L11a191

L11a190

L11a192

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a191 at Knotilus!

Link Presentations

[edit Notes on L11a191's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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