L11a405

L11a404

L11a406

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Link Presentations

[edit Notes on L11a405's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-t(2)^2 t(3)^4+t(1) t(3)^4-t(1) t(2) t(3)^4+2 t(2) t(3)^4-t(3)^4-t(1) t(2)^2 t(3)^3+2 t(2)^2 t(3)^3-2 t(1) t(3)^3+4 t(1) t(2) t(3)^3-4 t(2) t(3)^3+t(3)^3+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+2 t(1) t(3)^2-4 t(1) t(2) t(3)^2+4 t(2) t(3)^2-t(3)^2-t(1) t(2)^2 t(3)+2 t(2)^2 t(3)-2 t(1) t(3)+4 t(1) t(2) t(3)-4 t(2) t(3)+t(3)+t(1) t(2)^2-t(2)^2+t(1)-2 t(1) t(2)+t(2)}{\sqrt{t(1)} t(2) t(3)^2}$ (db)
Jones polynomial	$1-3 q^{-1} +7 q^{-2} -10 q^{-3} +16 q^{-4} -16 q^{-5} +18 q^{-6} -15 q^{-7} +11 q^{-8} -7 q^{-9} +3 q^{-10} - q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-z^2 a^{10}-a^{10} z^{-2} -2 a^{10}+3 z^4 a^8+9 z^2 a^8+4 a^8 z^{-2} +9 a^8-2 z^6 a^6-8 z^4 a^6-14 z^2 a^6-5 a^6 z^{-2} -14 a^6-z^6 a^4-z^4 a^4+5 z^2 a^4+2 a^4 z^{-2} +7 a^4+z^4 a^2+2 z^2 a^2$ (db)
Kauffman polynomial	$a^{13} z^5-2 a^{13} z^3+a^{13} z+3 a^{12} z^6-5 a^{12} z^4+a^{12} z^2+5 a^{11} z^7-8 a^{11} z^5+4 a^{11} z^3-4 a^{11} z+a^{11} z^{-1} +6 a^{10} z^8-11 a^{10} z^6+11 a^{10} z^4-10 a^{10} z^2-a^{10} z^{-2} +5 a^{10}+4 a^9 z^9-18 a^9 z^5+33 a^9 z^3-21 a^9 z+5 a^9 z^{-1} +a^8 z^{10}+13 a^8 z^8-48 a^8 z^6+72 a^8 z^4-48 a^8 z^2-4 a^8 z^{-2} +18 a^8+8 a^7 z^9-14 a^7 z^7-6 a^7 z^5+36 a^7 z^3-29 a^7 z+9 a^7 z^{-1} +a^6 z^{10}+12 a^6 z^8-50 a^6 z^6+77 a^6 z^4-57 a^6 z^2-5 a^6 z^{-2} +21 a^6+4 a^5 z^9-6 a^5 z^7-5 a^5 z^5+13 a^5 z^3-13 a^5 z+5 a^5 z^{-1} +5 a^4 z^8-15 a^4 z^6+18 a^4 z^4-18 a^4 z^2-2 a^4 z^{-2} +9 a^4+3 a^3 z^7-8 a^3 z^5+4 a^3 z^3+a^2 z^6-3 a^2 z^4+2 a^2 z^2$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ 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 1 2 χ 1                       1 1 -1                     2   -2 -3                   5 1   4 -5                 6 3     -3 -7               10 4       6 -9             8 8         0 -11           10 8           2 -13         6 9             3 -15       5 9               -4 -17     2 6                 4 -19   1 5                   -4 -21   2                     2 -23 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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