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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X18,16,19,15 X16,9,17,10 X8,17,9,18 X22,19,5,20 X20,13,21,14 X14,21,15,22 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 7q-11/2 - 12q-9/2 + 17q-7/2 - 21q-5/2 + 20q-3/2 - 19q-1/2 + 14q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 2q-18 + 3q-14 - 3q-12 + 4q-10 + q-8 + q-6 + 4q-4 - 3q-2 + 5 - 2q2 + 3q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - 2a-1z3 - a-1z5 + 2az-1 + 7az + 8az3 + 4az5 + az7 - 3a3z-1 - 10a3z - 9a3z3 - 3a3z5 + 3a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

a-3z3 - a-3z5 - a-2z2 + 5a-2z4 - 4a-2z6 - a-1z-1 + 4a-1z - 8a-1z3 + 13a-1z5 - 8a-1z7 + 2z2 - 6z4 + 13z6 - 9z8 - 2az-1 + 15az - 33az3 + 30az5 - 6az7 - 5az9 - 2a2 + 10a2z2 - 32a2z4 + 40a2z6 - 17a2z8 - a2z10 - 3a3z-1 + 22a3z - 44a3z3 + 32a3z5 + a3z7 - 8a3z9 + 5a4z2 - 22a4z4 + 30a4z6 - 12a4z8 - a4z10 - 3a5z-1 + 14a5z - 27a5z3 + 24a5z5 - 4a5z7 - 3a5z9 + 2a6 - 5a6z2 + 2a6z4 + 6a6z6 - 4a6z8 - a7z-1 + 3a7z - 7a7z3 + 8a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

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 = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 8 3     j = 0               11 6       j = -2             11 10         j = -4           10 9           j = -6         7 11             j = -8       5 10               j = -10     2 7                 j = -12   1 5                   j = -14   2                     j = -16 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, 20]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 20]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[18, 16, 19, 15], > X[16, 9, 17, 10], X[8, 17, 9, 18], X[22, 19, 5, 20], X[20, 13, 21, 14], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, 8, -9, 4, -5, 6, -4, > 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 7 12 17 21 20 19 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 2 3 3 4 -8 -6 4 3 2 5 - q - q + q - --- + --- - --- + --- + q + q + -- - -- - 2 q + 18 14 12 10 4 2 q q q q q q 6 8 10 > 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 20]][a, z]
Out[8]=	3 5 7 1 2 a 3 a 3 a a 2 z 3 5 7 -(---) + --- - ---- + ---- - -- - --- + 7 a z - 10 a z + 6 a z - a z - a z z z z z a 3 5 2 z 3 3 3 5 3 z 5 3 5 7 > ---- + 8 a z - 9 a z + 3 a z - -- + 4 a z - 3 a z + a z a a
In[9]:=	Kauffman[Link[11, Alternating, 20]][a, z]
Out[9]=	3 5 7 2 6 8 1 2 a 3 a 3 a a 4 z 3 -2 a + 2 a + a - --- - --- - ---- - ---- - -- + --- + 15 a z + 22 a z + a z z z z z a 2 5 7 2 z 2 2 4 2 6 2 8 2 > 14 a z + 3 a z + 2 z - -- + 10 a z + 5 a z - 5 a z - 3 a z + 2 a 3 3 4 z 8 z 3 3 3 5 3 7 3 4 5 z > -- - ---- - 33 a z - 44 a z - 27 a z - 7 a z - 6 z + ---- - 3 a 2 a a 5 5 2 4 4 4 6 4 8 4 z 13 z 5 3 5 > 32 a z - 22 a z + 2 a z + 3 a z - -- + ----- + 30 a z + 32 a z + 3 a a 6 5 5 7 5 6 4 z 2 6 4 6 6 6 8 6 > 24 a z + 8 a z + 13 z - ---- + 40 a z + 30 a z + 6 a z - a z - 2 a 7 8 z 7 3 7 5 7 7 7 8 2 8 4 8 > ---- - 6 a z + a z - 4 a z - 3 a z - 9 z - 17 a z - 12 a z - a 6 8 9 3 9 5 9 2 10 4 10 > 4 a z - 5 a z - 8 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 5 2 7 5 10 11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 7 11 10 9 11 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 8 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a20

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