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DrawMorseLink

PD Presentation:

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

Gauss Code:

{{1, -11, 2, -10, 3, -6, 5, -9}, {7, -1, 10, -2, 11, -3, 4, -5, 8, -7, 9, -4, 6, -8}}

Jones Polynomial:

- q-19/2 + 3q-17/2 - 6q-15/2 + 11q-13/2 - 16q-11/2 + 18q-9/2 - 19q-7/2 + 16q-5/2 - 13q-3/2 + 8q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - q-26 + q-24 - 3q-20 + 3q-18 - 2q-16 + 3q-14 + 3q-12 + 5q-8 - 3q-6 + 2q-4 - 1 + 2q2 - q4

HOMFLY-PT Polynomial:

2az3 + az5 - a3z-1 - a3z - 2a3z3 - 3a3z5 - a3z7 + a5z-1 - 3a5z - 6a5z3 - 4a5z5 - a5z7 + 2a7z + 3a7z3 + a7z5

Kauffman Polynomial:

- z2 + 2z4 - z6 - 6az3 + 10az5 - 4az7 - a2z2 - 4a2z4 + 13a2z6 - 6a2z8 - a3z-1 + a3z + 2a3z5 + 7a3z7 - 5a3z9 + a4 + 2a4z2 - 10a4z4 + 17a4z6 - 5a4z8 - 2a4z10 - a5z-1 + 12a5z3 - 25a5z5 + 24a5z7 - 10a5z9 + 6a6z2 - 21a6z4 + 18a6z6 - 5a6z8 - 2a6z10 + a7z - 3a7z3 - 6a7z5 + 8a7z7 - 5a7z9 + 2a8z2 - 11a8z4 + 12a8z6 - 6a8z8 + 2a9z - 7a9z3 + 10a9z5 - 5a9z7 - 2a10z2 + 6a10z4 - 3a10z6 + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     3   j = 0                   5 1   j = -2                 8 3     j = -4               9 6       j = -6             10 7         j = -8           8 9           j = -10         8 10             j = -12       4 9               j = -14     2 7                 j = -16   1 4                   j = -18   2                     j = -20 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 257]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 257]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[14, 5, 15, 6], X[20, 16, 21, 15], > X[16, 8, 17, 7], X[6, 21, 7, 22], X[18, 9, 19, 10], X[22, 17, 9, 18], > X[8, 20, 1, 19], X[4, 11, 5, 12], X[2, 13, 3, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -10, 3, -6, 5, -9}, > {7, -1, 10, -2, 11, -3, 4, -5, 8, -7, 9, -4, 6, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 6 11 16 18 19 16 13 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 8 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 -24 3 3 2 3 3 5 3 2 2 4 -1 + q - q + q - --- + --- - --- + --- + --- + -- - -- + -- + 2 q - q 20 18 16 14 12 8 6 4 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 257]][a, z]
Out[8]=	3 5 a a 3 5 7 3 3 3 5 3 7 3 -(--) + -- - a z - 3 a z + 2 a z + 2 a z - 2 a z - 6 a z + 3 a z + z z 5 3 5 5 5 7 5 3 7 5 7 > a z - 3 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 257]][a, z]
Out[9]=	3 5 4 a a 3 7 9 2 2 2 4 2 6 2 a - -- - -- + a z + a z + 2 a z - z - a z + 2 a z + 6 a z + z z 8 2 10 2 3 5 3 7 3 9 3 11 3 > 2 a z - 2 a z - 6 a z + 12 a z - 3 a z - 7 a z + 2 a z + 4 2 4 4 4 6 4 8 4 10 4 5 > 2 z - 4 a z - 10 a z - 21 a z - 11 a z + 6 a z + 10 a z + 3 5 5 5 7 5 9 5 11 5 6 2 6 > 2 a z - 25 a z - 6 a z + 10 a z - a z - z + 13 a z + 4 6 6 6 8 6 10 6 7 3 7 5 7 > 17 a z + 18 a z + 12 a z - 3 a z - 4 a z + 7 a z + 24 a z + 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 8 a z - 5 a z - 6 a z - 5 a z - 5 a z - 6 a z - 5 a z - 5 9 7 9 4 10 6 10 > 10 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 8 1 2 1 4 2 7 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 9 8 10 8 9 10 7 9 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 5 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 3 t 2 2 2 4 3 > --- + t + 3 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a257

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