L11a149

L11a148

L11a150

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a149 at Knotilus!

Link Presentations

[edit Notes on L11a149's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,17,15,18 X16,7,17,8 X22,15,7,16 X18,13,19,14 X6,20,1,19 X20,12,21,11 X12,6,13,5 X4,21,5,22
Gauss code	{1, -2, 3, -11, 10, -8}, {5, -1, 2, -3, 9, -10, 7, -4, 6, -5, 4, -7, 8, -9, 11, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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