L11a457

L11a456

L11a458

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a457 at Knotilus!

Link Presentations

[edit Notes on L11a457's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,12,21,11 X18,8,19,7 X22,9,13,10 X16,21,17,22 X8,18,9,17 X10,15,11,16 X12,20,5,19 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 5, -8, 3, -9}, {11, -2, 8, -6, 7, -4, 9, -3, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(3)^{2}t(2)^{3}+t(1)t(2)^{3}-t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-2t(3)^{3}t(2)^{2}-4t(1)t(3)^{2}t(2)^{2}+5t(3)^{2}t(2)^{2}-3t(1)t(2)^{2}+6t(1)t(3)t(2)^{2}-6t(3)t(2)^{2}+2t(2)^{2}-2t(1)t(3)^{3}t(2)+3t(3)^{3}t(2)+6t(1)t(3)^{2}t(2)-6t(3)^{2}t(2)+2t(1)t(2)-5t(1)t(3)t(2)+4t(3)t(2)-t(2)+t(1)t(3)^{3}-t(3)^{3}-2t(1)t(3)^{2}+t(3)^{2}+t(1)t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	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 q^3-4 q^2+10 q-15+21 q^{-1} -22 q^{-2} +23 q^{-3} -19 q^{-4} +14 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} } (db)
Signature	-2 (db)
HOMFLY-PT polynomial	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 -a^8+3 a^6 z^2+2 a^6-3 a^4 z^4-2 a^4 z^2+a^4 z^{-2} +a^4+a^2 z^6-3 a^2 z^2-2 a^2 z^{-2} +z^2 a^{-2} -5 a^2-2 z^4+ z^{-2} +3} (db)
Kauffman polynomial	$a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+3a^{8}z^{6}-5a^{8}z^{4}+3a^{8}z^{2}-a^{8}+5a^{7}z^{7}-5a^{7}z^{5}+a^{7}z+7a^{6}z^{8}-9a^{6}z^{6}+6a^{6}z^{4}+6a^{5}z^{9}-3a^{5}z^{7}-6a^{5}z^{5}+10a^{5}z^{3}-3a^{5}z+2a^{4}z^{10}+15a^{4}z^{8}-45a^{4}z^{6}+52a^{4}z^{4}-30a^{4}z^{2}-a^{4}z^{-2}+9a^{4}+13a^{3}z^{9}-21a^{3}z^{7}+3a^{3}z^{5}+10a^{3}z^{3}-8a^{3}z+2a^{3}z^{-1}+2a^{2}z^{10}+16a^{2}z^{8}-53a^{2}z^{6}+z^{6}a^{-2}+59a^{2}z^{4}-2z^{4}a^{-2}-39a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+7az^{9}-9az^{7}+4z^{7}a^{-1}-5az^{5}-8z^{5}a^{-1}+6az^{3}+4z^{3}a^{-1}-5az+2az^{-1}+8z^{8}-19z^{6}+16z^{4}-11z^{2}-z^{-2}+6$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   7 1   6 1                 8 3     -5 -1               13 7       6 -3             12 11         -1 -5           11 10           1 -7         8 12             4 -9       6 11               -5 -11     2 9                 7 -13   1 5                   -4 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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