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DrawMorseLink

PD Presentation:

X8192 X14,3,15,4 X12,15,7,16 X10,21,11,22 X16,9,17,10 X22,11,13,12 X18,6,19,5 X20,18,21,17 X2738 X4,13,5,14 X6,20,1,19

Gauss Code:

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

Jones Polynomial:

q-9 - 3q-8 + 8q-7 - 13q-6 + 19q-5 - 21q-4 + 23q-3 - 19q-2 + 15q-1 - 9 + 4q - q2

A2 (sl(3)) Invariant:

q-28 + q-26 + 5q-22 + 2q-18 + 7q-16 + 7q-12 + q-10 + 2q-8 + 3q-6 - 4q-4 + 4q-2 - 2 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 - a2 - a2z2 + a2z4 + a2z6 + a4z-2 + 7a4 + 11a4z2 + 7a4z4 + 2a4z6 - 2a6z-2 - 8a6 - 8a6z2 - 3a6z4 + a8z-2 + 2a8 + a8z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + z2 - 5z4 + 4z6 - az + 5az3 - 12az5 + 8az7 + 2a2 - 7a2z2 + 13a2z4 - 17a2z6 + 10a2z8 - 3a3z + 11a3z3 - 9a3z5 - 4a3z7 + 7a3z9 - a4z-2 + 10a4 - 31a4z2 + 52a4z4 - 45a4z6 + 15a4z8 + 2a4z10 + 2a5z-1 - 8a5z + 7a5z3 + 8a5z5 - 21a5z7 + 12a5z9 - 2a6z-2 + 12a6 - 31a6z2 + 43a6z4 - 35a6z6 + 10a6z8 + 2a6z10 + 2a7z-1 - 6a7z + 6a7z3 - 3a7z5 - 6a7z7 + 5a7z9 - a8z-2 + 4a8 - 5a8z2 + 6a8z4 - 10a8z6 + 5a8z8 + 4a9z3 - 7a9z5 + 3a9z7 - a10 + 3a10z2 - 3a10z4 + a10z6

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 = 5                       1 j = 3                     3   j = 1                   6 1   j = -1                 9 3     j = -3               11 7       j = -5             12 8         j = -7           10 12           j = -9         9 11             j = -11       5 11               j = -13     3 8                 j = -15   1 6                   j = -17   2                     j = -19 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 527]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 527]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 3, 15, 4], X[12, 15, 7, 16], X[10, 21, 11, 22], > X[16, 9, 17, 10], X[22, 11, 13, 12], X[18, 6, 19, 5], X[20, 18, 21, 17], > X[2, 7, 3, 8], X[4, 13, 5, 14], X[6, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 7, -11}, {9, -1, 5, -4, 6, -3}, > {10, -2, 3, -5, 8, -7, 11, -8, 4, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-9 3 8 13 19 21 23 19 15 2 -9 + q - -- + -- - -- + -- - -- + -- - -- + -- + 4 q - q 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 5 2 7 7 -10 2 3 4 4 2 -2 + q + q + --- + --- + --- + --- + q + -- + -- - -- + -- - q + 22 18 16 12 8 6 4 2 q q q q q q q q 4 6 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 527]][a, z]
Out[8]=	4 6 8 2 4 6 8 a 2 a a 2 2 2 4 2 6 2 -a + 7 a - 8 a + 2 a + -- - ---- + -- - z - a z + 11 a z - 8 a z + 2 2 2 z z z 8 2 4 2 4 4 4 6 4 2 6 4 6 > a z - z + a z + 7 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 527]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 10 a 2 a a 2 a 2 a 2 a + 10 a + 12 a + 4 a - a - -- - ---- - -- + ---- + ---- - a z - 2 2 2 z z z z z 3 5 7 2 2 2 4 2 6 2 8 2 > 3 a z - 8 a z - 6 a z + z - 7 a z - 31 a z - 31 a z - 5 a z + 3 10 2 z 3 3 3 5 3 7 3 9 3 4 > 3 a z - -- + 5 a z + 11 a z + 7 a z + 6 a z + 4 a z - 5 z + a 5 2 4 4 4 6 4 8 4 10 4 z 5 > 13 a z + 52 a z + 43 a z + 6 a z - 3 a z + -- - 12 a z - a 3 5 5 5 7 5 9 5 6 2 6 4 6 > 9 a z + 8 a z - 3 a z - 7 a z + 4 z - 17 a z - 45 a z - 6 6 8 6 10 6 7 3 7 5 7 7 7 > 35 a z - 10 a z + a z + 8 a z - 4 a z - 21 a z - 6 a z + 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > 3 a z + 10 a z + 15 a z + 10 a z + 5 a z + 7 a z + 12 a z + 7 9 4 10 6 10 > 5 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 9 1 2 1 6 3 8 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 11 9 11 10 12 12 8 11 3 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a527

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