L11a159

L11a158

L11a160

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a159 at Knotilus!

Link Presentations

[edit Notes on L11a159's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-4u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-2uv^{4}+9uv^{3}-15uv^{2}+9uv-2u+v^{4}-4v^{3}+6v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+8q^{5/2}-14q^{3/2}+19{\sqrt {q}}-{\frac {22}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {20}{q^{5/2}}}+{\frac {14}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-5a^{3}z^{3}+z^{3}a^{-3}-6a^{3}z+za^{-3}-2a^{3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+9az^{3}-5z^{3}a^{-1}+8az-5za^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+8a^{5}z^{7}-13a^{5}z^{5}+9a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+9a^{4}z^{8}-12a^{4}z^{6}+z^{6}a^{-4}+6a^{4}z^{4}-2z^{4}a^{-4}-2a^{4}z^{2}+z^{2}a^{-4}+5a^{3}z^{9}+9a^{3}z^{7}+4z^{7}a^{-3}-36a^{3}z^{5}-10z^{5}a^{-3}+36a^{3}z^{3}+8z^{3}a^{-3}-14a^{3}z-2za^{-3}+2a^{3}z^{-1}+a^{2}z^{10}+20a^{2}z^{8}+6z^{8}a^{-2}-46a^{2}z^{6}-12z^{6}a^{-2}+35a^{2}z^{4}+6z^{4}a^{-2}-10a^{2}z^{2}-z^{2}a^{-2}+a^{2}+9az^{9}+4z^{9}a^{-1}+2az^{7}+5z^{7}a^{-1}-43az^{5}-31z^{5}a^{-1}+48az^{3}+30z^{3}a^{-1}-18az-10za^{-1}+2az^{-1}+a^{-1}z^{-1}+z^{10}+17z^{8}-43z^{6}+32z^{4}-9z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   5 1   -4 4                 9 3     6 2               10 5       -5 0             12 9         3 -2           11 11           0 -4         9 11             -2 -6       6 12               6 -8     3 8                 -5 -10   1 6                   5 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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