L11n457

L11n456

L11n458

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n457 at Knotilus!

Link Presentations

[edit Notes on L11n457's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X15,18,16,11 X20,9,21,10 X22,13,19,14 X14,21,15,22 X10,19,5,20 X17,8,18,9 X7,16,8,17 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {7, -4, 6, -5}, {10, -1, -9, 8, 4, -7}, {11, -2, 5, -6, -3, 9, -8, 3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{t(1) t(4)^2 t(3)^2+t(2) t(4)^2 t(3)^2-t(4)^2 t(3)^2-t(1) t(4) t(3)^2+t(1) t(2) t(4) t(3)^2-t(2) t(4) t(3)^2+t(4) t(3)^2-t(1) t(4)^2 t(3)+t(1) t(2) t(4)^2 t(3)-t(2) t(4)^2 t(3)+t(4)^2 t(3)-t(1) t(3)+t(1) t(2) t(3)-t(2) t(3)+2 t(1) t(4) t(3)-3 t(1) t(2) t(4) t(3)+2 t(2) t(4) t(3)-3 t(4) t(3)+t(3)+t(1)-t(1) t(2)+t(2)-t(1) t(4)+t(1) t(2) t(4)-t(2) t(4)+t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}} (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {12}{q^{13/2}}}+{\frac {10}{q^{15/2}}}-{\frac {11}{q^{17/2}}}+{\frac {6}{q^{19/2}}}-{\frac {5}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}z^{-3}-2za^{11}-5a^{11}z^{-1}-3a^{11}z^{-3}+5z^{3}a^{9}+14za^{9}+10a^{9}z^{-1}+3a^{9}z^{-3}-3z^{5}a^{7}-11z^{3}a^{7}-12za^{7}-5a^{7}z^{-1}-a^{7}z^{-3}-z^{5}a^{5}-2z^{3}a^{5}$ (db)
Kauffman polynomial	$-z^{4}a^{14}-5z^{5}a^{13}+5z^{3}a^{13}+4za^{13}-5a^{13}z^{-1}+a^{13}z^{-3}-z^{8}a^{12}-2z^{6}a^{12}+3z^{4}a^{12}-6z^{2}a^{12}-3a^{12}z^{-2}+10a^{12}-z^{9}a^{11}-z^{7}a^{11}-z^{5}a^{11}-3z^{3}a^{11}+15za^{11}-12a^{11}z^{-1}+3a^{11}z^{-3}-5z^{8}a^{10}+6z^{6}a^{10}+4z^{4}a^{10}-20z^{2}a^{10}-6a^{10}z^{-2}+19a^{10}-z^{9}a^{9}-7z^{7}a^{9}+22z^{5}a^{9}-29z^{3}a^{9}+23za^{9}-12a^{9}z^{-1}+3a^{9}z^{-3}-4z^{8}a^{8}+5z^{6}a^{8}+5z^{4}a^{8}-14z^{2}a^{8}-3a^{8}z^{-2}+10a^{8}-6z^{7}a^{7}+17z^{5}a^{7}-19z^{3}a^{7}+12za^{7}-5a^{7}z^{-1}+a^{7}z^{-3}-3z^{6}a^{6}+5z^{4}a^{6}-z^{5}a^{5}+2z^{3}a^{5}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ).

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                 3 1 -2 -8               4     4 -10             4 3     -1 -12           8 4       4 -14         4 6         2 -16       7 6           1 -18     3 8             5 -20   2 3               -1 -22   4                 4 -24 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-6$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{7}$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}} ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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