L11a118

L11a117

L11a119

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a118 at Knotilus!

Link Presentations

[edit Notes on L11a118's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3uv^{4}-12uv^{3}+14uv^{2}-7uv+u+v^{5}-7v^{4}+14v^{3}-12v^{2}+3v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {16}{q^{5/2}}}-{\frac {22}{q^{7/2}}}+{\frac {24}{q^{9/2}}}-{\frac {24}{q^{11/2}}}+{\frac {20}{q^{13/2}}}-{\frac {15}{q^{15/2}}}+{\frac {8}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-6a^{7}z^{3}-8a^{7}z-3a^{7}z^{-1}+3a^{5}z^{5}+6a^{5}z^{3}+6a^{5}z+2a^{5}z^{-1}+a^{3}z^{5}-2a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}-az^{3}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-3z^{2}a^{12}+a^{12}-3z^{7}a^{11}+7z^{5}a^{11}-6z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-5z^{8}a^{10}+8z^{6}a^{10}-z^{4}a^{10}-4z^{2}a^{10}+2a^{10}-5z^{9}a^{9}+2z^{7}a^{9}+13z^{5}a^{9}-17z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-2z^{10}a^{8}-13z^{8}a^{8}+36z^{6}a^{8}-28z^{4}a^{8}+9z^{2}a^{8}-13z^{9}a^{7}+15z^{7}a^{7}+13z^{5}a^{7}-26z^{3}a^{7}+15za^{7}-3a^{7}z^{-1}-2z^{10}a^{6}-20z^{8}a^{6}+52z^{6}a^{6}-44z^{4}a^{6}+15z^{2}a^{6}-2a^{6}-8z^{9}a^{5}+z^{7}a^{5}+22z^{5}a^{5}-26z^{3}a^{5}+11za^{5}-2a^{5}z^{-1}-12z^{8}a^{4}+21z^{6}a^{4}-16z^{4}a^{4}+5z^{2}a^{4}-9z^{7}a^{3}+14z^{5}a^{3}-10z^{3}a^{3}+5za^{3}-a^{3}z^{-1}-4z^{6}a^{2}+4z^{4}a^{2}-z^{5}a+z^{3}a$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     3   -3 -2                   7 1   6 -4                 10 4     -6 -6               12 6       6 -8             12 10         -2 -10           12 12           0 -12         9 13             4 -14       6 11               -5 -16     2 9                 7 -18   1 6                   -5 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-4$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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