L11a24

L11a23

L11a25

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a24 at Knotilus!

Link Presentations

[edit Notes on L11a24's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X18,11,19,12 X22,19,5,20 X20,15,21,16 X16,21,17,22 X12,17,13,18 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3uv^{4}-10uv^{3}+12uv^{2}-7uv+u+v^{5}-7v^{4}+12v^{3}-10v^{2}+3v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {14}{q^{5/2}}}-{\frac {19}{q^{7/2}}}+{\frac {21}{q^{9/2}}}-{\frac {22}{q^{11/2}}}+{\frac {18}{q^{13/2}}}-{\frac {14}{q^{15/2}}}+{\frac {8}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-6a^{7}z^{3}-9a^{7}z-3a^{7}z^{-1}+3a^{5}z^{5}+7a^{5}z^{3}+7a^{5}z+2a^{5}z^{-1}+a^{3}z^{5}-a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-3z^{2}a^{12}+a^{12}-3z^{7}a^{11}+7z^{5}a^{11}-6z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-5z^{8}a^{10}+9z^{6}a^{10}-3z^{4}a^{10}-2z^{2}a^{10}+2a^{10}-4z^{9}a^{9}-2z^{7}a^{9}+21z^{5}a^{9}-21z^{3}a^{9}+11za^{9}-3a^{9}z^{-1}-z^{10}a^{8}-16z^{8}a^{8}+42z^{6}a^{8}-34z^{4}a^{8}+12z^{2}a^{8}-9z^{9}a^{7}+2z^{7}a^{7}+31z^{5}a^{7}-36z^{3}a^{7}+15za^{7}-3a^{7}z^{-1}-z^{10}a^{6}-19z^{8}a^{6}+48z^{6}a^{6}-45z^{4}a^{6}+18z^{2}a^{6}-2a^{6}-5z^{9}a^{5}-5z^{7}a^{5}+27z^{5}a^{5}-31z^{3}a^{5}+13za^{5}-2a^{5}z^{-1}-8z^{8}a^{4}+13z^{6}a^{4}-13z^{4}a^{4}+6z^{2}a^{4}-6z^{7}a^{3}+9z^{5}a^{3}-8z^{3}a^{3}+5za^{3}-a^{3}z^{-1}-3z^{6}a^{2}+4z^{4}a^{2}-z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     2   -2 -2                   6 1   5 -4                 9 3     -6 -6               10 5       5 -8             11 9         -2 -10           11 10           1 -12         8 12             4 -14       6 10               -4 -16     2 8                 6 -18   1 6                   -5 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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