L11a51

L11a50

L11a52

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a51 at Knotilus!

Link Presentations

[edit Notes on L11a51's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(v^{4}-v^{3}+v^{2}-v+1\right)}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-5q^{9/2}+8q^{7/2}-11q^{5/2}+12q^{3/2}-13{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{9}a^{-1}-az^{7}+7z^{7}a^{-1}-z^{7}a^{-3}-5az^{5}+17z^{5}a^{-1}-5z^{5}a^{-3}-7az^{3}+16z^{3}a^{-1}-7z^{3}a^{-3}-2az+4za^{-1}-2za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-4az^{9}-8z^{9}a^{-1}-4z^{9}a^{-3}-4a^{2}z^{8}+4z^{8}a^{-2}-4z^{8}a^{-4}+4z^{8}-3a^{3}z^{7}+15az^{7}+36z^{7}a^{-1}+14z^{7}a^{-3}-4z^{7}a^{-5}-a^{4}z^{6}+13a^{2}z^{6}+9z^{6}a^{-4}-3z^{6}a^{-6}+2z^{6}+10a^{3}z^{5}-21az^{5}-65z^{5}a^{-1}-24z^{5}a^{-3}+9z^{5}a^{-5}-z^{5}a^{-7}+3a^{4}z^{4}-9a^{2}z^{4}-11z^{4}a^{-2}-6z^{4}a^{-4}+7z^{4}a^{-6}-10z^{4}-6a^{3}z^{3}+16az^{3}+44z^{3}a^{-1}+16z^{3}a^{-3}-4z^{3}a^{-5}+2z^{3}a^{-7}-a^{4}z^{2}+a^{2}z^{2}+5z^{2}a^{-2}-2z^{2}a^{-6}+5z^{2}-4az-8za^{-1}-4za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     2   -2 10                   3 1   2 8                 5 2     -3 6               6 3       3 4             6 5         -1 2           7 6           1 0         5 8             3 -2       3 5               -2 -4     2 5                 3 -6   1 3                   -2 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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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