L11n176

L11n175

L11n177

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n176 at Knotilus!

Link Presentations

[edit Notes on L11n176's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X14,5,15,6 X11,18,12,19 X3,10,4,11 X12,7,13,8 X16,13,17,14 X22,17,7,18 X6,15,1,16 X4,21,5,22 X19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {6, -1, 2, 5, -4, -6, 7, -3, 9, -7, 8, 4, -11, -2, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{4}-u^{2}v^{3}-uv^{4}+uv^{3}+uv^{2}+uv-u-v+2}{uv^{2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{23/2}}}+{\frac {2}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}(-z)+a^{11}z^{5}+5a^{11}z^{3}+4a^{11}z-a^{11}z^{-1}-a^{9}z^{7}-5a^{9}z^{5}-5a^{9}z^{3}+2a^{9}z+3a^{9}z^{-1}-a^{7}z^{7}-6a^{7}z^{5}-11a^{7}z^{3}-8a^{7}z-2a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{13}+5z^{5}a^{13}-6z^{3}a^{13}+za^{13}-2z^{8}a^{12}+11z^{6}a^{12}-17z^{4}a^{12}+8z^{2}a^{12}+a^{12}-z^{9}a^{11}+4z^{7}a^{11}-2z^{5}a^{11}-z^{3}a^{11}-a^{11}z^{-1}-3z^{8}a^{10}+15z^{6}a^{10}-18z^{4}a^{10}+3z^{2}a^{10}+3a^{10}-z^{9}a^{9}+4z^{7}a^{9}-z^{5}a^{9}-6z^{3}a^{9}+7za^{9}-3a^{9}z^{-1}-z^{8}a^{8}+4z^{6}a^{8}-z^{4}a^{8}-5z^{2}a^{8}+3a^{8}-z^{7}a^{7}+6z^{5}a^{7}-11z^{3}a^{7}+8za^{7}-2a^{7}z^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                 1 1 -8               1 1 0 -10             2     2 -12         1 1 1     1 -14         3 2       1 -16     1 2 1         0 -18     2 3           -1 -20   1 1             0 -22   1               -1 -24 1                 1

Integral Khovanov Homology (db, data source)

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

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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