L11a139

L11a138

L11a140

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a139 at Knotilus!

Link Presentations

[edit Notes on L11a139's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-5u^{2}v^{3}+8u^{2}v^{2}-5u^{2}v+u^{2}-2uv^{4}+10uv^{3}-15uv^{2}+10uv-2u+v^{4}-5v^{3}+8v^{2}-5v+1}{uv^{2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-9q^{9/2}+16q^{7/2}-22q^{5/2}+25q^{3/2}-26{\sqrt {q}}+{\frac {22}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+3az^{3}-5z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-3za^{-1}+3za^{-3}-za^{-5}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-5z^{4}a^{-6}+2z^{2}a^{-6}+8z^{7}a^{-5}-11z^{5}a^{-5}+7z^{3}a^{-5}-2za^{-5}+10z^{8}a^{-4}+a^{4}z^{6}-12z^{6}a^{-4}-a^{4}z^{4}+5z^{4}a^{-4}+7z^{9}a^{-3}+5a^{3}z^{7}+4z^{7}a^{-3}-10a^{3}z^{5}-25z^{5}a^{-3}+5a^{3}z^{3}+22z^{3}a^{-3}+a^{3}z-6za^{-3}-a^{3}z^{-1}+2z^{10}a^{-2}+9a^{2}z^{8}+21z^{8}a^{-2}-19a^{2}z^{6}-49z^{6}a^{-2}+11a^{2}z^{4}+34z^{4}a^{-2}-2a^{2}z^{2}-8z^{2}a^{-2}+a^{2}+7az^{9}+14z^{9}a^{-1}-2az^{7}-11z^{7}a^{-1}-22az^{5}-25z^{5}a^{-1}+17az^{3}+26z^{3}a^{-1}-az-6za^{-1}-az^{-1}+2z^{10}+20z^{8}-53z^{6}+36z^{4}-8z^{2}$ (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                     3   -3 10                   6 1   5 8                 10 3     -7 6               12 6       6 4             13 10         -3 2           13 12           1 0         10 14             4 -2       7 12               -5 -4     4 11                 7 -6   1 6                   -5 -8   4                     4 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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