L11a249

L11a248

L11a250

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a249 at Knotilus!

Link Presentations

[edit Notes on L11a249's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,8,17,7 X18,12,19,11 X2,19,3,20 X12,4,13,3 X20,13,21,14 X14,5,15,6 X6,9,7,10 X22,16,9,15 X8,18,1,17 X4,22,5,21
Gauss code	{1, -4, 5, -11, 7, -8, 2, -10}, {8, -1, 3, -5, 6, -7, 9, -2, 10, -3, 4, -6, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{4}-2u^{2}v^{3}+u^{2}v^{2}-uv^{4}+3uv^{3}-6uv^{2}+3uv-u+v^{2}-2v+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+5q^{11/2}-11q^{9/2}+18q^{7/2}-25q^{5/2}+28q^{3/2}-29{\sqrt {q}}+{\frac {24}{\sqrt {q}}}-{\frac {18}{q^{3/2}}}+{\frac {11}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{9}a^{-1}-az^{7}+5z^{7}a^{-1}-z^{7}a^{-3}-3az^{5}+7z^{5}a^{-1}-3z^{5}a^{-3}-az^{3}-z^{3}a^{-3}+2az-4za^{-1}+2za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}+5z^{6}a^{-6}-4z^{4}a^{-6}+11z^{7}a^{-5}-14z^{5}a^{-5}+4z^{3}a^{-5}+14z^{8}a^{-4}+a^{4}z^{6}-18z^{6}a^{-4}-a^{4}z^{4}+4z^{4}a^{-4}+z^{2}a^{-4}+11z^{9}a^{-3}+5a^{3}z^{7}-6z^{7}a^{-3}-9a^{3}z^{5}-14z^{5}a^{-3}+3a^{3}z^{3}+13z^{3}a^{-3}-4za^{-3}+4z^{10}a^{-2}+10a^{2}z^{8}+18z^{8}a^{-2}-21a^{2}z^{6}-48z^{6}a^{-2}+11a^{2}z^{4}+30z^{4}a^{-2}-a^{2}z^{2}-3z^{2}a^{-2}+10az^{9}+21z^{9}a^{-1}-15az^{7}-37z^{7}a^{-1}-2az^{5}+8z^{5}a^{-1}+9az^{3}+15z^{3}a^{-1}-4az-8za^{-1}+az^{-1}+a^{-1}z^{-1}+4z^{10}+14z^{8}-47z^{6}+34z^{4}-5z^{2}-1$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed 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 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     4   -4 10                   7 1   6 8                 11 4     -7 6               14 7       7 4             14 11         -3 2           15 14           1 0         11 16             5 -2       7 13               -6 -4     4 11                 7 -6   1 7                   -6 -8   4                     4 -10 1                       -1

Integral Khovanov Homology (db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ 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 r=-2} ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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