L11a47

L11a46

L11a48

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a47 at Knotilus!

Link Presentations

[edit Notes on L11a47's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$3q^{9/2}-{\frac {4}{q^{9/2}}}-6q^{7/2}+{\frac {8}{q^{7/2}}}+10q^{5/2}-{\frac {13}{q^{5/2}}}-15q^{3/2}+{\frac {16}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+17{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-2a^{3}z^{3}-3z^{3}a^{-3}-2za^{-3}+a^{3}z^{-1}-a^{-3}z^{-1}+az^{7}+z^{7}a^{-1}+3az^{5}+4z^{5}a^{-1}+az^{3}+6z^{3}a^{-1}-3az+5za^{-1}-2az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}+z^{7}a^{-5}+4a^{5}z^{5}-4z^{5}a^{-5}-2a^{5}z^{3}+4z^{3}a^{-5}+3z^{8}a^{-4}+8a^{4}z^{6}-12z^{6}a^{-4}-7a^{4}z^{4}+14z^{4}a^{-4}+a^{4}z^{2}-4z^{2}a^{-4}+4z^{9}a^{-3}+11a^{3}z^{7}-15z^{7}a^{-3}-15a^{3}z^{5}+20z^{5}a^{-3}+6a^{3}z^{3}-14z^{3}a^{-3}+a^{3}z+5za^{-3}-a^{3}z^{-1}-a^{-3}z^{-1}+2z^{10}a^{-2}+10a^{2}z^{8}-14a^{2}z^{6}-14z^{6}a^{-2}+4a^{2}z^{4}+16z^{4}a^{-2}+a^{2}z^{2}-5z^{2}a^{-2}+6az^{9}+10z^{9}a^{-1}-4az^{7}-31z^{7}a^{-1}-6az^{5}+37z^{5}a^{-1}-3az^{3}-29z^{3}a^{-1}+8az+12za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{10}+7z^{8}-24z^{6}+14z^{4}-z^{2}-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     2   -2 8                   4 1   3 6                 6 2     -4 4               9 4       5 2             8 6         -2 0           10 9           1 -2         8 10             2 -4       5 8               -3 -6     3 8                 5 -8   1 5                   -4 -10   3                     3 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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