L10a51

L10a50

L10a52

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a51 at Knotilus!

Link Presentations

[edit Notes on L10a51's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X4,15,5,16 X12,5,13,6 X16,12,17,11 X6,18,1,17 X14,19,15,20 X18,13,19,14
Gauss code	{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 10, -9, 5, -7, 8, -10, 9, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+t(2)+t(1)-1\right)\left(t(1)t(2)^{2}-t(2)^{2}-t(1)t(2)+2t(2)+t(1)-1\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}-q^{7/2}+{\frac {12}{q^{7/2}}}+4q^{5/2}-{\frac {16}{q^{5/2}}}-8q^{3/2}+{\frac {16}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+12{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-2a^{3}z^{5}-5a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+6az^{3}-2z^{3}a^{-1}+2az-za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-6a^{6}z^{4}+2a^{6}z^{2}+7a^{5}z^{7}-12a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+6a^{4}z^{8}-4a^{4}z^{6}-6a^{4}z^{4}+4a^{4}z^{2}+2a^{3}z^{9}+13a^{3}z^{7}-33a^{3}z^{5}+z^{5}a^{-3}+24a^{3}z^{3}-z^{3}a^{-3}-5a^{3}z-a^{3}z^{-1}+12a^{2}z^{8}-16a^{2}z^{6}+4z^{6}a^{-2}-6z^{4}a^{-2}+4a^{2}z^{2}+2z^{2}a^{-2}+a^{2}+2az^{9}+13az^{7}+7z^{7}a^{-1}-33az^{5}-12z^{5}a^{-1}+24az^{3}+7z^{3}a^{-1}-5az-2za^{-1}-az^{-1}+6z^{8}-4z^{6}-6z^{4}+4z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   3   -3 4                 5 1   4 2               7 3     -4 0             9 5       4 -2           8 8         0 -4         8 8           0 -6       5 9             4 -8     3 7               -4 -10   1 5                 4 -12   3                   -3 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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