L11a239

L11a238

L11a240

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a239 at Knotilus!

Link Presentations

[edit Notes on L11a239's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,10,21,9 X22,13,7,14 X14,21,15,22 X10,16,11,15 X18,5,19,6 X16,20,17,19 X2738 X4,11,5,12 X6,17,1,18
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 3, -6, 10, -2, 4, -5, 6, -8, 11, -7, 8, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{3}-6u^{2}v^{2}+7u^{2}v-2u^{2}+uv^{4}-7uv^{3}+13uv^{2}-7uv+u-2v^{4}+7v^{3}-6v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {17}{q^{9/2}}}+{\frac {20}{q^{7/2}}}+q^{5/2}-{\frac {20}{q^{5/2}}}-4q^{3/2}+{\frac {18}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {8}{q^{13/2}}}+{\frac {12}{q^{11/2}}}+8{\sqrt {q}}-{\frac {14}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-4a^{7}z-2a^{7}z^{-1}+5a^{5}z^{3}+6a^{5}z+2a^{5}z^{-1}-2a^{3}z^{5}-3a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}-az^{5}+z^{3}a^{-1}-az$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+6a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-10a^{8}z^{6}+10a^{8}z^{4}-3a^{8}z^{2}+4a^{7}z^{9}-9a^{7}z^{7}-2a^{7}z^{5}+12a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+2a^{6}z^{10}+6a^{6}z^{8}-35a^{6}z^{6}+39a^{6}z^{4}-15a^{6}z^{2}+a^{6}+12a^{5}z^{9}-28a^{5}z^{7}+6a^{5}z^{5}+15a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+2a^{4}z^{10}+16a^{4}z^{8}-57a^{4}z^{6}+53a^{4}z^{4}-17a^{4}z^{2}+8a^{3}z^{9}-6a^{3}z^{7}-16a^{3}z^{5}+17a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}+13a^{2}z^{8}-24a^{2}z^{6}+17a^{2}z^{4}+z^{4}a^{-2}-5a^{2}z^{2}+12az^{7}-16az^{5}+4z^{5}a^{-1}+6az^{3}-2z^{3}a^{-1}+az+8z^{6}-6z^{4}$ (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 1 2 3 χ 6                       1 -1 4                     3   3 2                   5 1   -4 0                 9 3     6 -2               10 6       -4 -4             10 8         2 -6           10 10           0 -8         7 10             -3 -10       5 10               5 -12     3 7                 -4 -14   1 6                   5 -16   2                     -2 -18 1                       1

Integral Khovanov Homology (db, data source)

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

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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