L11a461

L11a460

L11a462

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a461 at Knotilus!

Link Presentations

[edit Notes on L11a461's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X18,15,19,16 X16,6,17,5 X12,18,5,17 X8,22,9,21 X20,8,21,7 X22,10,13,9 X10,14,11,13 X2,11,3,12 X4,20,1,19
Gauss code	{1, -10, 2, -11}, {4, -1, 7, -6, 8, -9, 10, -5}, {9, -2, 3, -4, 5, -3, 11, -7, 6, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}-2t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}-t(3)t(2)^{3}-2t(1)t(3)^{3}t(2)^{2}+2t(3)^{3}t(2)^{2}+4t(1)t(3)^{2}t(2)^{2}-4t(3)^{2}t(2)^{2}-2t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-3t(1)t(3)^{2}t(2)+2t(3)^{2}t(2)-2t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+2t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{10}-3q^{9}+7q^{8}-10q^{7}+14q^{6}-15q^{5}+16q^{4}-13q^{3}+10q^{2}-6q+4-q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{4}a^{-8}+3z^{2}a^{-8}+a^{-8}z^{-2}+2a^{-8}-2z^{6}a^{-6}-8z^{4}a^{-6}-9z^{2}a^{-6}-2a^{-6}z^{-2}-5a^{-6}+z^{8}a^{-4}+5z^{6}a^{-4}+8z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}-z^{6}a^{-2}-3z^{4}a^{-2}+2a^{-2}$ (db)
Kauffman 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 z^4 a^{-12} -z^2 a^{-12} +3 z^5 a^{-11} -2 z^3 a^{-11} +6 z^6 a^{-10} -8 z^4 a^{-10} +6 z^2 a^{-10} -2 a^{-10} +7 z^7 a^{-9} -8 z^5 a^{-9} +2 z^3 a^{-9} +z a^{-9} +7 z^8 a^{-8} -11 z^6 a^{-8} +8 z^4 a^{-8} -6 z^2 a^{-8} - a^{-8} z^{-2} +3 a^{-8} +5 z^9 a^{-7} -6 z^7 a^{-7} -4 z^5 a^{-7} +9 z^3 a^{-7} -8 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +5 z^8 a^{-6} -29 z^6 a^{-6} +37 z^4 a^{-6} -24 z^2 a^{-6} -2 a^{-6} z^{-2} +9 a^{-6} +10 z^9 a^{-5} -33 z^7 a^{-5} +29 z^5 a^{-5} -z^3 a^{-5} -8 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +2 z^8 a^{-4} -28 z^6 a^{-4} +36 z^4 a^{-4} -13 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +5 z^9 a^{-3} -19 z^7 a^{-3} +19 z^5 a^{-3} -5 z^3 a^{-3} +z a^{-3} +4 z^8 a^{-2} -16 z^6 a^{-2} +16 z^4 a^{-2} -2 z^2 a^{-2} -2 a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +z^3 a^{-1} } (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums 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 \chi} (fixed $j$, alternation over $r$).

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3 1 -2 17                   4     4 15                 6 3     -3 13               8 4       4 11             8 7         -1 9           8 7           1 7         5 8             3 5       5 8               -3 3     3 7                 4 1   1 3                   -2 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a460 L11a462