L11a155

L11a154

L11a156

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a155 at Knotilus!

Link Presentations

[edit Notes on L11a155's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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