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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X22,12,19,11 X10,4,11,3 X20,5,21,6 X18,21,5,22 X12,20,13,19 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-8 + 3q-7 - 8q-6 + 13q-5 - 17q-4 + 21q-3 - 20q-2 + 19q-1 - 12 + 9q - 4q2 + q3

A2 (sl(3)) Invariant:

- q-24 - q-20 - 4q-18 + q-16 - 4q-14 + 2q-12 + 4q-10 + 2q-8 + 10q-6 + 2q-4 + 9q-2 + 4 + q2 + 3q4 - 2q6 + q8

HOMFLY-PT Polynomial:

2z-2 + 4 + 3z2 + 3z4 + z6 - 5a2z-2 - 11a2 - 12a2z2 - 10a2z4 - 5a2z6 - a2z8 + 4a4z-2 + 10a4 + 12a4z2 + 8a4z4 + 2a4z6 - a6z-2 - 3a6 - 3a6z2 - a6z4

Kauffman Polynomial:

- 2a-2z4 + a-2z6 + 2a-1z3 - 9a-1z5 + 4a-1z7 - 2z-2 + 7 - 12z2 + 23z4 - 24z6 + 8z8 + 5az-1 - 10az + 5az3 + 8az5 - 16az7 + 7az9 - 5a2z-2 + 17a2 - 40a2z2 + 65a2z4 - 50a2z6 + 12a2z8 + 2a2z10 + 9a3z-1 - 26a3z + 25a3z3 + 8a3z5 - 25a3z7 + 12a3z9 - 4a4z-2 + 16a4 - 38a4z2 + 52a4z4 - 38a4z6 + 11a4z8 + 2a4z10 + 5a5z-1 - 22a5z + 32a5z3 - 20a5z5 + a5z7 + 5a5z9 - a6z-2 + 5a6 - 9a6z2 + 8a6z4 - 10a6z6 + 7a6z8 + a7z-1 - 5a7z + 8a7z3 - 10a7z5 + 6a7z7 + a8z2 - 4a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     3   j = 3                   6 1   j = 1                 6 3     j = -1               13 6       j = -3             11 10         j = -5           10 9           j = -7         7 11             j = -9       6 10               j = -11     2 7                 j = -13   1 6                   j = -15   2                     j = -17 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, 494]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 494]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[16, 9, 17, 10], X[8, 15, 9, 16], > X[4, 17, 1, 18], X[22, 12, 19, 11], X[10, 4, 11, 3], X[20, 5, 21, 6], > X[18, 21, 5, 22], X[12, 20, 13, 19], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -5}, {10, -8, 9, -6}, > {8, -1, 2, -4, 3, -7, 6, -10, 11, -2, 4, -3, 5, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 8 13 17 21 20 19 2 3 -12 - q + -- - -- + -- - -- + -- - -- + -- + 9 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -20 4 -16 4 2 4 2 10 2 9 2 4 - q - q - --- + q - --- + --- + --- + -- + -- + -- + -- + q + 18 14 12 10 8 6 4 2 q q q q q q q q 4 6 8 > 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 494]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 2 4 2 4 - 11 a + 10 a - 3 a + -- - ---- + ---- - -- + 3 z - 12 a z + 12 a z - 2 2 2 2 z z z z 6 2 4 2 4 4 4 6 4 6 2 6 4 6 2 8 > 3 a z + 3 z - 10 a z + 8 a z - a z + z - 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 494]][a, z]
Out[9]=	2 4 6 3 5 7 2 4 6 2 5 a 4 a a 5 a 9 a 5 a a 7 + 17 a + 16 a + 5 a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 2 2 2 2 z z z z z z z z 3 5 7 9 2 2 2 4 2 > 10 a z - 26 a z - 22 a z - 5 a z + a z - 12 z - 40 a z - 38 a z - 3 6 2 8 2 2 z 3 3 3 5 3 7 3 9 3 > 9 a z + a z + ---- + 5 a z + 25 a z + 32 a z + 8 a z - 2 a z + a 4 5 4 2 z 2 4 4 4 6 4 8 4 9 z 5 > 23 z - ---- + 65 a z + 52 a z + 8 a z - 4 a z - ---- + 8 a z + 2 a a 6 3 5 5 5 7 5 9 5 6 z 2 6 4 6 > 8 a z - 20 a z - 10 a z + a z - 24 z + -- - 50 a z - 38 a z - 2 a 7 6 6 8 6 4 z 7 3 7 5 7 7 7 8 > 10 a z + 3 a z + ---- - 16 a z - 25 a z + a z + 6 a z + 8 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 12 a z + 11 a z + 7 a z + 7 a z + 12 a z + 5 a z + 2 a z + 4 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 13 1 2 1 6 2 7 6 10 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 7 11 10 9 11 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 6 q t + 3 q t + 6 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a494

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