L11a127

L11a126

L11a128

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a127 at Knotilus!

Link Presentations

[edit Notes on L11a127's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3uv^{4}-10uv^{3}+14uv^{2}-7uv+u+v^{5}-7v^{4}+14v^{3}-10v^{2}+3v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {19}{q^{9/2}}}+{\frac {22}{q^{7/2}}}+q^{5/2}-{\frac {23}{q^{5/2}}}-4q^{3/2}+{\frac {20}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {8}{q^{13/2}}}+{\frac {14}{q^{11/2}}}+9{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-4za^{7}-3a^{7}z^{-1}+6z^{3}a^{5}+9za^{5}+4a^{5}z^{-1}-3z^{5}a^{3}-7z^{3}a^{3}-8za^{3}-2a^{3}z^{-1}-z^{5}a+z^{3}a+za+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+6a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-10a^{8}z^{6}+12a^{8}z^{4}-6a^{8}z^{2}+a^{8}+4a^{7}z^{9}-7a^{7}z^{7}-8a^{7}z^{5}+23a^{7}z^{3}-15a^{7}z+3a^{7}z^{-1}+2a^{6}z^{10}+9a^{6}z^{8}-42a^{6}z^{6}+44a^{6}z^{4}-17a^{6}z^{2}+3a^{6}+13a^{5}z^{9}-22a^{5}z^{7}-19a^{5}z^{5}+44a^{5}z^{3}-24a^{5}z+4a^{5}z^{-1}+2a^{4}z^{10}+22a^{4}z^{8}-68a^{4}z^{6}+53a^{4}z^{4}-15a^{4}z^{2}+2a^{4}+9a^{3}z^{9}+a^{3}z^{7}-41a^{3}z^{5}+42a^{3}z^{3}-16a^{3}z+2a^{3}z^{-1}+16a^{2}z^{8}-27a^{2}z^{6}+14a^{2}z^{4}+z^{4}a^{-2}-3a^{2}z^{2}+a^{2}+15az^{7}-22az^{5}+4z^{5}a^{-1}+14az^{3}-z^{3}a^{-1}-3az+9z^{6}-6z^{4}+z^{2}$ (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 6                       1 -1 4                     3   3 2                   6 1   -5 0                 10 3     7 -2               11 7       -4 -4             12 9         3 -6           10 11           1 -8         9 12             -3 -10       6 11               5 -12     2 8                 -6 -14   1 6                   5 -16   2                     -2 -18 1                       1

Integral Khovanov Homology (db, data source)

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

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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