-3}+10z^{5}a^{-5}-z^{5}a^{-7}+a^{2}z^{4}-16z^{4}a^{-2}-7z^{4}a^{-4}+7z^{4}a^{-6}-z^{4}+4a^{3}z^{3}+7az^{3}+7z^{3}a^{-1}-4z^{3}a^{-3}-6z^{3}a^{-5}+2z^{3}a^{-7}+3a^{2}z^{2}+7z^{2}a^{-2}+3z^{2}a^{-4}-2z^{2}a^{-6}+5z^{2}-5a^{3}z-7az-3za^{-1}+za^{-5}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   2   -2 10                 3 1   2 8               4 2     -2 6             4 3       1 4           5 4         -1 2         3 4           -1 0       3 6             3 -2     1 2               -1 -4     3                 3 -6 1 1                   0 -8 1                     1

Integral Khovanov Homology (db, data source)

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

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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