L11a189

L11a188

L11a190

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a189 at Knotilus!

Link Presentations

[edit Notes on L11a189's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X16,6,17,5 X18,11,19,12 X22,15,7,16 X12,21,13,22 X20,13,21,14 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, 7, -8, 5, -3, 9, -4, 8, -7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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