L11a145

L11a144

L11a146

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a145 at Knotilus!

Link Presentations

[edit Notes on L11a145's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X18,12,19,11 X12,6,13,5 X4,19,5,20 X14,7,15,8 X20,13,21,14 X22,15,7,16 X16,21,17,22 X6,18,1,17
Gauss code	{1, -2, 3, -6, 5, -11}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -10, 11, -4, 6, -8, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+7t(1)t(2)^{3}-2t(2)^{3}+5t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+5t(2)^{2}-2t(1)^{2}t(2)+7t(1)t(2)-5t(2)-2t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {17}{q^{9/2}}}-{\frac {18}{q^{7/2}}}+{\frac {15}{q^{5/2}}}+q^{3/2}-{\frac {12}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {11}{q^{13/2}}}-{\frac {15}{q^{11/2}}}-4{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+2za^{7}-z^{7}a^{5}-4z^{5}a^{5}-6z^{3}a^{5}-4za^{5}-z^{7}a^{3}-3z^{5}a^{3}-z^{3}a^{3}+2za^{3}+a^{3}z^{-1}+z^{5}a+2z^{3}a-za-az^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-3z^{6}a^{10}+6z^{4}a^{10}-2z^{2}a^{10}-5z^{7}a^{9}+10z^{5}a^{9}-6z^{3}a^{9}+za^{9}-6z^{8}a^{8}+13z^{6}a^{8}-13z^{4}a^{8}+4z^{2}a^{8}-5z^{9}a^{7}+10z^{7}a^{7}-12z^{5}a^{7}+5z^{3}a^{7}-za^{7}-2z^{10}a^{6}-4z^{8}a^{6}+19z^{6}a^{6}-28z^{4}a^{6}+12z^{2}a^{6}-10z^{9}a^{5}+29z^{7}a^{5}-38z^{5}a^{5}+23z^{3}a^{5}-5za^{5}-2z^{10}a^{4}-4z^{8}a^{4}+20z^{6}a^{4}-21z^{4}a^{4}+8z^{2}a^{4}-5z^{9}a^{3}+10z^{7}a^{3}-4z^{5}a^{3}+4z^{3}a^{3}-4za^{3}+a^{3}z^{-1}-6z^{8}a^{2}+16z^{6}a^{2}-10z^{4}a^{2}+2z^{2}a^{2}-a^{2}-4z^{7}a+11z^{5}a-6z^{3}a-za+az^{-1}-z^{6}+2z^{4}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   4 1   -3 -2                 8 3     5 -4               8 5       -3 -6             10 7         3 -8           8 9           1 -10         7 9             -2 -12       4 8               4 -14     2 7                 -5 -16   1 4                   3 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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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