L11n453

L11n452

L11n454

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n453 at Knotilus!

Link Presentations

[edit Notes on L11n453's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X15,2,16,3 X16,7,17,8 X19,22,20,15 X21,14,22,11 X13,20,14,21 X9,18,10,19 X11,10,12,5 X4,17,1,18
Gauss code	{1, 4, -3, -11}, {-10, 2, -8, 7}, {-2, -1, 5, 3, -9, 10}, {-4, -5, 11, 9, -6, 8, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uvw^{2}x^{2}-uvw^{2}x-uvwx^{2}-uw^{2}x^{2}+uw^{2}x+uwx^{2}-uwx-vwx+vw+vx-v-w-x+2}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{25/2}}}-{\frac {3}{q^{23/2}}}+{\frac {3}{q^{21/2}}}-{\frac {6}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {4}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}+a^{13}z^{-3}+z^{5}a^{11}+4z^{3}a^{11}+za^{11}-5a^{11}z^{-1}-3a^{11}z^{-3}-z^{7}a^{9}-4z^{5}a^{9}+11za^{9}+10a^{9}z^{-1}+3a^{9}z^{-3}-z^{7}a^{7}-6z^{5}a^{7}-12z^{3}a^{7}-11za^{7}-5a^{7}z^{-1}-a^{7}z^{-3}$ (db)
Kauffman polynomial	$-z^{2}a^{16}-3z^{3}a^{15}+3za^{15}-z^{6}a^{14}+z^{4}a^{14}-z^{2}a^{14}-3z^{7}a^{13}+12z^{5}a^{13}-18z^{3}a^{13}+12za^{13}-5a^{13}z^{-1}+a^{13}z^{-3}-3z^{8}a^{12}+12z^{6}a^{12}-12z^{4}a^{12}-3z^{2}a^{12}-3a^{12}z^{-2}+10a^{12}-z^{9}a^{11}+14z^{5}a^{11}-25z^{3}a^{11}+21za^{11}-12a^{11}z^{-1}+3a^{11}z^{-3}-4z^{8}a^{10}+16z^{6}a^{10}-10z^{4}a^{10}-15z^{2}a^{10}-6a^{10}z^{-2}+19a^{10}-z^{9}a^{9}+2z^{7}a^{9}+8z^{5}a^{9}-22z^{3}a^{9}+23za^{9}-12a^{9}z^{-1}+3a^{9}z^{-3}-z^{8}a^{8}+3z^{6}a^{8}+3z^{4}a^{8}-12z^{2}a^{8}-3a^{8}z^{-2}+10a^{8}-z^{7}a^{7}+6z^{5}a^{7}-12z^{3}a^{7}+11za^{7}-5a^{7}z^{-1}+a^{7}z^{-3}$ (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 χ -6                   1 1 -8                 1 1 0 -10               3     3 -12           1 1 1     1 -14           6 3       3 -16       2 2 2         2 -18       6 4           2 -20   1 2 4             3 -22   2 2               0 -24   2                 2 -26 1                   -1

Integral Khovanov Homology (db, data source)

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

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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