L11a162

L11a161

L11a163

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a162 at Knotilus!

Link Presentations

[edit Notes on L11a162's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,11,17,12 X14,6,15,5 X4,16,5,15 X20,13,21,14 X22,17,7,18 X18,21,19,22 X12,19,13,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -11, 8, -6, 7, -5, 9, -10, 11, -8, 10, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-4t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-2t(2)^{3}+4t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+4t(2)^{2}-2t(1)^{2}t(2)+4t(1)t(2)-4t(2)-2t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {7}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {3}{q^{21/2}}}+{\frac {6}{q^{19/2}}}-{\frac {9}{q^{17/2}}}+{\frac {12}{q^{15/2}}}-{\frac {13}{q^{13/2}}}+{\frac {12}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-3z^{3}a^{9}-2za^{9}-a^{9}z^{-1}+z^{7}a^{7}+4z^{5}a^{7}+5z^{3}a^{7}+4za^{7}+2a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+4z^{3}a^{5}+za^{5}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{4}-a^{14}z^{2}+3a^{13}z^{5}-3a^{13}z^{3}+5a^{12}z^{6}-6a^{12}z^{4}+2a^{12}z^{2}+6a^{11}z^{7}-9a^{11}z^{5}+6a^{11}z^{3}-a^{11}z+6a^{10}z^{8}-13a^{10}z^{6}+15a^{10}z^{4}-8a^{10}z^{2}+2a^{10}+4a^{9}z^{9}-8a^{9}z^{7}+8a^{9}z^{5}-6a^{9}z^{3}+3a^{9}z-a^{9}z^{-1}+a^{8}z^{10}+6a^{8}z^{8}-27a^{8}z^{6}+36a^{8}z^{4}-22a^{8}z^{2}+5a^{8}+6a^{7}z^{9}-20a^{7}z^{7}+23a^{7}z^{5}-16a^{7}z^{3}+7a^{7}z-2a^{7}z^{-1}+a^{6}z^{10}+2a^{6}z^{8}-17a^{6}z^{6}+22a^{6}z^{4}-12a^{6}z^{2}+3a^{6}+2a^{5}z^{9}-5a^{5}z^{7}-2a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+2a^{4}z^{8}-8a^{4}z^{6}+8a^{4}z^{4}-a^{4}z^{2}-a^{4}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                     1   -1 -4                   4 1   3 -6                 4 2     -2 -8               7 3       4 -10             6 5         -1 -12           7 6           1 -14         5 6             1 -16       4 7               -3 -18     2 5                 3 -20   1 4                   -3 -22   2                     2 -24 1                       -1

Integral Khovanov Homology (db, data source)

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

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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