L11a470

L11a469

L11a471

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Link Presentations

[edit Notes on L11a470's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(w-1) \left(2 u v^2 w^2-2 u v^2 w-3 u v w^2+3 u v w-u v+u w^2-2 u w+u+v^2 w^3-2 v^2 w^2+v^2 w-v w^3+3 v w^2-3 v w-2 w^2+2 w\right)}{\sqrt{u} v w^2}$ (db)
Jones polynomial	$1-3 q^{-1} +8 q^{-2} -12 q^{-3} +17 q^{-4} -19 q^{-5} +20 q^{-6} -16 q^{-7} +13 q^{-8} -7 q^{-9} +3 q^{-10} - q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-z^2 a^{10}-2 a^{10}+3 z^4 a^8+8 z^2 a^8+a^8 z^{-2} +6 a^8-2 z^6 a^6-7 z^4 a^6-10 z^2 a^6-2 a^6 z^{-2} -8 a^6-z^6 a^4-z^4 a^4+3 z^2 a^4+a^4 z^{-2} +3 a^4+z^4 a^2+2 z^2 a^2+a^2$ (db)
Kauffman polynomial	$a^{13} z^5-2 a^{13} z^3+a^{13} z+3 a^{12} z^6-5 a^{12} z^4+3 a^{12} z^2-a^{12}+5 a^{11} z^7-6 a^{11} z^5+a^{11} z^3+a^{11} z+6 a^{10} z^8-7 a^{10} z^6+4 a^{10} z^4-2 a^{10} z^2+4 a^9 z^9+3 a^9 z^7-15 a^9 z^5+14 a^9 z^3-3 a^9 z+a^8 z^{10}+14 a^8 z^8-38 a^8 z^6+46 a^8 z^4-31 a^8 z^2-a^8 z^{-2} +10 a^8+8 a^7 z^9-7 a^7 z^7-12 a^7 z^5+19 a^7 z^3-10 a^7 z+2 a^7 z^{-1} +a^6 z^{10}+13 a^6 z^8-40 a^6 z^6+48 a^6 z^4-36 a^6 z^2-2 a^6 z^{-2} +14 a^6+4 a^5 z^9-2 a^5 z^7-11 a^5 z^5+12 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +5 a^4 z^8-11 a^4 z^6+8 a^4 z^4-7 a^4 z^2-a^4 z^{-2} +5 a^4+3 a^3 z^7-7 a^3 z^5+4 a^3 z^3+a^2 z^6-3 a^2 z^4+3 a^2 z^2-a^2$ (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                   6 1   5 -5                 7 3     -4 -7               10 5       5 -9             10 8         -2 -11           10 9           1 -13         7 11             4 -15       6 9               -3 -17     2 8                 6 -19   1 5                   -4 -21   2                     2 -23 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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