L11a123

L11a122

L11a124

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a123 at Knotilus!

Link Presentations

[edit Notes on L11a123's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{5}-2t(2)^{5}-4t(1)t(2)^{4}+4t(2)^{4}+4t(1)t(2)^{3}-4t(2)^{3}-4t(1)t(2)^{2}+4t(2)^{2}+4t(1)t(2)-4t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {5}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {11}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {12}{q^{13/2}}}+{\frac {10}{q^{15/2}}}-{\frac {8}{q^{17/2}}}+{\frac {5}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}(-z)-2a^{11}z^{-1}+3a^{9}z^{3}+9a^{9}z+5a^{9}z^{-1}-3a^{7}z^{5}-11a^{7}z^{3}-10a^{7}z-3a^{7}z^{-1}+a^{5}z^{7}+4a^{5}z^{5}+4a^{5}z^{3}+a^{5}z-a^{3}z^{5}-3a^{3}z^{3}-a^{3}z$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-a^{14}-2z^{5}a^{13}+2z^{3}a^{13}-3z^{6}a^{12}+2z^{4}a^{12}-4z^{7}a^{11}+6z^{5}a^{11}-8z^{3}a^{11}+5za^{11}-2a^{11}z^{-1}-3z^{8}a^{10}+9z^{4}a^{10}-13z^{2}a^{10}+5a^{10}-2z^{9}a^{9}-2z^{7}a^{9}+14z^{5}a^{9}-20z^{3}a^{9}+15za^{9}-5a^{9}z^{-1}-z^{10}a^{8}-2z^{8}a^{8}+6z^{6}a^{8}+4z^{4}a^{8}-9z^{2}a^{8}+5a^{8}-5z^{9}a^{7}+14z^{7}a^{7}-8z^{5}a^{7}-4z^{3}a^{7}+9za^{7}-3a^{7}z^{-1}-z^{10}a^{6}-2z^{8}a^{6}+16z^{6}a^{6}-17z^{4}a^{6}+5z^{2}a^{6}-3z^{9}a^{5}+11z^{7}a^{5}-10z^{5}a^{5}+2z^{3}a^{5}-3z^{8}a^{4}+13z^{6}a^{4}-15z^{4}a^{4}+3z^{2}a^{4}-z^{7}a^{3}+4z^{5}a^{3}-4z^{3}a^{3}+za^{3}$ (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 0                       1 1 -2                     2   -2 -4                   3 1   2 -6                 5 3     -2 -8               6 2       4 -10             5 5         0 -12           7 6           1 -14         4 6             2 -16       4 6               -2 -18     1 4                 3 -20   1 4                   -3 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

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

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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